Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.4% → 95.4%

Time: 9.5s

Alternatives: 24

Speedup: 12.5×

• 27.3% 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: 55.4% 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: 95.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{\left(\cos k \cdot \ell\right) \cdot 2}{k} \cdot \frac{\frac{\frac{\ell}{{\sin k}^{2}}}{k}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell}}}{\left(\frac{\sin k \cdot t\_m}{\ell} \cdot t\_m\right) \cdot \left(\left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right) \cdot \tan k\right)}\\ \end{array} \end{array} \]

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 9.6e-10)
    (* (/ (* (* (cos k) l) 2.0) k) (/ (/ (/ l (pow (sin k) 2.0)) k) t_m))
    (/
     (/ 2.0 (/ t_m l))
     (*
      (* (/ (* (sin k) t_m) l) t_m)
      (* (+ (pow (/ k t_m) 2.0) 2.0) (tan 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 (t_m <= 9.6e-10) {
		tmp = (((cos(k) * l) * 2.0) / k) * (((l / pow(sin(k), 2.0)) / k) / t_m);
	} else {
		tmp = (2.0 / (t_m / l)) / ((((sin(k) * t_m) / l) * t_m) * ((pow((k / t_m), 2.0) + 2.0) * tan(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 (t_m <= 9.6d-10) then
        tmp = (((cos(k) * l) * 2.0d0) / k) * (((l / (sin(k) ** 2.0d0)) / k) / t_m)
    else
        tmp = (2.0d0 / (t_m / l)) / ((((sin(k) * t_m) / l) * t_m) * ((((k / t_m) ** 2.0d0) + 2.0d0) * tan(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 (t_m <= 9.6e-10) {
		tmp = (((Math.cos(k) * l) * 2.0) / k) * (((l / Math.pow(Math.sin(k), 2.0)) / k) / t_m);
	} else {
		tmp = (2.0 / (t_m / l)) / ((((Math.sin(k) * t_m) / l) * t_m) * ((Math.pow((k / t_m), 2.0) + 2.0) * Math.tan(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 t_m <= 9.6e-10:
		tmp = (((math.cos(k) * l) * 2.0) / k) * (((l / math.pow(math.sin(k), 2.0)) / k) / t_m)
	else:
		tmp = (2.0 / (t_m / l)) / ((((math.sin(k) * t_m) / l) * t_m) * ((math.pow((k / t_m), 2.0) + 2.0) * math.tan(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 (t_m <= 9.6e-10)
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * 2.0) / k) * Float64(Float64(Float64(l / (sin(k) ^ 2.0)) / k) / t_m));
	else
		tmp = Float64(Float64(2.0 / Float64(t_m / l)) / Float64(Float64(Float64(Float64(sin(k) * t_m) / l) * t_m) * Float64(Float64((Float64(k / t_m) ^ 2.0) + 2.0) * tan(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 (t_m <= 9.6e-10)
		tmp = (((cos(k) * l) * 2.0) / k) * (((l / (sin(k) ^ 2.0)) / k) / t_m);
	else
		tmp = (2.0 / (t_m / l)) / ((((sin(k) * t_m) / l) * t_m) * ((((k / t_m) ^ 2.0) + 2.0) * tan(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[t$95$m, 9.6e-10], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / k), $MachinePrecision] * N[(N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.5999999999999999e-10

   1. Initial program 46.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 58.6

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

   5. Applied rewrites 58.6%

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

   7. Applied rewrites 69.4%

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

   9. Applied rewrites 79.3%

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

   if 9.5999999999999999e-10 < t

   1. Initial program 67.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-*l/ N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. lift-*.f64 N/A

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

      9. frac-times N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 92.7

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

   6. Applied rewrites 92.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 93.8

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

   8. Applied rewrites 93.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 96.3

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

   10. Applied rewrites 96.3%

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

Alternative 2: 93.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.85 \cdot 10^{-39}:\\ \;\;\;\;\frac{\left(\cos k \cdot \ell\right) \cdot 2}{k} \cdot \frac{\frac{\frac{\ell}{{\sin k}^{2}}}{k}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right)}}{\tan k \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)}\\ \end{array} \end{array} \]

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 2.85e-39)
    (* (/ (* (* (cos k) l) 2.0) k) (/ (/ (/ l (pow (sin k) 2.0)) k) t_m))
    (/
     (/ 2.0 (* (/ t_m l) (* (* (sin k) t_m) (/ t_m l))))
     (* (tan k) (+ (pow (/ k t_m) 2.0) 2.0))))))

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 <= 2.85e-39) {
		tmp = (((cos(k) * l) * 2.0) / k) * (((l / pow(sin(k), 2.0)) / k) / t_m);
	} else {
		tmp = (2.0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (tan(k) * (pow((k / t_m), 2.0) + 2.0));
	}
	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 <= 2.85d-39) then
        tmp = (((cos(k) * l) * 2.0d0) / k) * (((l / (sin(k) ** 2.0d0)) / k) / t_m)
    else
        tmp = (2.0d0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (tan(k) * (((k / t_m) ** 2.0d0) + 2.0d0))
    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 <= 2.85e-39) {
		tmp = (((Math.cos(k) * l) * 2.0) / k) * (((l / Math.pow(Math.sin(k), 2.0)) / k) / t_m);
	} else {
		tmp = (2.0 / ((t_m / l) * ((Math.sin(k) * t_m) * (t_m / l)))) / (Math.tan(k) * (Math.pow((k / t_m), 2.0) + 2.0));
	}
	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 <= 2.85e-39:
		tmp = (((math.cos(k) * l) * 2.0) / k) * (((l / math.pow(math.sin(k), 2.0)) / k) / t_m)
	else:
		tmp = (2.0 / ((t_m / l) * ((math.sin(k) * t_m) * (t_m / l)))) / (math.tan(k) * (math.pow((k / t_m), 2.0) + 2.0))
	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 <= 2.85e-39)
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * 2.0) / k) * Float64(Float64(Float64(l / (sin(k) ^ 2.0)) / k) / t_m));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(sin(k) * t_m) * Float64(t_m / l)))) / Float64(tan(k) * Float64((Float64(k / t_m) ^ 2.0) + 2.0)));
	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 <= 2.85e-39)
		tmp = (((cos(k) * l) * 2.0) / k) * (((l / (sin(k) ^ 2.0)) / k) / t_m);
	else
		tmp = (2.0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (tan(k) * (((k / t_m) ^ 2.0) + 2.0));
	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, 2.85e-39], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / k), $MachinePrecision] * N[(N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.8499999999999998e-39

   1. Initial program 45.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 58.2

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

   5. Applied rewrites 58.2%

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

   7. Applied rewrites 69.4%

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

   9. Applied rewrites 79.7%

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

   if 2.8499999999999998e-39 < t

   1. Initial program 69.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 77.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-*l/ N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. lift-*.f64 N/A

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

      9. frac-times N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 93.1

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

   6. Applied rewrites 93.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 94.2

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

   8. Applied rewrites 94.2%

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

Alternative 3: 93.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\left(\cos k \cdot \ell\right) \cdot 2}{k} \cdot \frac{\frac{\frac{\ell}{{\sin k}^{2}}}{k}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \sin k\right)\right)}}{\tan k \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)}\\ \end{array} \end{array} \]

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 6e-37)
    (* (/ (* (* (cos k) l) 2.0) k) (/ (/ (/ l (pow (sin k) 2.0)) k) t_m))
    (/
     (/ 2.0 (* t_m (* (/ t_m l) (* (/ t_m l) (sin k)))))
     (* (tan k) (+ (pow (/ k t_m) 2.0) 2.0))))))

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 <= 6e-37) {
		tmp = (((cos(k) * l) * 2.0) / k) * (((l / pow(sin(k), 2.0)) / k) / t_m);
	} else {
		tmp = (2.0 / (t_m * ((t_m / l) * ((t_m / l) * sin(k))))) / (tan(k) * (pow((k / t_m), 2.0) + 2.0));
	}
	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 <= 6d-37) then
        tmp = (((cos(k) * l) * 2.0d0) / k) * (((l / (sin(k) ** 2.0d0)) / k) / t_m)
    else
        tmp = (2.0d0 / (t_m * ((t_m / l) * ((t_m / l) * sin(k))))) / (tan(k) * (((k / t_m) ** 2.0d0) + 2.0d0))
    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 <= 6e-37) {
		tmp = (((Math.cos(k) * l) * 2.0) / k) * (((l / Math.pow(Math.sin(k), 2.0)) / k) / t_m);
	} else {
		tmp = (2.0 / (t_m * ((t_m / l) * ((t_m / l) * Math.sin(k))))) / (Math.tan(k) * (Math.pow((k / t_m), 2.0) + 2.0));
	}
	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 <= 6e-37:
		tmp = (((math.cos(k) * l) * 2.0) / k) * (((l / math.pow(math.sin(k), 2.0)) / k) / t_m)
	else:
		tmp = (2.0 / (t_m * ((t_m / l) * ((t_m / l) * math.sin(k))))) / (math.tan(k) * (math.pow((k / t_m), 2.0) + 2.0))
	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 <= 6e-37)
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * 2.0) / k) * Float64(Float64(Float64(l / (sin(k) ^ 2.0)) / k) / t_m));
	else
		tmp = Float64(Float64(2.0 / Float64(t_m * Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * sin(k))))) / Float64(tan(k) * Float64((Float64(k / t_m) ^ 2.0) + 2.0)));
	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 <= 6e-37)
		tmp = (((cos(k) * l) * 2.0) / k) * (((l / (sin(k) ^ 2.0)) / k) / t_m);
	else
		tmp = (2.0 / (t_m * ((t_m / l) * ((t_m / l) * sin(k))))) / (tan(k) * (((k / t_m) ^ 2.0) + 2.0));
	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, 6e-37], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / k), $MachinePrecision] * N[(N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6e-37

   1. Initial program 45.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 58.4

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

   5. Applied rewrites 58.4%

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

   7. Applied rewrites 69.5%

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

   9. Applied rewrites 79.8%

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

   if 6e-37 < t

   1. Initial program 69.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 76.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-*l/ N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. lift-*.f64 N/A

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

      9. frac-times N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 93.1

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

   6. Applied rewrites 93.1%

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

Alternative 4: 92.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-9}:\\ \;\;\;\;\frac{\left(\cos k \cdot \ell\right) \cdot 2}{k} \cdot \frac{\frac{\frac{\ell}{{\sin k}^{2}}}{k}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{t\_m}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \left(t\_m \cdot \tan k\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 1e-9)
    (* (/ (* (* (cos k) l) 2.0) k) (/ (/ (/ l (pow (sin k) 2.0)) k) t_m))
    (/
     2.0
     (*
      (* (* (pow (/ t_m l) 2.0) (sin k)) (* t_m (tan k)))
      (fma (/ k t_m) (/ k t_m) 2.0))))))

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 <= 1e-9) {
		tmp = (((cos(k) * l) * 2.0) / k) * (((l / pow(sin(k), 2.0)) / k) / t_m);
	} else {
		tmp = 2.0 / (((pow((t_m / l), 2.0) * sin(k)) * (t_m * tan(k))) * fma((k / t_m), (k / t_m), 2.0));
	}
	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 <= 1e-9)
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * 2.0) / k) * Float64(Float64(Float64(l / (sin(k) ^ 2.0)) / k) / t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(t_m / l) ^ 2.0) * sin(k)) * Float64(t_m * tan(k))) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	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, 1e-9], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / k), $MachinePrecision] * N[(N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(t$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.00000000000000006e-9

   1. Initial program 46.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 58.6

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

   5. Applied rewrites 58.6%

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

   7. Applied rewrites 69.4%

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

   9. Applied rewrites 79.3%

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

   if 1.00000000000000006e-9 < t

   1. Initial program 67.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow3 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 81.6

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

   4. Applied rewrites 81.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 81.6

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

   6. Applied rewrites 81.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 92.6%

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

Alternative 5: 77.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.0021:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right)}}{2 \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos k \cdot \ell\right) \cdot 2}{k} \cdot \frac{\frac{\frac{\ell}{{\sin k}^{2}}}{k}}{t\_m}\\ \end{array} \end{array} \]

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 (<= k 0.0021)
    (/ (/ 2.0 (* (/ t_m l) (* (* (sin k) t_m) (/ t_m l)))) (* 2.0 k))
    (* (/ (* (* (cos k) l) 2.0) k) (/ (/ (/ l (pow (sin k) 2.0)) k) 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 (k <= 0.0021) {
		tmp = (2.0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (2.0 * k);
	} else {
		tmp = (((cos(k) * l) * 2.0) / k) * (((l / pow(sin(k), 2.0)) / k) / 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 (k <= 0.0021d0) then
        tmp = (2.0d0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (2.0d0 * k)
    else
        tmp = (((cos(k) * l) * 2.0d0) / k) * (((l / (sin(k) ** 2.0d0)) / k) / 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 (k <= 0.0021) {
		tmp = (2.0 / ((t_m / l) * ((Math.sin(k) * t_m) * (t_m / l)))) / (2.0 * k);
	} else {
		tmp = (((Math.cos(k) * l) * 2.0) / k) * (((l / Math.pow(Math.sin(k), 2.0)) / k) / 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 k <= 0.0021:
		tmp = (2.0 / ((t_m / l) * ((math.sin(k) * t_m) * (t_m / l)))) / (2.0 * k)
	else:
		tmp = (((math.cos(k) * l) * 2.0) / k) * (((l / math.pow(math.sin(k), 2.0)) / k) / 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 (k <= 0.0021)
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(sin(k) * t_m) * Float64(t_m / l)))) / Float64(2.0 * k));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * 2.0) / k) * Float64(Float64(Float64(l / (sin(k) ^ 2.0)) / k) / 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 (k <= 0.0021)
		tmp = (2.0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (2.0 * k);
	else
		tmp = (((cos(k) * l) * 2.0) / k) * (((l / (sin(k) ^ 2.0)) / k) / 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[k, 0.0021], N[(N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / k), $MachinePrecision] * N[(N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.00209999999999999987

   1. Initial program 52.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-*l/ N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. lift-*.f64 N/A

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

      9. frac-times N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 78.2

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

   6. Applied rewrites 78.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.2

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

   8. Applied rewrites 79.2%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 69.3

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

   11. Applied rewrites 69.3%

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

   if 0.00209999999999999987 < k

   1. Initial program 55.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 63.0

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

   5. Applied rewrites 63.0%

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

   7. Applied rewrites 72.4%

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

   9. Applied rewrites 90.4%

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

Alternative 6: 90.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-9}:\\ \;\;\;\;\frac{\left(\cos k \cdot \ell\right) \cdot 2}{k} \cdot \frac{\frac{\ell}{{\sin k}^{2}}}{k \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \left(\left(\sin k \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 1e-9)
    (* (/ (* (* (cos k) l) 2.0) k) (/ (/ l (pow (sin k) 2.0)) (* k t_m)))
    (/
     2.0
     (*
      (* (* (/ t_m l) t_m) (* (* (sin k) (/ t_m l)) (tan k)))
      (fma (/ k t_m) (/ k t_m) 2.0))))))

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 <= 1e-9) {
		tmp = (((cos(k) * l) * 2.0) / k) * ((l / pow(sin(k), 2.0)) / (k * t_m));
	} else {
		tmp = 2.0 / ((((t_m / l) * t_m) * ((sin(k) * (t_m / l)) * tan(k))) * fma((k / t_m), (k / t_m), 2.0));
	}
	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 <= 1e-9)
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * 2.0) / k) * Float64(Float64(l / (sin(k) ^ 2.0)) / Float64(k * t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(Float64(sin(k) * Float64(t_m / l)) * tan(k))) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	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, 1e-9], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.00000000000000006e-9

   1. Initial program 46.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 58.6

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

   5. Applied rewrites 58.6%

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

   7. Applied rewrites 69.4%

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

   9. Applied rewrites 78.2%

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

   if 1.00000000000000006e-9 < t

   1. Initial program 67.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow3 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 81.6

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

   4. Applied rewrites 81.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 81.6

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

   6. Applied rewrites 81.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 90.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 90.3

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

   8. Applied rewrites 90.3%

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

Alternative 7: 87.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{-126}:\\ \;\;\;\;\left(\left(\cos k \cdot \ell\right) \cdot 2\right) \cdot \frac{\frac{\ell}{{\sin k}^{2}}}{\left(k \cdot t\_m\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \left(\left(\sin k \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 5e-126)
    (* (* (* (cos k) l) 2.0) (/ (/ l (pow (sin k) 2.0)) (* (* k t_m) k)))
    (/
     2.0
     (*
      (* (* (/ t_m l) t_m) (* (* (sin k) (/ t_m l)) (tan k)))
      (fma (/ k t_m) (/ k t_m) 2.0))))))

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 <= 5e-126) {
		tmp = ((cos(k) * l) * 2.0) * ((l / pow(sin(k), 2.0)) / ((k * t_m) * k));
	} else {
		tmp = 2.0 / ((((t_m / l) * t_m) * ((sin(k) * (t_m / l)) * tan(k))) * fma((k / t_m), (k / t_m), 2.0));
	}
	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 <= 5e-126)
		tmp = Float64(Float64(Float64(cos(k) * l) * 2.0) * Float64(Float64(l / (sin(k) ^ 2.0)) / Float64(Float64(k * t_m) * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(Float64(sin(k) * Float64(t_m / l)) * tan(k))) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	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, 5e-126], N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.00000000000000006e-126

   1. Initial program 42.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lower-sin.f64 57.1

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

   5. Applied rewrites 57.1%

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

   7. Applied rewrites 68.2%

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

   9. Applied rewrites 73.6%

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

   if 5.00000000000000006e-126 < t

   1. Initial program 69.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow3 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 80.9

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

   4. Applied rewrites 80.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 80.9

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

   6. Applied rewrites 80.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 88.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 88.2

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

   8. Applied rewrites 88.2%

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

Alternative 8: 87.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{-126}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(k \cdot t\_m\right) \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot t\_m\right) \cdot \left(\left(\sin k \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 5e-126)
    (/ 2.0 (/ (* (* (/ (sin k) l) (tan k)) (* (* k t_m) k)) l))
    (/
     2.0
     (*
      (* (* (/ t_m l) t_m) (* (* (sin k) (/ t_m l)) (tan k)))
      (fma (/ k t_m) (/ k t_m) 2.0))))))

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 <= 5e-126) {
		tmp = 2.0 / ((((sin(k) / l) * tan(k)) * ((k * t_m) * k)) / l);
	} else {
		tmp = 2.0 / ((((t_m / l) * t_m) * ((sin(k) * (t_m / l)) * tan(k))) * fma((k / t_m), (k / t_m), 2.0));
	}
	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 <= 5e-126)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) / l) * tan(k)) * Float64(Float64(k * t_m) * k)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * t_m) * Float64(Float64(sin(k) * Float64(t_m / l)) * tan(k))) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	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, 5e-126], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.00000000000000006e-126

   1. Initial program 42.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow3 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 54.9

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

   4. Applied rewrites 54.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 54.9

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

   6. Applied rewrites 54.9%

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

   7. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 62.9

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

   9. Applied rewrites 62.9%

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

   11. Applied rewrites 73.3%

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

   if 5.00000000000000006e-126 < t

   1. Initial program 69.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow3 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 80.9

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

   4. Applied rewrites 80.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 80.9

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

   6. Applied rewrites 80.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 88.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 88.2

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

   8. Applied rewrites 88.2%

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

Alternative 9: 85.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{-126}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(k \cdot t\_m\right) \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot t\_m\right)\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 5e-126)
    (/ 2.0 (/ (* (* (/ (sin k) l) (tan k)) (* (* k t_m) k)) l))
    (/
     2.0
     (*
      (* (* (* (/ t_m l) (* (/ t_m l) t_m)) (sin k)) (tan k))
      (fma (/ k t_m) (/ k t_m) 2.0))))))

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 <= 5e-126) {
		tmp = 2.0 / ((((sin(k) / l) * tan(k)) * ((k * t_m) * k)) / l);
	} else {
		tmp = 2.0 / (((((t_m / l) * ((t_m / l) * t_m)) * sin(k)) * tan(k)) * fma((k / t_m), (k / t_m), 2.0));
	}
	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 <= 5e-126)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) / l) * tan(k)) * Float64(Float64(k * t_m) * k)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * t_m)) * sin(k)) * tan(k)) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	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, 5e-126], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.00000000000000006e-126

   1. Initial program 42.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow3 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 54.9

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

   4. Applied rewrites 54.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 54.9

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

   6. Applied rewrites 54.9%

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

   7. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 62.9

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

   9. Applied rewrites 62.9%

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

   11. Applied rewrites 73.3%

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

   if 5.00000000000000006e-126 < t

   1. Initial program 69.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow3 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 80.9

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

   4. Applied rewrites 80.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 80.9

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

   6. Applied rewrites 80.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 80.9

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 87.3

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

   8. Applied rewrites 87.3%

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

Alternative 10: 75.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.0021:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right)}}{2 \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(k \cdot t\_m\right) \cdot k\right)}{\ell}}\\ \end{array} \end{array} \]

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 0.0021)
    (/ (/ 2.0 (* (/ t_m l) (* (* (sin k) t_m) (/ t_m l)))) (* 2.0 k))
    (/ 2.0 (/ (* (* (/ (sin k) l) (tan k)) (* (* k 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) {
	double tmp;
	if (k <= 0.0021) {
		tmp = (2.0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (2.0 * k);
	} else {
		tmp = 2.0 / ((((sin(k) / l) * tan(k)) * ((k * t_m) * k)) / l);
	}
	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 <= 0.0021d0) then
        tmp = (2.0d0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (2.0d0 * k)
    else
        tmp = 2.0d0 / ((((sin(k) / l) * tan(k)) * ((k * t_m) * k)) / l)
    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 <= 0.0021) {
		tmp = (2.0 / ((t_m / l) * ((Math.sin(k) * t_m) * (t_m / l)))) / (2.0 * k);
	} else {
		tmp = 2.0 / ((((Math.sin(k) / l) * Math.tan(k)) * ((k * t_m) * k)) / l);
	}
	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 <= 0.0021:
		tmp = (2.0 / ((t_m / l) * ((math.sin(k) * t_m) * (t_m / l)))) / (2.0 * k)
	else:
		tmp = 2.0 / ((((math.sin(k) / l) * math.tan(k)) * ((k * t_m) * k)) / l)
	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 <= 0.0021)
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(sin(k) * t_m) * Float64(t_m / l)))) / Float64(2.0 * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) / l) * tan(k)) * Float64(Float64(k * t_m) * k)) / l));
	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 <= 0.0021)
		tmp = (2.0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (2.0 * k);
	else
		tmp = 2.0 / ((((sin(k) / l) * tan(k)) * ((k * t_m) * k)) / l);
	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, 0.0021], N[(N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.00209999999999999987

   1. Initial program 52.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-*l/ N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. lift-*.f64 N/A

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

      9. frac-times N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 78.2

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

   6. Applied rewrites 78.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.2

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

   8. Applied rewrites 79.2%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 69.3

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

   11. Applied rewrites 69.3%

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

   if 0.00209999999999999987 < k

   1. Initial program 55.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow3 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 67.2

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

   4. Applied rewrites 67.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 67.2

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

   6. Applied rewrites 67.2%

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

   7. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 71.7

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

   9. Applied rewrites 71.7%

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

   11. Applied rewrites 80.2%

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

Alternative 11: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.0021:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right)}}{2 \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot \left(k \cdot t\_m\right)\right) \cdot k}\\ \end{array} \end{array} \]

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 0.0021)
    (/ (/ 2.0 (* (/ t_m l) (* (* (sin k) t_m) (/ t_m l)))) (* 2.0 k))
    (/ 2.0 (* (* (* (tan k) (/ (sin k) (* l l))) (* k t_m)) 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 <= 0.0021) {
		tmp = (2.0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (2.0 * k);
	} else {
		tmp = 2.0 / (((tan(k) * (sin(k) / (l * l))) * (k * t_m)) * 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 <= 0.0021d0) then
        tmp = (2.0d0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (2.0d0 * k)
    else
        tmp = 2.0d0 / (((tan(k) * (sin(k) / (l * l))) * (k * t_m)) * 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 <= 0.0021) {
		tmp = (2.0 / ((t_m / l) * ((Math.sin(k) * t_m) * (t_m / l)))) / (2.0 * k);
	} else {
		tmp = 2.0 / (((Math.tan(k) * (Math.sin(k) / (l * l))) * (k * t_m)) * 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 <= 0.0021:
		tmp = (2.0 / ((t_m / l) * ((math.sin(k) * t_m) * (t_m / l)))) / (2.0 * k)
	else:
		tmp = 2.0 / (((math.tan(k) * (math.sin(k) / (l * l))) * (k * t_m)) * 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 <= 0.0021)
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(sin(k) * t_m) * Float64(t_m / l)))) / Float64(2.0 * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(sin(k) / Float64(l * l))) * Float64(k * t_m)) * 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 <= 0.0021)
		tmp = (2.0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (2.0 * k);
	else
		tmp = 2.0 / (((tan(k) * (sin(k) / (l * l))) * (k * t_m)) * 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, 0.0021], N[(N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.00209999999999999987

   1. Initial program 52.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-*l/ N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. lift-*.f64 N/A

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

      9. frac-times N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 78.2

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

   6. Applied rewrites 78.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.2

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

   8. Applied rewrites 79.2%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 69.3

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

   11. Applied rewrites 69.3%

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

   if 0.00209999999999999987 < k

   1. Initial program 55.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow3 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 67.2

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

   4. Applied rewrites 67.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 67.2

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

   6. Applied rewrites 67.2%

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

   7. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 71.7

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

   9. Applied rewrites 71.7%

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

   11. Applied rewrites 71.8%

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

Alternative 12: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.0021:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right)}}{2 \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot k\right) \cdot \left(k \cdot t\_m\right)}\\ \end{array} \end{array} \]

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 0.0021)
    (/ (/ 2.0 (* (/ t_m l) (* (* (sin k) t_m) (/ t_m l)))) (* 2.0 k))
    (/ 2.0 (* (* (* (tan k) (/ (sin k) (* l l))) k) (* k 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 (k <= 0.0021) {
		tmp = (2.0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (2.0 * k);
	} else {
		tmp = 2.0 / (((tan(k) * (sin(k) / (l * l))) * k) * (k * 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 (k <= 0.0021d0) then
        tmp = (2.0d0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (2.0d0 * k)
    else
        tmp = 2.0d0 / (((tan(k) * (sin(k) / (l * l))) * k) * (k * 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 (k <= 0.0021) {
		tmp = (2.0 / ((t_m / l) * ((Math.sin(k) * t_m) * (t_m / l)))) / (2.0 * k);
	} else {
		tmp = 2.0 / (((Math.tan(k) * (Math.sin(k) / (l * l))) * k) * (k * 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 k <= 0.0021:
		tmp = (2.0 / ((t_m / l) * ((math.sin(k) * t_m) * (t_m / l)))) / (2.0 * k)
	else:
		tmp = 2.0 / (((math.tan(k) * (math.sin(k) / (l * l))) * k) * (k * 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 (k <= 0.0021)
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(sin(k) * t_m) * Float64(t_m / l)))) / Float64(2.0 * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(sin(k) / Float64(l * l))) * k) * Float64(k * 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 (k <= 0.0021)
		tmp = (2.0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (2.0 * k);
	else
		tmp = 2.0 / (((tan(k) * (sin(k) / (l * l))) * k) * (k * 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[k, 0.0021], N[(N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.00209999999999999987

   1. Initial program 52.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. associate-*l/ N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow3 N/A

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

      8. lift-*.f64 N/A

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

      9. frac-times N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 78.2

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

   6. Applied rewrites 78.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.2

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

   8. Applied rewrites 79.2%

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

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)}}{\color{blue}{2 \cdot k}} \]
   10. Step-by-step derivation

       1. lower-*.f64 69.3

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)}}{\color{blue}{2 \cdot k}} \]

   11. Applied rewrites 69.3%

       \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)}}{\color{blue}{2 \cdot k}} \]

   if 0.00209999999999999987 < k

   1. Initial program 55.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. unpow3 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. times-frac N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-/.f64 67.2

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 67.2%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 2\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 2\right)} \]

      8. lower-fma.f64 67.2

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   6. Applied rewrites 67.2%

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   7. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(t \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot {k}^{2}\right)} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot t\right)} \cdot k\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot t\right) \cdot k\right)} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot t\right)} \cdot k\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

      16. associate-*r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right)} \cdot \ell}} \]

      19. lower-cos.f64 71.7

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}} \]

   9. Applied rewrites 71.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 73.4%

       \[\leadsto \frac{2}{\left(\left(\tan k \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 74.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.0021:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right)}}{2 \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(t\_m \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot k\right)\right)}\\ \end{array} \end{array} \]

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 0.0021)
    (/ (/ 2.0 (* (/ t_m l) (* (* (sin k) t_m) (/ t_m l)))) (* 2.0 k))
    (/ 2.0 (* k (* t_m (* (* (tan k) (/ (sin 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 <= 0.0021) {
		tmp = (2.0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (2.0 * k);
	} else {
		tmp = 2.0 / (k * (t_m * ((tan(k) * (sin(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 <= 0.0021d0) then
        tmp = (2.0d0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (2.0d0 * k)
    else
        tmp = 2.0d0 / (k * (t_m * ((tan(k) * (sin(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 <= 0.0021) {
		tmp = (2.0 / ((t_m / l) * ((Math.sin(k) * t_m) * (t_m / l)))) / (2.0 * k);
	} else {
		tmp = 2.0 / (k * (t_m * ((Math.tan(k) * (Math.sin(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 <= 0.0021:
		tmp = (2.0 / ((t_m / l) * ((math.sin(k) * t_m) * (t_m / l)))) / (2.0 * k)
	else:
		tmp = 2.0 / (k * (t_m * ((math.tan(k) * (math.sin(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 <= 0.0021)
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(sin(k) * t_m) * Float64(t_m / l)))) / Float64(2.0 * k));
	else
		tmp = Float64(2.0 / Float64(k * Float64(t_m * Float64(Float64(tan(k) * Float64(sin(k) / Float64(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 <= 0.0021)
		tmp = (2.0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (2.0 * k);
	else
		tmp = 2.0 / (k * (t_m * ((tan(k) * (sin(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, 0.0021], N[(N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(t$95$m * N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.00209999999999999987

   1. Initial program 52.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   4. Applied rewrites 61.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\frac{\sin k}{\ell}}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell}} \cdot \frac{\sin k}{\ell}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      4. frac-times N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      7. unpow3 N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      9. frac-times N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      13. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\ell}} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\color{blue}{t \cdot t}}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      15. associate-/l* N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      16. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\left(t \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      17. associate-*l* N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      20. lower-*.f64 78.2

          \[\leadsto \frac{\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \sin k\right)}\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

   6. Applied rewrites 78.2%

      \[\leadsto \frac{\frac{2}{\color{blue}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot t}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot t}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot t\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \sin k\right)}\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(t \cdot \color{blue}{\left(\sin k \cdot \frac{t}{\ell}\right)}\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t}{\ell}\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      12. lower-*.f64 79.2

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t}{\ell}\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

   8. Applied rewrites 79.2%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)}}{\color{blue}{2 \cdot k}} \]
   10. Step-by-step derivation

       1. lower-*.f64 69.3

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)}}{\color{blue}{2 \cdot k}} \]

   11. Applied rewrites 69.3%

       \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)}}{\color{blue}{2 \cdot k}} \]

   if 0.00209999999999999987 < k

   1. Initial program 55.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. unpow3 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. times-frac N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-/.f64 67.2

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 67.2%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 2\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 2\right)} \]

      8. lower-fma.f64 67.2

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   6. Applied rewrites 67.2%

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   7. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(t \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot {k}^{2}\right)} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot t\right)} \cdot k\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot t\right) \cdot k\right)} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot t\right)} \cdot k\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

      16. associate-*r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right)} \cdot \ell}} \]

      19. lower-cos.f64 71.7

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}} \]

   9. Applied rewrites 71.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 73.5%

       \[\leadsto \frac{2}{k \cdot \color{blue}{\left(t \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 74.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-132}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\left(0.26666666666666666 + \frac{0.3333333333333333}{t\_m \cdot t\_m}\right) \cdot k, k, 0.6666666666666666 + \frac{1}{t\_m \cdot t\_m}\right), k \cdot k, 2\right) \cdot k}\\ \end{array} \end{array} \]

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 (<= t_m 1.1e-132)
    (/ 2.0 (* (/ (pow k 4.0) l) (/ t_m l)))
    (/
     (/ 2.0 (* (/ t_m l) (* (* (sin k) t_m) (/ t_m l))))
     (*
      (fma
       (fma
        (* (+ 0.26666666666666666 (/ 0.3333333333333333 (* t_m t_m))) k)
        k
        (+ 0.6666666666666666 (/ 1.0 (* t_m t_m))))
       (* k k)
       2.0)
      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 (t_m <= 1.1e-132) {
		tmp = 2.0 / ((pow(k, 4.0) / l) * (t_m / l));
	} else {
		tmp = (2.0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (fma(fma(((0.26666666666666666 + (0.3333333333333333 / (t_m * t_m))) * k), k, (0.6666666666666666 + (1.0 / (t_m * t_m)))), (k * k), 2.0) * 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 (t_m <= 1.1e-132)
		tmp = Float64(2.0 / Float64(Float64((k ^ 4.0) / l) * Float64(t_m / l)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(sin(k) * t_m) * Float64(t_m / l)))) / Float64(fma(fma(Float64(Float64(0.26666666666666666 + Float64(0.3333333333333333 / Float64(t_m * t_m))) * k), k, Float64(0.6666666666666666 + Float64(1.0 / Float64(t_m * t_m)))), Float64(k * k), 2.0) * k));
	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, 1.1e-132], N[(2.0 / N[(N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(0.26666666666666666 + N[(0.3333333333333333 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k + N[(0.6666666666666666 + N[(1.0 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + 2.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.09999999999999995e-132

   1. Initial program 42.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. unpow3 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. times-frac N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-/.f64 54.7

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 54.7%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 2\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 2\right)} \]

      8. lower-fma.f64 54.7

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   6. Applied rewrites 54.7%

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   7. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(t \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot {k}^{2}\right)} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot t\right)} \cdot k\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot t\right) \cdot k\right)} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot t\right)} \cdot k\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

      16. associate-*r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right)} \cdot \ell}} \]

      19. lower-cos.f64 62.8

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}} \]

   9. Applied rewrites 62.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{{\ell}^{2}}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 54.0%

       \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \color{blue}{\frac{t}{\ell}}} \]

   if 1.09999999999999995e-132 < t

   1. Initial program 68.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   4. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\frac{\sin k}{\ell}}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell}} \cdot \frac{\sin k}{\ell}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      4. frac-times N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      7. unpow3 N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      9. frac-times N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      13. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\ell}} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\color{blue}{t \cdot t}}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      15. associate-/l* N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      16. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\left(t \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      17. associate-*l* N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      20. lower-*.f64 89.2

          \[\leadsto \frac{\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \sin k\right)}\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

   6. Applied rewrites 89.2%

      \[\leadsto \frac{\frac{2}{\color{blue}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot t}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot t}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot t\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \sin k\right)}\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(t \cdot \color{blue}{\left(\sin k \cdot \frac{t}{\ell}\right)}\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t}{\ell}\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      12. lower-*.f64 90.9

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t}{\ell}\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

   8. Applied rewrites 90.9%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)}}{\color{blue}{k \cdot \left(2 + {k}^{2} \cdot \left(\frac{2}{3} + \left({k}^{2} \cdot \left(\frac{4}{15} + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right) + \frac{1}{{t}^{2}}\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)}}{\color{blue}{\left(2 + {k}^{2} \cdot \left(\frac{2}{3} + \left({k}^{2} \cdot \left(\frac{4}{15} + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right) + \frac{1}{{t}^{2}}\right)\right)\right) \cdot k}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)}}{\color{blue}{\left(2 + {k}^{2} \cdot \left(\frac{2}{3} + \left({k}^{2} \cdot \left(\frac{4}{15} + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right) + \frac{1}{{t}^{2}}\right)\right)\right) \cdot k}} \]

   11. Applied rewrites 83.1%

       \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(0.26666666666666666 + \frac{0.3333333333333333}{t \cdot t}\right) \cdot k, k, 0.6666666666666666 + \frac{1}{t \cdot t}\right), k \cdot k, 2\right) \cdot k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 74.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-132}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t\_m}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \sin k\right)\right)}}{\mathsf{fma}\left(0.6666666666666666 + \frac{1}{t\_m \cdot t\_m}, k \cdot k, 2\right) \cdot k}\\ \end{array} \end{array} \]

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 (<= t_m 1.1e-132)
    (/ 2.0 (* (/ (pow k 4.0) l) (/ t_m l)))
    (/
     (/ 2.0 (* t_m (* (/ t_m l) (* (/ t_m l) (sin k)))))
     (* (fma (+ 0.6666666666666666 (/ 1.0 (* t_m t_m))) (* k k) 2.0) 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 (t_m <= 1.1e-132) {
		tmp = 2.0 / ((pow(k, 4.0) / l) * (t_m / l));
	} else {
		tmp = (2.0 / (t_m * ((t_m / l) * ((t_m / l) * sin(k))))) / (fma((0.6666666666666666 + (1.0 / (t_m * t_m))), (k * k), 2.0) * 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 (t_m <= 1.1e-132)
		tmp = Float64(2.0 / Float64(Float64((k ^ 4.0) / l) * Float64(t_m / l)));
	else
		tmp = Float64(Float64(2.0 / Float64(t_m * Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * sin(k))))) / Float64(fma(Float64(0.6666666666666666 + Float64(1.0 / Float64(t_m * t_m))), Float64(k * k), 2.0) * k));
	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, 1.1e-132], N[(2.0 / N[(N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.6666666666666666 + N[(1.0 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + 2.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.09999999999999995e-132

   1. Initial program 42.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. unpow3 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. times-frac N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-/.f64 54.7

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 54.7%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 2\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 2\right)} \]

      8. lower-fma.f64 54.7

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   6. Applied rewrites 54.7%

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   7. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(t \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot {k}^{2}\right)} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot t\right)} \cdot k\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot t\right) \cdot k\right)} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot t\right)} \cdot k\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

      16. associate-*r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right)} \cdot \ell}} \]

      19. lower-cos.f64 62.8

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}} \]

   9. Applied rewrites 62.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{{\ell}^{2}}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 54.0%

       \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \color{blue}{\frac{t}{\ell}}} \]

   if 1.09999999999999995e-132 < t

   1. Initial program 68.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   4. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\frac{\sin k}{\ell}}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell}} \cdot \frac{\sin k}{\ell}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      4. frac-times N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      7. unpow3 N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      9. frac-times N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      13. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\ell}} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\color{blue}{t \cdot t}}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      15. associate-/l* N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      16. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\left(t \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      17. associate-*l* N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      20. lower-*.f64 89.2

          \[\leadsto \frac{\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \sin k\right)}\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

   6. Applied rewrites 89.2%

      \[\leadsto \frac{\frac{2}{\color{blue}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}{\color{blue}{k \cdot \left(2 + {k}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}{\color{blue}{\left(2 + {k}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right) \cdot k}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}{\color{blue}{\left(2 + {k}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right)\right) \cdot k}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}{\color{blue}{\left({k}^{2} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + 2\right)} \cdot k} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}{\left(\color{blue}{\left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) \cdot {k}^{2}} + 2\right) \cdot k} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}{\color{blue}{\mathsf{fma}\left(\frac{2}{3} + \frac{1}{{t}^{2}}, {k}^{2}, 2\right)} \cdot k} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}{\mathsf{fma}\left(\color{blue}{\frac{2}{3} + \frac{1}{{t}^{2}}}, {k}^{2}, 2\right) \cdot k} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}{\mathsf{fma}\left(\frac{2}{3} + \color{blue}{\frac{1}{{t}^{2}}}, {k}^{2}, 2\right) \cdot k} \]

      8. unpow2 N/A

         \[\leadsto \frac{\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}{\mathsf{fma}\left(\frac{2}{3} + \frac{1}{\color{blue}{t \cdot t}}, {k}^{2}, 2\right) \cdot k} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}{\mathsf{fma}\left(\frac{2}{3} + \frac{1}{\color{blue}{t \cdot t}}, {k}^{2}, 2\right) \cdot k} \]

      10. unpow2 N/A

          \[\leadsto \frac{\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}{\mathsf{fma}\left(\frac{2}{3} + \frac{1}{t \cdot t}, \color{blue}{k \cdot k}, 2\right) \cdot k} \]

      11. lower-*.f64 82.3

          \[\leadsto \frac{\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}{\mathsf{fma}\left(0.6666666666666666 + \frac{1}{t \cdot t}, \color{blue}{k \cdot k}, 2\right) \cdot k} \]

   9. Applied rewrites 82.3%

      \[\leadsto \frac{\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}{\color{blue}{\mathsf{fma}\left(0.6666666666666666 + \frac{1}{t \cdot t}, k \cdot k, 2\right) \cdot k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 69.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.1 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(\sin k \cdot t\_m\right) \cdot \frac{t\_m}{\ell}\right)}}{2 \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot t\_m\right) \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\\ \end{array} \end{array} \]

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 4.1e+17)
    (/ (/ 2.0 (* (/ t_m l) (* (* (sin k) t_m) (/ t_m l)))) (* 2.0 k))
    (/ 2.0 (* (* (* k t_m) k) (* (/ k l) (/ 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) {
	double tmp;
	if (k <= 4.1e+17) {
		tmp = (2.0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (2.0 * k);
	} else {
		tmp = 2.0 / (((k * t_m) * k) * ((k / l) * (k / l)));
	}
	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 <= 4.1d+17) then
        tmp = (2.0d0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (2.0d0 * k)
    else
        tmp = 2.0d0 / (((k * t_m) * k) * ((k / l) * (k / l)))
    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 <= 4.1e+17) {
		tmp = (2.0 / ((t_m / l) * ((Math.sin(k) * t_m) * (t_m / l)))) / (2.0 * k);
	} else {
		tmp = 2.0 / (((k * t_m) * k) * ((k / l) * (k / l)));
	}
	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 <= 4.1e+17:
		tmp = (2.0 / ((t_m / l) * ((math.sin(k) * t_m) * (t_m / l)))) / (2.0 * k)
	else:
		tmp = 2.0 / (((k * t_m) * k) * ((k / l) * (k / l)))
	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 <= 4.1e+17)
		tmp = Float64(Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(sin(k) * t_m) * Float64(t_m / l)))) / Float64(2.0 * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * t_m) * k) * Float64(Float64(k / l) * Float64(k / l))));
	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 <= 4.1e+17)
		tmp = (2.0 / ((t_m / l) * ((sin(k) * t_m) * (t_m / l)))) / (2.0 * k);
	else
		tmp = 2.0 / (((k * t_m) * k) * ((k / l) * (k / l)));
	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, 4.1e+17], N[(N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.1e17

   1. Initial program 52.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   4. Applied rewrites 61.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \color{blue}{\frac{\sin k}{\ell}}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell}} \cdot \frac{\sin k}{\ell}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      4. frac-times N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      5. associate-*l/ N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      7. unpow3 N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{\color{blue}{\left(t \cdot t\right)} \cdot t}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      9. frac-times N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      13. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t \cdot t}{\ell}} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{\color{blue}{t \cdot t}}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      15. associate-/l* N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{\left(t \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      16. lift-/.f64 N/A

          \[\leadsto \frac{\frac{2}{\left(t \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      17. associate-*l* N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\color{blue}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      20. lower-*.f64 78.1

          \[\leadsto \frac{\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \sin k\right)}\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

   6. Applied rewrites 78.1%

      \[\leadsto \frac{\frac{2}{\color{blue}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot t}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot t}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot t\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \sin k\right)}\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(t \cdot \color{blue}{\left(\sin k \cdot \frac{t}{\ell}\right)}\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t}{\ell}\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

      12. lower-*.f64 79.0

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(\color{blue}{\left(\sin k \cdot t\right)} \cdot \frac{t}{\ell}\right)}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

   8. Applied rewrites 79.0%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)}}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)}}{\color{blue}{2 \cdot k}} \]
   10. Step-by-step derivation

       1. lower-*.f64 69.1

          \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)}}{\color{blue}{2 \cdot k}} \]

   11. Applied rewrites 69.1%

       \[\leadsto \frac{\frac{2}{\frac{t}{\ell} \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right)}}{\color{blue}{2 \cdot k}} \]

   if 4.1e17 < k

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. unpow3 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. times-frac N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-/.f64 67.9

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 67.9%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 2\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 2\right)} \]

      8. lower-fma.f64 67.9

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   6. Applied rewrites 67.9%

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   7. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(t \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot {k}^{2}\right)} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot t\right)} \cdot k\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot t\right) \cdot k\right)} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot t\right)} \cdot k\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

      16. associate-*r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right)} \cdot \ell}} \]

      19. lower-cos.f64 71.7

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}} \]

   9. Applied rewrites 71.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{k}^{2}}{\color{blue}{{\ell}^{2}}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 56.4%

       \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \left(\frac{k}{\ell} \cdot \color{blue}{\frac{k}{\ell}}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 73.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.4 \cdot 10^{-81}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t\_m}{\ell}}\\ \mathbf{elif}\;t\_m \leq 4.8 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k \cdot t\_m} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m}}{\left(k \cdot t\_m\right) \cdot k}\\ \end{array} \end{array} \]

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 (<= t_m 6.4e-81)
    (/ 2.0 (* (/ (pow k 4.0) l) (/ t_m l)))
    (if (<= t_m 4.8e+130)
      (* (/ (/ l (* t_m t_m)) (* k t_m)) (/ l k))
      (/ (* (/ l t_m) (/ l t_m)) (* (* k t_m) 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 (t_m <= 6.4e-81) {
		tmp = 2.0 / ((pow(k, 4.0) / l) * (t_m / l));
	} else if (t_m <= 4.8e+130) {
		tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k);
	} else {
		tmp = ((l / t_m) * (l / t_m)) / ((k * t_m) * 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 (t_m <= 6.4d-81) then
        tmp = 2.0d0 / (((k ** 4.0d0) / l) * (t_m / l))
    else if (t_m <= 4.8d+130) then
        tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k)
    else
        tmp = ((l / t_m) * (l / t_m)) / ((k * t_m) * 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 (t_m <= 6.4e-81) {
		tmp = 2.0 / ((Math.pow(k, 4.0) / l) * (t_m / l));
	} else if (t_m <= 4.8e+130) {
		tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k);
	} else {
		tmp = ((l / t_m) * (l / t_m)) / ((k * t_m) * 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 t_m <= 6.4e-81:
		tmp = 2.0 / ((math.pow(k, 4.0) / l) * (t_m / l))
	elif t_m <= 4.8e+130:
		tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k)
	else:
		tmp = ((l / t_m) * (l / t_m)) / ((k * t_m) * 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 (t_m <= 6.4e-81)
		tmp = Float64(2.0 / Float64(Float64((k ^ 4.0) / l) * Float64(t_m / l)));
	elseif (t_m <= 4.8e+130)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / Float64(k * t_m)) * Float64(l / k));
	else
		tmp = Float64(Float64(Float64(l / t_m) * Float64(l / t_m)) / Float64(Float64(k * t_m) * 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 (t_m <= 6.4e-81)
		tmp = 2.0 / (((k ^ 4.0) / l) * (t_m / l));
	elseif (t_m <= 4.8e+130)
		tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k);
	else
		tmp = ((l / t_m) * (l / t_m)) / ((k * t_m) * 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[t$95$m, 6.4e-81], N[(2.0 / N[(N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.8e+130], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.4e-81

   1. Initial program 45.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{t}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. unpow3 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{\left(t \cdot t\right) \cdot t}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      5. times-frac N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{\color{blue}{t \cdot t}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      9. lower-/.f64 57.4

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 57.4%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} + 1\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} + 2\right)} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 2\right)} \]

      8. lower-fma.f64 57.4

         \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   6. Applied rewrites 57.4%

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}} \]

   7. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{2}{{k}^{2} \cdot \color{blue}{\left(t \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot {k}^{2}\right)} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(k \cdot k\right)}\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(t \cdot k\right) \cdot k\right)} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot t\right)} \cdot k\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot t\right) \cdot k\right)} \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot t\right)} \cdot k\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      13. lower-sin.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      14. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      15. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

      16. associate-*r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right)} \cdot \ell}} \]

      19. lower-cos.f64 65.0

          \[\leadsto \frac{2}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}} \]

   9. Applied rewrites 65.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot t\right) \cdot k\right) \cdot \frac{{\sin k}^{2}}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{{\ell}^{2}}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 56.8%

       \[\leadsto \frac{2}{\frac{{k}^{4}}{\ell} \cdot \color{blue}{\frac{t}{\ell}}} \]

   if 6.4e-81 < t < 4.80000000000000048e130

   1. Initial program 72.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 55.5

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]
   8. Step-by-step derivation

   9. Applied rewrites 73.0%

      \[\leadsto \frac{\frac{\ell}{t \cdot t}}{k \cdot t} \cdot \frac{\color{blue}{\ell}}{k} \]

   if 4.80000000000000048e130 < t

   1. Initial program 61.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 61.4

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 61.4%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 61.4%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 77.4%

      \[\leadsto \frac{\ell \cdot \frac{\ell}{t \cdot t}}{\color{blue}{\left(k \cdot t\right) \cdot k}} \]
   10. Step-by-step derivation

   11. Applied rewrites 92.5%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{\left(k \cdot t\right)} \cdot k} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 73.0% accurate, 7.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.4 \cdot 10^{-81}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t\_m} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;t\_m \leq 4.8 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k \cdot t\_m} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m}}{\left(k \cdot t\_m\right) \cdot k}\\ \end{array} \end{array} \]

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 (<= t_m 6.4e-81)
    (* (/ 2.0 (* (* k k) t_m)) (* (/ l k) (/ l k)))
    (if (<= t_m 4.8e+130)
      (* (/ (/ l (* t_m t_m)) (* k t_m)) (/ l k))
      (/ (* (/ l t_m) (/ l t_m)) (* (* k t_m) 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 (t_m <= 6.4e-81) {
		tmp = (2.0 / ((k * k) * t_m)) * ((l / k) * (l / k));
	} else if (t_m <= 4.8e+130) {
		tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k);
	} else {
		tmp = ((l / t_m) * (l / t_m)) / ((k * t_m) * 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 (t_m <= 6.4d-81) then
        tmp = (2.0d0 / ((k * k) * t_m)) * ((l / k) * (l / k))
    else if (t_m <= 4.8d+130) then
        tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k)
    else
        tmp = ((l / t_m) * (l / t_m)) / ((k * t_m) * 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 (t_m <= 6.4e-81) {
		tmp = (2.0 / ((k * k) * t_m)) * ((l / k) * (l / k));
	} else if (t_m <= 4.8e+130) {
		tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k);
	} else {
		tmp = ((l / t_m) * (l / t_m)) / ((k * t_m) * 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 t_m <= 6.4e-81:
		tmp = (2.0 / ((k * k) * t_m)) * ((l / k) * (l / k))
	elif t_m <= 4.8e+130:
		tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k)
	else:
		tmp = ((l / t_m) * (l / t_m)) / ((k * t_m) * 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 (t_m <= 6.4e-81)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * t_m)) * Float64(Float64(l / k) * Float64(l / k)));
	elseif (t_m <= 4.8e+130)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / Float64(k * t_m)) * Float64(l / k));
	else
		tmp = Float64(Float64(Float64(l / t_m) * Float64(l / t_m)) / Float64(Float64(k * t_m) * 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 (t_m <= 6.4e-81)
		tmp = (2.0 / ((k * k) * t_m)) * ((l / k) * (l / k));
	elseif (t_m <= 4.8e+130)
		tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k);
	else
		tmp = ((l / t_m) * (l / t_m)) / ((k * t_m) * 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[t$95$m, 6.4e-81], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.8e+130], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.4e-81

   1. Initial program 45.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot t}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\cos k \cdot {\ell}^{2}}}{{\sin k}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{\sin k}^{2}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}{{\sin k}^{2}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}{{\sin k}^{2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\left(\cos k \cdot \ell\right)} \cdot \ell}{{\sin k}^{2}} \]

      15. lower-cos.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}{{\sin k}^{2}} \]

      16. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\color{blue}{{\sin k}^{2}}} \]

      17. lower-sin.f64 59.5

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\color{blue}{\sin k}}^{2}} \]

   5. Applied rewrites 59.5%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{{\sin k}^{2}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.7%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{k} \cdot \color{blue}{\frac{\ell}{k}}\right) \]

   if 6.4e-81 < t < 4.80000000000000048e130

   1. Initial program 72.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 55.5

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]
   8. Step-by-step derivation

   9. Applied rewrites 73.0%

      \[\leadsto \frac{\frac{\ell}{t \cdot t}}{k \cdot t} \cdot \frac{\color{blue}{\ell}}{k} \]

   if 4.80000000000000048e130 < t

   1. Initial program 61.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 61.4

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 61.4%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 61.4%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 77.4%

      \[\leadsto \frac{\ell \cdot \frac{\ell}{t \cdot t}}{\color{blue}{\left(k \cdot t\right) \cdot k}} \]
   10. Step-by-step derivation

   11. Applied rewrites 92.5%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{\left(k \cdot t\right)} \cdot k} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 66.6% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k \cdot t\_m} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m}}{\left(k \cdot t\_m\right) \cdot k}\\ \end{array} \end{array} \]

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 (<= l 2.1e-44)
    (* (/ (/ l (* t_m t_m)) (* k t_m)) (/ l k))
    (/ (* (/ l t_m) (/ l t_m)) (* (* k t_m) 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 (l <= 2.1e-44) {
		tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k);
	} else {
		tmp = ((l / t_m) * (l / t_m)) / ((k * t_m) * 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 (l <= 2.1d-44) then
        tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k)
    else
        tmp = ((l / t_m) * (l / t_m)) / ((k * t_m) * 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 (l <= 2.1e-44) {
		tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k);
	} else {
		tmp = ((l / t_m) * (l / t_m)) / ((k * t_m) * 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 l <= 2.1e-44:
		tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k)
	else:
		tmp = ((l / t_m) * (l / t_m)) / ((k * t_m) * 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 (l <= 2.1e-44)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / Float64(k * t_m)) * Float64(l / k));
	else
		tmp = Float64(Float64(Float64(l / t_m) * Float64(l / t_m)) / Float64(Float64(k * t_m) * 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 (l <= 2.1e-44)
		tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k);
	else
		tmp = ((l / t_m) * (l / t_m)) / ((k * t_m) * 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[l, 2.1e-44], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.10000000000000001e-44

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 56.5

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 56.5%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]
   8. Step-by-step derivation

   9. Applied rewrites 66.5%

      \[\leadsto \frac{\frac{\ell}{t \cdot t}}{k \cdot t} \cdot \frac{\color{blue}{\ell}}{k} \]

   if 2.10000000000000001e-44 < l

   1. Initial program 50.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 46.1

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 46.1%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 46.1%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 57.2%

      \[\leadsto \frac{\ell \cdot \frac{\ell}{t \cdot t}}{\color{blue}{\left(k \cdot t\right) \cdot k}} \]
   10. Step-by-step derivation

   11. Applied rewrites 63.9%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{\left(k \cdot t\right)} \cdot k} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 66.2% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k \cdot t\_m} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{\ell}{t\_m}}{t\_m}}{\left(k \cdot t\_m\right) \cdot k}\\ \end{array} \end{array} \]

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 (<= l 2.5e-44)
    (* (/ (/ l (* t_m t_m)) (* k t_m)) (/ l k))
    (/ (* l (/ (/ l t_m) t_m)) (* (* k t_m) 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 (l <= 2.5e-44) {
		tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k);
	} else {
		tmp = (l * ((l / t_m) / t_m)) / ((k * t_m) * 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 (l <= 2.5d-44) then
        tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k)
    else
        tmp = (l * ((l / t_m) / t_m)) / ((k * t_m) * 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 (l <= 2.5e-44) {
		tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k);
	} else {
		tmp = (l * ((l / t_m) / t_m)) / ((k * t_m) * 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 l <= 2.5e-44:
		tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k)
	else:
		tmp = (l * ((l / t_m) / t_m)) / ((k * t_m) * 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 (l <= 2.5e-44)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / Float64(k * t_m)) * Float64(l / k));
	else
		tmp = Float64(Float64(l * Float64(Float64(l / t_m) / t_m)) / Float64(Float64(k * t_m) * 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 (l <= 2.5e-44)
		tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k);
	else
		tmp = (l * ((l / t_m) / t_m)) / ((k * t_m) * 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[l, 2.5e-44], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(N[(l / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.50000000000000019e-44

   1. Initial program 54.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 56.5

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 56.5%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]
   8. Step-by-step derivation

   9. Applied rewrites 66.5%

      \[\leadsto \frac{\frac{\ell}{t \cdot t}}{k \cdot t} \cdot \frac{\color{blue}{\ell}}{k} \]

   if 2.50000000000000019e-44 < l

   1. Initial program 50.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 46.1

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 46.1%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 46.1%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 57.2%

      \[\leadsto \frac{\ell \cdot \frac{\ell}{t \cdot t}}{\color{blue}{\left(k \cdot t\right) \cdot k}} \]
   10. Step-by-step derivation

   11. Applied rewrites 63.9%

       \[\leadsto \frac{\ell \cdot \frac{\frac{\ell}{t}}{t}}{\left(k \cdot \color{blue}{t}\right) \cdot k} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 64.7% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k \cdot t\_m} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t\_m} \cdot \frac{\frac{\ell}{k \cdot k}}{t\_m \cdot t\_m}\\ \end{array} \end{array} \]

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 (<= k 8e-145)
    (* (/ (/ l (* t_m t_m)) (* k t_m)) (/ l k))
    (* (/ l t_m) (/ (/ l (* k k)) (* t_m 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 (k <= 8e-145) {
		tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k);
	} else {
		tmp = (l / t_m) * ((l / (k * k)) / (t_m * 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 (k <= 8d-145) then
        tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k)
    else
        tmp = (l / t_m) * ((l / (k * k)) / (t_m * 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 (k <= 8e-145) {
		tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k);
	} else {
		tmp = (l / t_m) * ((l / (k * k)) / (t_m * 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 k <= 8e-145:
		tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k)
	else:
		tmp = (l / t_m) * ((l / (k * k)) / (t_m * 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 (k <= 8e-145)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / Float64(k * t_m)) * Float64(l / k));
	else
		tmp = Float64(Float64(l / t_m) * Float64(Float64(l / Float64(k * k)) / Float64(t_m * 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 (k <= 8e-145)
		tmp = ((l / (t_m * t_m)) / (k * t_m)) * (l / k);
	else
		tmp = (l / t_m) * ((l / (k * k)) / (t_m * 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[k, 8e-145], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.99999999999999932e-145

   1. Initial program 52.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 49.6

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 49.6%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 57.4%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}}}{k} \cdot \frac{\ell}{k}} \]
   8. Step-by-step derivation

   9. Applied rewrites 64.0%

      \[\leadsto \frac{\frac{\ell}{t \cdot t}}{k \cdot t} \cdot \frac{\color{blue}{\ell}}{k} \]

   if 7.99999999999999932e-145 < k

   1. Initial program 54.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 60.7

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 63.9%

      \[\leadsto \frac{\ell}{t} \cdot \color{blue}{\frac{\frac{\ell}{k \cdot k}}{t \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 62.0% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell}{t\_m \cdot t\_m} \cdot \frac{\ell}{\left(k \cdot t\_m\right) \cdot k}\right) \end{array} \]

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 (* t_m t_m)) (/ l (* (* k t_m) k)))))

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 / (t_m * t_m)) * (l / ((k * t_m) * k)));
}

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 / (t_m * t_m)) * (l / ((k * t_m) * k)))
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 / (t_m * t_m)) * (l / ((k * t_m) * k)));
}

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 / (t_m * t_m)) * (l / ((k * t_m) * k)))

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(t_m * t_m)) * Float64(l / Float64(Float64(k * t_m) * k))))
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 / (t_m * t_m)) * (l / ((k * t_m) * k)));
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[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   9. lower-*.f64 53.6

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

5. Applied rewrites 53.6%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
6. Step-by-step derivation

7. Applied rewrites 53.6%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
8. Step-by-step derivation

9. Applied rewrites 60.1%

   \[\leadsto \frac{\ell \cdot \frac{\ell}{t \cdot t}}{\color{blue}{\left(k \cdot t\right) \cdot k}} \]
10. Step-by-step derivation

11. Applied rewrites 62.2%

    \[\leadsto \frac{\ell}{t \cdot t} \cdot \color{blue}{\frac{\ell}{\left(k \cdot t\right) \cdot k}} \]
12. Add Preprocessing

Alternative 23: 57.4% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\ell \cdot \ell}{\left(t\_m \cdot t\_m\right) \cdot \left(\left(k \cdot t\_m\right) \cdot k\right)} \end{array} \]

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 l) (* (* t_m t_m) (* (* k t_m) k)))))

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 * l) / ((t_m * t_m) * ((k * t_m) * k)));
}

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 * l) / ((t_m * t_m) * ((k * t_m) * k)))
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 * l) / ((t_m * t_m) * ((k * t_m) * k)));
}

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 * l) / ((t_m * t_m) * ((k * t_m) * k)))

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 * l) / Float64(Float64(t_m * t_m) * Float64(Float64(k * t_m) * k))))
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 * l) / ((t_m * t_m) * ((k * t_m) * k)));
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 * l), $MachinePrecision] / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   9. lower-*.f64 53.6

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

5. Applied rewrites 53.6%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
6. Step-by-step derivation

7. Applied rewrites 50.6%

   \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({t}^{3} \cdot k\right) \cdot k}} \]
8. Step-by-step derivation

9. Applied rewrites 55.9%

   \[\leadsto \frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \color{blue}{\left(\left(k \cdot t\right) \cdot k\right)}} \]
10. Add Preprocessing

Alternative 24: 53.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\ell \cdot \ell}{t\_m \cdot \left(\left(t\_m \cdot t\_m\right) \cdot \left(k \cdot k\right)\right)} \end{array} \]

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 l) (* t_m (* (* t_m t_m) (* k k))))))

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 * l) / (t_m * ((t_m * t_m) * (k * k))));
}

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 * l) / (t_m * ((t_m * t_m) * (k * k))))
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 * l) / (t_m * ((t_m * t_m) * (k * k))));
}

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 * l) / (t_m * ((t_m * t_m) * (k * k))))

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 * l) / Float64(t_m * Float64(Float64(t_m * t_m) * Float64(k * k)))))
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 * l) / (t_m * ((t_m * t_m) * (k * k))));
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 * l), $MachinePrecision] / N[(t$95$m * N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   9. lower-*.f64 53.6

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

5. Applied rewrites 53.6%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
6. Step-by-step derivation

7. Applied rewrites 50.6%

   \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left({t}^{3} \cdot k\right) \cdot k}} \]
8. Step-by-step derivation

9. Applied rewrites 51.6%

   \[\leadsto \frac{\ell \cdot \ell}{t \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(k \cdot k\right)\right)}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025015 
(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))))