Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.4% → 92.5%

Time: 9.1s

Alternatives: 19

Speedup: 10.7×

• 27.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% 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: 92.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.1 \cdot 10^{-57}:\\ \;\;\;\;\frac{\left(\left(\frac{\ell}{{\sin k}^{2}} \cdot \cos k\right) \cdot \frac{\ell}{k}\right) \cdot 2}{t\_m \cdot k}\\ \mathbf{elif}\;t\_m \leq 9.5 \cdot 10^{+183}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{t\_2 + 2}}{\tan k}}{\frac{{t\_m}^{1.5}}{\ell}}}{\sin k \cdot \frac{{\left({t\_m}^{0.75}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \sin k\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + t\_2\right) + 1\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 6.1e-57)
      (/ (* (* (* (/ l (pow (sin k) 2.0)) (cos k)) (/ l k)) 2.0) (* t_m k))
      (if (<= t_m 9.5e+183)
        (/
         (/ (/ (/ 2.0 (+ t_2 2.0)) (tan k)) (/ (pow t_m 1.5) l))
         (* (sin k) (/ (pow (pow t_m 0.75) 2.0) l)))
        (/
         2.0
         (*
          (* (* t_m (* (/ t_m l) (* (/ t_m l) (sin k)))) (tan k))
          (+ (+ 1.0 t_2) 1.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 6.1e-57) {
		tmp = ((((l / pow(sin(k), 2.0)) * cos(k)) * (l / k)) * 2.0) / (t_m * k);
	} else if (t_m <= 9.5e+183) {
		tmp = (((2.0 / (t_2 + 2.0)) / tan(k)) / (pow(t_m, 1.5) / l)) / (sin(k) * (pow(pow(t_m, 0.75), 2.0) / l));
	} else {
		tmp = 2.0 / (((t_m * ((t_m / l) * ((t_m / l) * sin(k)))) * tan(k)) * ((1.0 + t_2) + 1.0));
	}
	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) :: t_2
    real(8) :: tmp
    t_2 = (k / t_m) ** 2.0d0
    if (t_m <= 6.1d-57) then
        tmp = ((((l / (sin(k) ** 2.0d0)) * cos(k)) * (l / k)) * 2.0d0) / (t_m * k)
    else if (t_m <= 9.5d+183) then
        tmp = (((2.0d0 / (t_2 + 2.0d0)) / tan(k)) / ((t_m ** 1.5d0) / l)) / (sin(k) * (((t_m ** 0.75d0) ** 2.0d0) / l))
    else
        tmp = 2.0d0 / (((t_m * ((t_m / l) * ((t_m / l) * sin(k)))) * tan(k)) * ((1.0d0 + t_2) + 1.0d0))
    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 t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 6.1e-57) {
		tmp = ((((l / Math.pow(Math.sin(k), 2.0)) * Math.cos(k)) * (l / k)) * 2.0) / (t_m * k);
	} else if (t_m <= 9.5e+183) {
		tmp = (((2.0 / (t_2 + 2.0)) / Math.tan(k)) / (Math.pow(t_m, 1.5) / l)) / (Math.sin(k) * (Math.pow(Math.pow(t_m, 0.75), 2.0) / l));
	} else {
		tmp = 2.0 / (((t_m * ((t_m / l) * ((t_m / l) * Math.sin(k)))) * Math.tan(k)) * ((1.0 + t_2) + 1.0));
	}
	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):
	t_2 = math.pow((k / t_m), 2.0)
	tmp = 0
	if t_m <= 6.1e-57:
		tmp = ((((l / math.pow(math.sin(k), 2.0)) * math.cos(k)) * (l / k)) * 2.0) / (t_m * k)
	elif t_m <= 9.5e+183:
		tmp = (((2.0 / (t_2 + 2.0)) / math.tan(k)) / (math.pow(t_m, 1.5) / l)) / (math.sin(k) * (math.pow(math.pow(t_m, 0.75), 2.0) / l))
	else:
		tmp = 2.0 / (((t_m * ((t_m / l) * ((t_m / l) * math.sin(k)))) * math.tan(k)) * ((1.0 + t_2) + 1.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 6.1e-57)
		tmp = Float64(Float64(Float64(Float64(Float64(l / (sin(k) ^ 2.0)) * cos(k)) * Float64(l / k)) * 2.0) / Float64(t_m * k));
	elseif (t_m <= 9.5e+183)
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(t_2 + 2.0)) / tan(k)) / Float64((t_m ^ 1.5) / l)) / Float64(sin(k) * Float64(((t_m ^ 0.75) ^ 2.0) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * sin(k)))) * tan(k)) * Float64(Float64(1.0 + t_2) + 1.0)));
	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)
	t_2 = (k / t_m) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 6.1e-57)
		tmp = ((((l / (sin(k) ^ 2.0)) * cos(k)) * (l / k)) * 2.0) / (t_m * k);
	elseif (t_m <= 9.5e+183)
		tmp = (((2.0 / (t_2 + 2.0)) / tan(k)) / ((t_m ^ 1.5) / l)) / (sin(k) * (((t_m ^ 0.75) ^ 2.0) / l));
	else
		tmp = 2.0 / (((t_m * ((t_m / l) * ((t_m / l) * sin(k)))) * tan(k)) * ((1.0 + t_2) + 1.0));
	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_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.1e-57], N[(N[(N[(N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.5e+183], N[(N[(N[(N[(2.0 / N[(t$95$2 + 2.0), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[Power[t$95$m, 0.75], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(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] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 6.0999999999999998e-57

   1. Initial program 51.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 70.3%

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

   9. Applied rewrites 75.1%

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

   if 6.0999999999999998e-57 < t < 9.5000000000000003e183

   1. Initial program 77.1%

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

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 87.8

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 87.8%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\tan k}}{\frac{{t}^{1.5}}{\ell}}}{\sin k \cdot \frac{{t}^{1.5}}{\ell}}} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 96.1

         \[\leadsto \frac{\frac{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\tan k}}{\frac{{t}^{1.5}}{\ell}}}{\sin k \cdot \frac{{\left({t}^{\color{blue}{0.75}}\right)}^{2}}{\ell}} \]

   8. Applied rewrites 96.1%

      \[\leadsto \frac{\frac{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\tan k}}{\frac{{t}^{1.5}}{\ell}}}{\sin k \cdot \frac{\color{blue}{{\left({t}^{0.75}\right)}^{2}}}{\ell}} \]

   if 9.5000000000000003e183 < t

   1. Initial program 58.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 70.0

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

   4. Applied rewrites 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 95.0

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

   6. Applied rewrites 95.0%

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

Alternative 2: 92.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{t\_m}^{1.5}}{\ell}\\ t_3 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.1 \cdot 10^{-57}:\\ \;\;\;\;\frac{\left(\left(\frac{\ell}{{\sin k}^{2}} \cdot \cos k\right) \cdot \frac{\ell}{k}\right) \cdot 2}{t\_m \cdot k}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+173}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{t\_3 + 2}}{\tan k}}{t\_2}}{\sin k \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \sin k\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + t\_3\right) + 1\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (pow t_m 1.5) l)) (t_3 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 6.1e-57)
      (/ (* (* (* (/ l (pow (sin k) 2.0)) (cos k)) (/ l k)) 2.0) (* t_m k))
      (if (<= t_m 2e+173)
        (/ (/ (/ (/ 2.0 (+ t_3 2.0)) (tan k)) t_2) (* (sin k) t_2))
        (/
         2.0
         (*
          (* (* t_m (* (/ t_m l) (* (/ t_m l) (sin k)))) (tan k))
          (+ (+ 1.0 t_3) 1.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(t_m, 1.5) / l;
	double t_3 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 6.1e-57) {
		tmp = ((((l / pow(sin(k), 2.0)) * cos(k)) * (l / k)) * 2.0) / (t_m * k);
	} else if (t_m <= 2e+173) {
		tmp = (((2.0 / (t_3 + 2.0)) / tan(k)) / t_2) / (sin(k) * t_2);
	} else {
		tmp = 2.0 / (((t_m * ((t_m / l) * ((t_m / l) * sin(k)))) * tan(k)) * ((1.0 + t_3) + 1.0));
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (t_m ** 1.5d0) / l
    t_3 = (k / t_m) ** 2.0d0
    if (t_m <= 6.1d-57) then
        tmp = ((((l / (sin(k) ** 2.0d0)) * cos(k)) * (l / k)) * 2.0d0) / (t_m * k)
    else if (t_m <= 2d+173) then
        tmp = (((2.0d0 / (t_3 + 2.0d0)) / tan(k)) / t_2) / (sin(k) * t_2)
    else
        tmp = 2.0d0 / (((t_m * ((t_m / l) * ((t_m / l) * sin(k)))) * tan(k)) * ((1.0d0 + t_3) + 1.0d0))
    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 t_2 = Math.pow(t_m, 1.5) / l;
	double t_3 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 6.1e-57) {
		tmp = ((((l / Math.pow(Math.sin(k), 2.0)) * Math.cos(k)) * (l / k)) * 2.0) / (t_m * k);
	} else if (t_m <= 2e+173) {
		tmp = (((2.0 / (t_3 + 2.0)) / Math.tan(k)) / t_2) / (Math.sin(k) * t_2);
	} else {
		tmp = 2.0 / (((t_m * ((t_m / l) * ((t_m / l) * Math.sin(k)))) * Math.tan(k)) * ((1.0 + t_3) + 1.0));
	}
	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):
	t_2 = math.pow(t_m, 1.5) / l
	t_3 = math.pow((k / t_m), 2.0)
	tmp = 0
	if t_m <= 6.1e-57:
		tmp = ((((l / math.pow(math.sin(k), 2.0)) * math.cos(k)) * (l / k)) * 2.0) / (t_m * k)
	elif t_m <= 2e+173:
		tmp = (((2.0 / (t_3 + 2.0)) / math.tan(k)) / t_2) / (math.sin(k) * t_2)
	else:
		tmp = 2.0 / (((t_m * ((t_m / l) * ((t_m / l) * math.sin(k)))) * math.tan(k)) * ((1.0 + t_3) + 1.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64((t_m ^ 1.5) / l)
	t_3 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 6.1e-57)
		tmp = Float64(Float64(Float64(Float64(Float64(l / (sin(k) ^ 2.0)) * cos(k)) * Float64(l / k)) * 2.0) / Float64(t_m * k));
	elseif (t_m <= 2e+173)
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(t_3 + 2.0)) / tan(k)) / t_2) / Float64(sin(k) * t_2));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * sin(k)))) * tan(k)) * Float64(Float64(1.0 + t_3) + 1.0)));
	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)
	t_2 = (t_m ^ 1.5) / l;
	t_3 = (k / t_m) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 6.1e-57)
		tmp = ((((l / (sin(k) ^ 2.0)) * cos(k)) * (l / k)) * 2.0) / (t_m * k);
	elseif (t_m <= 2e+173)
		tmp = (((2.0 / (t_3 + 2.0)) / tan(k)) / t_2) / (sin(k) * t_2);
	else
		tmp = 2.0 / (((t_m * ((t_m / l) * ((t_m / l) * sin(k)))) * tan(k)) * ((1.0 + t_3) + 1.0));
	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_] := Block[{t$95$2 = N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.1e-57], N[(N[(N[(N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e+173], N[(N[(N[(N[(2.0 / N[(t$95$3 + 2.0), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(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] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + t$95$3), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 6.0999999999999998e-57

   1. Initial program 51.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 70.3%

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

   9. Applied rewrites 75.1%

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

   if 6.0999999999999998e-57 < t < 2e173

   1. Initial program 76.6%

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

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 87.6

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 87.6%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\tan k}}{\frac{{t}^{1.5}}{\ell}}}{\sin k \cdot \frac{{t}^{1.5}}{\ell}}} \]

   if 2e173 < t

   1. Initial program 60.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 71.5

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

   4. Applied rewrites 71.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 95.2

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

   6. Applied rewrites 95.2%

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

Alternative 3: 91.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t_3 := \frac{{t\_m}^{1.5}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\left(\left(\frac{\ell}{{\sin k}^{2}} \cdot \cos k\right) \cdot \frac{\ell}{k}\right) \cdot 2}{t\_m \cdot k}\\ \mathbf{elif}\;t\_m \leq 1.02 \cdot 10^{+157}:\\ \;\;\;\;\frac{2}{t\_3 \cdot \left(\left(\sin k \cdot t\_3\right) \cdot \left(\left(t\_2 + 2\right) \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \sin k\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + t\_2\right) + 1\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)) (t_3 (/ (pow t_m 1.5) l)))
   (*
    t_s
    (if (<= t_m 6.2e-57)
      (/ (* (* (* (/ l (pow (sin k) 2.0)) (cos k)) (/ l k)) 2.0) (* t_m k))
      (if (<= t_m 1.02e+157)
        (/ 2.0 (* t_3 (* (* (sin k) t_3) (* (+ t_2 2.0) (tan k)))))
        (/
         2.0
         (*
          (* (* t_m (* (/ t_m l) (* (/ t_m l) (sin k)))) (tan k))
          (+ (+ 1.0 t_2) 1.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double t_3 = pow(t_m, 1.5) / l;
	double tmp;
	if (t_m <= 6.2e-57) {
		tmp = ((((l / pow(sin(k), 2.0)) * cos(k)) * (l / k)) * 2.0) / (t_m * k);
	} else if (t_m <= 1.02e+157) {
		tmp = 2.0 / (t_3 * ((sin(k) * t_3) * ((t_2 + 2.0) * tan(k))));
	} else {
		tmp = 2.0 / (((t_m * ((t_m / l) * ((t_m / l) * sin(k)))) * tan(k)) * ((1.0 + t_2) + 1.0));
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (k / t_m) ** 2.0d0
    t_3 = (t_m ** 1.5d0) / l
    if (t_m <= 6.2d-57) then
        tmp = ((((l / (sin(k) ** 2.0d0)) * cos(k)) * (l / k)) * 2.0d0) / (t_m * k)
    else if (t_m <= 1.02d+157) then
        tmp = 2.0d0 / (t_3 * ((sin(k) * t_3) * ((t_2 + 2.0d0) * tan(k))))
    else
        tmp = 2.0d0 / (((t_m * ((t_m / l) * ((t_m / l) * sin(k)))) * tan(k)) * ((1.0d0 + t_2) + 1.0d0))
    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 t_2 = Math.pow((k / t_m), 2.0);
	double t_3 = Math.pow(t_m, 1.5) / l;
	double tmp;
	if (t_m <= 6.2e-57) {
		tmp = ((((l / Math.pow(Math.sin(k), 2.0)) * Math.cos(k)) * (l / k)) * 2.0) / (t_m * k);
	} else if (t_m <= 1.02e+157) {
		tmp = 2.0 / (t_3 * ((Math.sin(k) * t_3) * ((t_2 + 2.0) * Math.tan(k))));
	} else {
		tmp = 2.0 / (((t_m * ((t_m / l) * ((t_m / l) * Math.sin(k)))) * Math.tan(k)) * ((1.0 + t_2) + 1.0));
	}
	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):
	t_2 = math.pow((k / t_m), 2.0)
	t_3 = math.pow(t_m, 1.5) / l
	tmp = 0
	if t_m <= 6.2e-57:
		tmp = ((((l / math.pow(math.sin(k), 2.0)) * math.cos(k)) * (l / k)) * 2.0) / (t_m * k)
	elif t_m <= 1.02e+157:
		tmp = 2.0 / (t_3 * ((math.sin(k) * t_3) * ((t_2 + 2.0) * math.tan(k))))
	else:
		tmp = 2.0 / (((t_m * ((t_m / l) * ((t_m / l) * math.sin(k)))) * math.tan(k)) * ((1.0 + t_2) + 1.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	t_3 = Float64((t_m ^ 1.5) / l)
	tmp = 0.0
	if (t_m <= 6.2e-57)
		tmp = Float64(Float64(Float64(Float64(Float64(l / (sin(k) ^ 2.0)) * cos(k)) * Float64(l / k)) * 2.0) / Float64(t_m * k));
	elseif (t_m <= 1.02e+157)
		tmp = Float64(2.0 / Float64(t_3 * Float64(Float64(sin(k) * t_3) * Float64(Float64(t_2 + 2.0) * tan(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * sin(k)))) * tan(k)) * Float64(Float64(1.0 + t_2) + 1.0)));
	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)
	t_2 = (k / t_m) ^ 2.0;
	t_3 = (t_m ^ 1.5) / l;
	tmp = 0.0;
	if (t_m <= 6.2e-57)
		tmp = ((((l / (sin(k) ^ 2.0)) * cos(k)) * (l / k)) * 2.0) / (t_m * k);
	elseif (t_m <= 1.02e+157)
		tmp = 2.0 / (t_3 * ((sin(k) * t_3) * ((t_2 + 2.0) * tan(k))));
	else
		tmp = 2.0 / (((t_m * ((t_m / l) * ((t_m / l) * sin(k)))) * tan(k)) * ((1.0 + t_2) + 1.0));
	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_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.2e-57], N[(N[(N[(N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.02e+157], N[(2.0 / N[(t$95$3 * N[(N[(N[Sin[k], $MachinePrecision] * t$95$3), $MachinePrecision] * N[(N[(t$95$2 + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(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] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 6.19999999999999952e-57

   1. Initial program 51.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 70.3%

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

   9. Applied rewrites 75.1%

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

   if 6.19999999999999952e-57 < t < 1.02000000000000003e157

   1. Initial program 80.8%

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

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 87.1

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 87.1%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 95.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{1.5}}{\ell} \cdot \left(\left(\sin k \cdot \frac{{t}^{1.5}}{\ell}\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right)\right)}} \]

   if 1.02000000000000003e157 < t

   1. Initial program 58.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 66.9

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

   4. Applied rewrites 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 93.9

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

   6. Applied rewrites 93.9%

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

Alternative 4: 68.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq \infty:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \frac{\ell}{t\_m}}{\left(k \cdot k\right) \cdot t\_m}\\ \end{array} \end{array} \]

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 (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       INFINITY)
    (* (/ (/ l (* t_m t_m)) k) (/ l (* t_m k)))
    (/ (* (/ l t_m) (/ l t_m)) (* (* 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 ((2.0 / ((((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= ((double) INFINITY)) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	} else {
		tmp = ((l / t_m) * (l / t_m)) / ((k * k) * t_m);
	}
	return t_s * tmp;
}

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 ((2.0 / ((((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= Double.POSITIVE_INFINITY) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	} else {
		tmp = ((l / t_m) * (l / t_m)) / ((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 (2.0 / ((((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= math.inf:
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k))
	else:
		tmp = ((l / t_m) * (l / t_m)) / ((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 (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= Inf)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(t_m * k)));
	else
		tmp = Float64(Float64(Float64(l / t_m) * Float64(l / t_m)) / Float64(Float64(k * 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 ((2.0 / (((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= Inf)
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	else
		tmp = ((l / t_m) * (l / t_m)) / ((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[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 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$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < +inf.0

   1. Initial program 85.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 75.3

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

   5. Applied rewrites 75.3%

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

   7. Applied rewrites 75.3%

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

   9. Applied rewrites 72.1%

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

   11. Applied rewrites 82.4%

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

   if +inf.0 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 0.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 15.7

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

   5. Applied rewrites 15.7%

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

   7. Applied rewrites 15.7%

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

   9. Applied rewrites 21.8%

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

   11. Applied rewrites 38.3%

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

Alternative 5: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\left(\left(\frac{\ell}{{\sin k}^{2}} \cdot \cos k\right) \cdot \frac{\ell}{k}\right) \cdot 2}{t\_m \cdot k}\\ \mathbf{elif}\;t\_m \leq 1.12 \cdot 10^{+83}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{\sin k}{\ell} \cdot {t\_m}^{3}\right) \cdot \left(\left(t\_2 + 2\right) \cdot \tan k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \sin k\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + t\_2\right) + 1\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 6.2e-57)
      (/ (* (* (* (/ l (pow (sin k) 2.0)) (cos k)) (/ l k)) 2.0) (* t_m k))
      (if (<= t_m 1.12e+83)
        (/
         2.0
         (/ (* (* (/ (sin k) l) (pow t_m 3.0)) (* (+ t_2 2.0) (tan k))) l))
        (/
         2.0
         (*
          (* (* t_m (* (/ t_m l) (* (/ t_m l) (sin k)))) (tan k))
          (+ (+ 1.0 t_2) 1.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 6.2e-57) {
		tmp = ((((l / pow(sin(k), 2.0)) * cos(k)) * (l / k)) * 2.0) / (t_m * k);
	} else if (t_m <= 1.12e+83) {
		tmp = 2.0 / ((((sin(k) / l) * pow(t_m, 3.0)) * ((t_2 + 2.0) * tan(k))) / l);
	} else {
		tmp = 2.0 / (((t_m * ((t_m / l) * ((t_m / l) * sin(k)))) * tan(k)) * ((1.0 + t_2) + 1.0));
	}
	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) :: t_2
    real(8) :: tmp
    t_2 = (k / t_m) ** 2.0d0
    if (t_m <= 6.2d-57) then
        tmp = ((((l / (sin(k) ** 2.0d0)) * cos(k)) * (l / k)) * 2.0d0) / (t_m * k)
    else if (t_m <= 1.12d+83) then
        tmp = 2.0d0 / ((((sin(k) / l) * (t_m ** 3.0d0)) * ((t_2 + 2.0d0) * tan(k))) / l)
    else
        tmp = 2.0d0 / (((t_m * ((t_m / l) * ((t_m / l) * sin(k)))) * tan(k)) * ((1.0d0 + t_2) + 1.0d0))
    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 t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 6.2e-57) {
		tmp = ((((l / Math.pow(Math.sin(k), 2.0)) * Math.cos(k)) * (l / k)) * 2.0) / (t_m * k);
	} else if (t_m <= 1.12e+83) {
		tmp = 2.0 / ((((Math.sin(k) / l) * Math.pow(t_m, 3.0)) * ((t_2 + 2.0) * Math.tan(k))) / l);
	} else {
		tmp = 2.0 / (((t_m * ((t_m / l) * ((t_m / l) * Math.sin(k)))) * Math.tan(k)) * ((1.0 + t_2) + 1.0));
	}
	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):
	t_2 = math.pow((k / t_m), 2.0)
	tmp = 0
	if t_m <= 6.2e-57:
		tmp = ((((l / math.pow(math.sin(k), 2.0)) * math.cos(k)) * (l / k)) * 2.0) / (t_m * k)
	elif t_m <= 1.12e+83:
		tmp = 2.0 / ((((math.sin(k) / l) * math.pow(t_m, 3.0)) * ((t_2 + 2.0) * math.tan(k))) / l)
	else:
		tmp = 2.0 / (((t_m * ((t_m / l) * ((t_m / l) * math.sin(k)))) * math.tan(k)) * ((1.0 + t_2) + 1.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 6.2e-57)
		tmp = Float64(Float64(Float64(Float64(Float64(l / (sin(k) ^ 2.0)) * cos(k)) * Float64(l / k)) * 2.0) / Float64(t_m * k));
	elseif (t_m <= 1.12e+83)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) / l) * (t_m ^ 3.0)) * Float64(Float64(t_2 + 2.0) * tan(k))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * sin(k)))) * tan(k)) * Float64(Float64(1.0 + t_2) + 1.0)));
	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)
	t_2 = (k / t_m) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 6.2e-57)
		tmp = ((((l / (sin(k) ^ 2.0)) * cos(k)) * (l / k)) * 2.0) / (t_m * k);
	elseif (t_m <= 1.12e+83)
		tmp = 2.0 / ((((sin(k) / l) * (t_m ^ 3.0)) * ((t_2 + 2.0) * tan(k))) / l);
	else
		tmp = 2.0 / (((t_m * ((t_m / l) * ((t_m / l) * sin(k)))) * tan(k)) * ((1.0 + t_2) + 1.0));
	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_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.2e-57], N[(N[(N[(N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.12e+83], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(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] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 6.19999999999999952e-57

   1. Initial program 51.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 70.3%

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

   9. Applied rewrites 75.1%

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

   if 6.19999999999999952e-57 < t < 1.12e83

   1. Initial program 81.6%

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

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 84.5

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 84.5%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

   6. Applied rewrites 96.7%

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

   if 1.12e83 < t

   1. Initial program 65.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 73.3

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

   4. Applied rewrites 73.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 93.6

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

   6. Applied rewrites 93.6%

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

Alternative 6: 90.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\left(\left(\frac{\ell}{{\sin k}^{2}} \cdot \cos k\right) \cdot \frac{\ell}{k}\right) \cdot 2}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \sin k\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

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 6.2e-57)
    (/ (* (* (* (/ l (pow (sin k) 2.0)) (cos k)) (/ l k)) 2.0) (* t_m k))
    (/
     2.0
     (*
      (* (* t_m (* (/ t_m l) (* (/ t_m l) (sin k)))) (tan k))
      (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0))))))

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 <= 6.2e-57) {
		tmp = ((((l / pow(sin(k), 2.0)) * cos(k)) * (l / k)) * 2.0) / (t_m * k);
	} else {
		tmp = 2.0 / (((t_m * ((t_m / l) * ((t_m / l) * sin(k)))) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	}
	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 <= 6.2d-57) then
        tmp = ((((l / (sin(k) ** 2.0d0)) * cos(k)) * (l / k)) * 2.0d0) / (t_m * k)
    else
        tmp = 2.0d0 / (((t_m * ((t_m / l) * ((t_m / l) * sin(k)))) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    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 <= 6.2e-57) {
		tmp = ((((l / Math.pow(Math.sin(k), 2.0)) * Math.cos(k)) * (l / k)) * 2.0) / (t_m * k);
	} else {
		tmp = 2.0 / (((t_m * ((t_m / l) * ((t_m / l) * Math.sin(k)))) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	}
	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 <= 6.2e-57:
		tmp = ((((l / math.pow(math.sin(k), 2.0)) * math.cos(k)) * (l / k)) * 2.0) / (t_m * k)
	else:
		tmp = 2.0 / (((t_m * ((t_m / l) * ((t_m / l) * math.sin(k)))) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	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 <= 6.2e-57)
		tmp = Float64(Float64(Float64(Float64(Float64(l / (sin(k) ^ 2.0)) * cos(k)) * Float64(l / k)) * 2.0) / Float64(t_m * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64(Float64(t_m / l) * Float64(Float64(t_m / l) * sin(k)))) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	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 <= 6.2e-57)
		tmp = ((((l / (sin(k) ^ 2.0)) * cos(k)) * (l / k)) * 2.0) / (t_m * k);
	else
		tmp = 2.0 / (((t_m * ((t_m / l) * ((t_m / l) * sin(k)))) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	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, 6.2e-57], N[(N[(N[(N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(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] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.19999999999999952e-57

   1. Initial program 51.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 70.3%

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

   9. Applied rewrites 75.1%

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

   if 6.19999999999999952e-57 < t

   1. Initial program 72.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 78.1

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

   4. Applied rewrites 78.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 89.6

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

   6. Applied rewrites 89.6%

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

Alternative 7: 74.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{{\sin k}^{2}}\\ t_3 := \frac{{t\_m}^{1.5}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\frac{1}{k}}{t\_3}}{\sin k \cdot t\_3}\\ \mathbf{elif}\;k \leq 9.6 \cdot 10^{+170}:\\ \;\;\;\;\left(\cos k \cdot \ell\right) \cdot \left(t\_2 \cdot \frac{2}{\left(k \cdot k\right) \cdot t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(t\_2 \cdot \cos k\right) \cdot \frac{\ell}{k}\right) \cdot 2}{t\_m \cdot k}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (pow (sin k) 2.0))) (t_3 (/ (pow t_m 1.5) l)))
   (*
    t_s
    (if (<= k 1.05e-9)
      (/ (/ (/ 1.0 k) t_3) (* (sin k) t_3))
      (if (<= k 9.6e+170)
        (* (* (cos k) l) (* t_2 (/ 2.0 (* (* k k) t_m))))
        (/ (* (* (* t_2 (cos k)) (/ l k)) 2.0) (* 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 t_2 = l / pow(sin(k), 2.0);
	double t_3 = pow(t_m, 1.5) / l;
	double tmp;
	if (k <= 1.05e-9) {
		tmp = ((1.0 / k) / t_3) / (sin(k) * t_3);
	} else if (k <= 9.6e+170) {
		tmp = (cos(k) * l) * (t_2 * (2.0 / ((k * k) * t_m)));
	} else {
		tmp = (((t_2 * cos(k)) * (l / k)) * 2.0) / (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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = l / (sin(k) ** 2.0d0)
    t_3 = (t_m ** 1.5d0) / l
    if (k <= 1.05d-9) then
        tmp = ((1.0d0 / k) / t_3) / (sin(k) * t_3)
    else if (k <= 9.6d+170) then
        tmp = (cos(k) * l) * (t_2 * (2.0d0 / ((k * k) * t_m)))
    else
        tmp = (((t_2 * cos(k)) * (l / k)) * 2.0d0) / (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 t_2 = l / Math.pow(Math.sin(k), 2.0);
	double t_3 = Math.pow(t_m, 1.5) / l;
	double tmp;
	if (k <= 1.05e-9) {
		tmp = ((1.0 / k) / t_3) / (Math.sin(k) * t_3);
	} else if (k <= 9.6e+170) {
		tmp = (Math.cos(k) * l) * (t_2 * (2.0 / ((k * k) * t_m)));
	} else {
		tmp = (((t_2 * Math.cos(k)) * (l / k)) * 2.0) / (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):
	t_2 = l / math.pow(math.sin(k), 2.0)
	t_3 = math.pow(t_m, 1.5) / l
	tmp = 0
	if k <= 1.05e-9:
		tmp = ((1.0 / k) / t_3) / (math.sin(k) * t_3)
	elif k <= 9.6e+170:
		tmp = (math.cos(k) * l) * (t_2 * (2.0 / ((k * k) * t_m)))
	else:
		tmp = (((t_2 * math.cos(k)) * (l / k)) * 2.0) / (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)
	t_2 = Float64(l / (sin(k) ^ 2.0))
	t_3 = Float64((t_m ^ 1.5) / l)
	tmp = 0.0
	if (k <= 1.05e-9)
		tmp = Float64(Float64(Float64(1.0 / k) / t_3) / Float64(sin(k) * t_3));
	elseif (k <= 9.6e+170)
		tmp = Float64(Float64(cos(k) * l) * Float64(t_2 * Float64(2.0 / Float64(Float64(k * k) * t_m))));
	else
		tmp = Float64(Float64(Float64(Float64(t_2 * cos(k)) * Float64(l / k)) * 2.0) / Float64(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)
	t_2 = l / (sin(k) ^ 2.0);
	t_3 = (t_m ^ 1.5) / l;
	tmp = 0.0;
	if (k <= 1.05e-9)
		tmp = ((1.0 / k) / t_3) / (sin(k) * t_3);
	elseif (k <= 9.6e+170)
		tmp = (cos(k) * l) * (t_2 * (2.0 / ((k * k) * t_m)));
	else
		tmp = (((t_2 * cos(k)) * (l / k)) * 2.0) / (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_] := Block[{t$95$2 = N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.05e-9], N[(N[(N[(1.0 / k), $MachinePrecision] / t$95$3), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 9.6e+170], N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * N[(t$95$2 * N[(2.0 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$2 * N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.0500000000000001e-9

   1. Initial program 60.3%

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

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 36.3

          \[\leadsto \frac{2}{\left(\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{\color{blue}{1.5}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied rewrites 36.3%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Applied rewrites 38.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 2}}{\tan k}}{\frac{{t}^{1.5}}{\ell}}}{\sin k \cdot \frac{{t}^{1.5}}{\ell}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{k}}}{\frac{{t}^{\frac{3}{2}}}{\ell}}}{\sin k \cdot \frac{{t}^{\frac{3}{2}}}{\ell}} \]
   8. Step-by-step derivation

      1. lower-/.f64 35.5

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{k}}}{\frac{{t}^{1.5}}{\ell}}}{\sin k \cdot \frac{{t}^{1.5}}{\ell}} \]

   9. Applied rewrites 35.5%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{k}}}{\frac{{t}^{1.5}}{\ell}}}{\sin k \cdot \frac{{t}^{1.5}}{\ell}} \]

   if 1.0500000000000001e-9 < k < 9.5999999999999999e170

   1. Initial program 57.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 79.9

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

   5. Applied rewrites 79.9%

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

   7. Applied rewrites 82.6%

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

   if 9.5999999999999999e170 < k

   1. Initial program 39.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 55.5

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

   5. Applied rewrites 55.5%

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

   7. Applied rewrites 65.7%

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

   9. Applied rewrites 90.6%

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

Alternative 8: 83.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.6 \cdot 10^{-57}:\\ \;\;\;\;\frac{\left(\left(\frac{\ell}{{\sin k}^{2}} \cdot \cos k\right) \cdot \frac{\ell}{k}\right) \cdot 2}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m \cdot t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 1\right) + 1\right)}\\ \end{array} \end{array} \]

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 6.6e-57)
    (/ (* (* (* (/ l (pow (sin k) 2.0)) (cos k)) (/ l k)) 2.0) (* t_m k))
    (/
     2.0
     (*
      (* (* (* (/ (* t_m t_m) l) (/ t_m l)) (sin k)) (tan k))
      (+ (fma k (/ k (* t_m t_m)) 1.0) 1.0))))))

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 <= 6.6e-57) {
		tmp = ((((l / pow(sin(k), 2.0)) * cos(k)) * (l / k)) * 2.0) / (t_m * k);
	} else {
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * tan(k)) * (fma(k, (k / (t_m * t_m)), 1.0) + 1.0));
	}
	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 <= 6.6e-57)
		tmp = Float64(Float64(Float64(Float64(Float64(l / (sin(k) ^ 2.0)) * cos(k)) * Float64(l / k)) * 2.0) / Float64(t_m * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l) * Float64(t_m / l)) * sin(k)) * tan(k)) * Float64(fma(k, Float64(k / Float64(t_m * t_m)), 1.0) + 1.0)));
	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, 6.6e-57], N[(N[(N[(N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.5999999999999997e-57

   1. Initial program 51.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 70.3%

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

   9. Applied rewrites 75.1%

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

   if 6.5999999999999997e-57 < t

   1. Initial program 72.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 78.1

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

   4. Applied rewrites 78.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 78.1

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

   6. Applied rewrites 78.1%

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

Alternative 9: 82.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.1 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\left(\cos k \cdot \ell\right) \cdot \frac{\frac{\ell}{{\sin k}^{2}}}{t\_m \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m \cdot t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 1\right) + 1\right)}\\ \end{array} \end{array} \]

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 6.1e-57)
    (* (/ 2.0 k) (* (* (cos k) l) (/ (/ l (pow (sin k) 2.0)) (* t_m k))))
    (/
     2.0
     (*
      (* (* (* (/ (* t_m t_m) l) (/ t_m l)) (sin k)) (tan k))
      (+ (fma k (/ k (* t_m t_m)) 1.0) 1.0))))))

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 <= 6.1e-57) {
		tmp = (2.0 / k) * ((cos(k) * l) * ((l / pow(sin(k), 2.0)) / (t_m * k)));
	} else {
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * tan(k)) * (fma(k, (k / (t_m * t_m)), 1.0) + 1.0));
	}
	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 <= 6.1e-57)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(cos(k) * l) * Float64(Float64(l / (sin(k) ^ 2.0)) / Float64(t_m * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l) * Float64(t_m / l)) * sin(k)) * tan(k)) * Float64(fma(k, Float64(k / Float64(t_m * t_m)), 1.0) + 1.0)));
	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, 6.1e-57], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.0999999999999998e-57

   1. Initial program 51.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 70.3%

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

   9. Applied rewrites 73.8%

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

   if 6.0999999999999998e-57 < t

   1. Initial program 72.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 78.1

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

   4. Applied rewrites 78.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 78.1

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

   6. Applied rewrites 78.1%

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

Alternative 10: 79.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.75 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)} \cdot \left(\cos k \cdot \ell\right)}{k \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m \cdot t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 1\right) + 1\right)}\\ \end{array} \end{array} \]

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 1.75e-57)
    (*
     (/ 2.0 k)
     (/ (* (/ l (- 0.5 (* 0.5 (cos (* 2.0 k))))) (* (cos k) l)) (* k t_m)))
    (/
     2.0
     (*
      (* (* (* (/ (* t_m t_m) l) (/ t_m l)) (sin k)) (tan k))
      (+ (fma k (/ k (* t_m t_m)) 1.0) 1.0))))))

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 <= 1.75e-57) {
		tmp = (2.0 / k) * (((l / (0.5 - (0.5 * cos((2.0 * k))))) * (cos(k) * l)) / (k * t_m));
	} else {
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * tan(k)) * (fma(k, (k / (t_m * t_m)), 1.0) + 1.0));
	}
	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 <= 1.75e-57)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(Float64(l / Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k))))) * Float64(cos(k) * l)) / Float64(k * t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l) * Float64(t_m / l)) * sin(k)) * tan(k)) * Float64(fma(k, Float64(k / Float64(t_m * t_m)), 1.0) + 1.0)));
	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, 1.75e-57], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l / N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.74999999999999996e-57

   1. Initial program 51.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 64.7

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

   5. Applied rewrites 64.7%

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

   7. Applied rewrites 70.3%

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

   9. Applied rewrites 67.3%

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

   if 1.74999999999999996e-57 < t

   1. Initial program 72.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 78.1

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

   4. Applied rewrites 78.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 78.1

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

   6. Applied rewrites 78.1%

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

Alternative 11: 71.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)} \cdot \left(\cos k \cdot \ell\right)}{k \cdot t\_m}\\ \end{array} \end{array} \]

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 1.45e-5)
    (* (/ (/ l (* t_m t_m)) k) (/ l (* t_m k)))
    (*
     (/ 2.0 k)
     (/ (* (/ l (- 0.5 (* 0.5 (cos (* 2.0 k))))) (* (cos k) l)) (* 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 <= 1.45e-5) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	} else {
		tmp = (2.0 / k) * (((l / (0.5 - (0.5 * cos((2.0 * k))))) * (cos(k) * l)) / (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 <= 1.45d-5) then
        tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k))
    else
        tmp = (2.0d0 / k) * (((l / (0.5d0 - (0.5d0 * cos((2.0d0 * k))))) * (cos(k) * l)) / (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 <= 1.45e-5) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	} else {
		tmp = (2.0 / k) * (((l / (0.5 - (0.5 * Math.cos((2.0 * k))))) * (Math.cos(k) * l)) / (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 <= 1.45e-5:
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k))
	else:
		tmp = (2.0 / k) * (((l / (0.5 - (0.5 * math.cos((2.0 * k))))) * (math.cos(k) * l)) / (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 <= 1.45e-5)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(t_m * k)));
	else
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(Float64(l / Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k))))) * Float64(cos(k) * l)) / 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 <= 1.45e-5)
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	else
		tmp = (2.0 / k) * (((l / (0.5 - (0.5 * cos((2.0 * k))))) * (cos(k) * l)) / (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, 1.45e-5], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l / N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.45e-5

   1. Initial program 60.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 57.9

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

   5. Applied rewrites 57.9%

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

   7. Applied rewrites 57.9%

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

   9. Applied rewrites 57.2%

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

   11. Applied rewrites 67.8%

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

   if 1.45e-5 < k

   1. Initial program 49.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 69.1

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

   5. Applied rewrites 69.1%

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

   7. Applied rewrites 73.6%

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

   9. Applied rewrites 73.7%

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

Alternative 12: 69.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot t\_m} \cdot \frac{\left(\cos k \cdot \ell\right) \cdot \ell}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

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 (<= k 1.45e-5)
    (* (/ (/ l (* t_m t_m)) k) (/ l (* t_m k)))
    (*
     (/ 2.0 (* (* k k) t_m))
     (/ (* (* (cos k) l) l) (- 0.5 (* 0.5 (cos (* 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 (k <= 1.45e-5) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	} else {
		tmp = (2.0 / ((k * k) * t_m)) * (((cos(k) * l) * l) / (0.5 - (0.5 * cos((2.0 * 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 <= 1.45d-5) then
        tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k))
    else
        tmp = (2.0d0 / ((k * k) * t_m)) * (((cos(k) * l) * l) / (0.5d0 - (0.5d0 * cos((2.0d0 * 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 <= 1.45e-5) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	} else {
		tmp = (2.0 / ((k * k) * t_m)) * (((Math.cos(k) * l) * l) / (0.5 - (0.5 * Math.cos((2.0 * 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 <= 1.45e-5:
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k))
	else:
		tmp = (2.0 / ((k * k) * t_m)) * (((math.cos(k) * l) * l) / (0.5 - (0.5 * math.cos((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 (k <= 1.45e-5)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(t_m * k)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * t_m)) * Float64(Float64(Float64(cos(k) * l) * l) / Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * 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 <= 1.45e-5)
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	else
		tmp = (2.0 / ((k * k) * t_m)) * (((cos(k) * l) * l) / (0.5 - (0.5 * cos((2.0 * 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, 1.45e-5], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] / N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.45e-5

   1. Initial program 60.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 57.9

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

   5. Applied rewrites 57.9%

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

   7. Applied rewrites 57.9%

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

   9. Applied rewrites 57.2%

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

   11. Applied rewrites 67.8%

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

   if 1.45e-5 < k

   1. Initial program 49.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 69.1

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

   5. Applied rewrites 69.1%

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

   7. Applied rewrites 69.1%

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

Alternative 13: 70.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.3 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\ell}{{k}^{3}} \cdot \frac{\ell}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{t\_m \cdot k}\\ \end{array} \end{array} \]

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 5.3e-58)
    (* (/ 2.0 k) (* (/ l (pow k 3.0)) (/ l t_m)))
    (* (/ (/ l (* t_m t_m)) k) (/ l (* 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 <= 5.3e-58) {
		tmp = (2.0 / k) * ((l / pow(k, 3.0)) * (l / t_m));
	} else {
		tmp = ((l / (t_m * t_m)) / k) * (l / (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 <= 5.3d-58) then
        tmp = (2.0d0 / k) * ((l / (k ** 3.0d0)) * (l / t_m))
    else
        tmp = ((l / (t_m * t_m)) / k) * (l / (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 <= 5.3e-58) {
		tmp = (2.0 / k) * ((l / Math.pow(k, 3.0)) * (l / t_m));
	} else {
		tmp = ((l / (t_m * t_m)) / k) * (l / (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 <= 5.3e-58:
		tmp = (2.0 / k) * ((l / math.pow(k, 3.0)) * (l / t_m))
	else:
		tmp = ((l / (t_m * t_m)) / k) * (l / (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 <= 5.3e-58)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(l / (k ^ 3.0)) * Float64(l / t_m)));
	else
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(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 <= 5.3e-58)
		tmp = (2.0 / k) * ((l / (k ^ 3.0)) * (l / t_m));
	else
		tmp = ((l / (t_m * t_m)) / k) * (l / (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, 5.3e-58], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(l / N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.3000000000000003e-58

   1. Initial program 51.5%

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

   3. Taylor expanded in 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 65.0

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

   5. Applied rewrites 65.0%

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

   7. Applied rewrites 70.7%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 59.3%

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

   if 5.3000000000000003e-58 < t

   1. Initial program 71.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 62.6

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

   5. Applied rewrites 62.6%

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

   7. Applied rewrites 62.6%

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

   9. Applied rewrites 58.8%

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

   11. Applied rewrites 69.5%

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

Alternative 14: 65.7% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \ell}{t\_m \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)}\\ \end{array} \end{array} \]

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 7.5e+181)
    (* (/ (/ l (* t_m t_m)) k) (/ l (* t_m k)))
    (/ (* (/ l t_m) l) (* t_m (* (* 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 <= 7.5e+181) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	} else {
		tmp = ((l / t_m) * l) / (t_m * ((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 <= 7.5d+181) then
        tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k))
    else
        tmp = ((l / t_m) * l) / (t_m * ((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 <= 7.5e+181) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	} else {
		tmp = ((l / t_m) * l) / (t_m * ((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 <= 7.5e+181:
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k))
	else:
		tmp = ((l / t_m) * l) / (t_m * ((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 <= 7.5e+181)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(t_m * k)));
	else
		tmp = Float64(Float64(Float64(l / t_m) * l) / Float64(t_m * Float64(Float64(k * 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 <= 7.5e+181)
		tmp = ((l / (t_m * t_m)) / k) * (l / (t_m * k));
	else
		tmp = ((l / t_m) * l) / (t_m * ((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, 7.5e+181], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(t$95$m * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.5000000000000005e181

   1. Initial program 59.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 57.4

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

   5. Applied rewrites 57.4%

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

   7. Applied rewrites 57.4%

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

   9. Applied rewrites 56.4%

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

   11. Applied rewrites 65.1%

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

   if 7.5000000000000005e181 < k

   1. Initial program 41.7%

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

   3. 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 42.8

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

   5. Applied rewrites 42.8%

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

   7. Applied rewrites 42.8%

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

   9. Applied rewrites 49.2%

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

   11. Applied rewrites 59.6%

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

Alternative 15: 65.4% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{+181}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{t\_m \cdot t\_m}}{t\_m \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \ell}{t\_m \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)}\\ \end{array} \end{array} \]

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 7.5e+181)
    (* (/ l k) (/ (/ l (* t_m t_m)) (* t_m k)))
    (/ (* (/ l t_m) l) (* t_m (* (* 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 <= 7.5e+181) {
		tmp = (l / k) * ((l / (t_m * t_m)) / (t_m * k));
	} else {
		tmp = ((l / t_m) * l) / (t_m * ((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 <= 7.5d+181) then
        tmp = (l / k) * ((l / (t_m * t_m)) / (t_m * k))
    else
        tmp = ((l / t_m) * l) / (t_m * ((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 <= 7.5e+181) {
		tmp = (l / k) * ((l / (t_m * t_m)) / (t_m * k));
	} else {
		tmp = ((l / t_m) * l) / (t_m * ((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 <= 7.5e+181:
		tmp = (l / k) * ((l / (t_m * t_m)) / (t_m * k))
	else:
		tmp = ((l / t_m) * l) / (t_m * ((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 <= 7.5e+181)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / Float64(t_m * t_m)) / Float64(t_m * k)));
	else
		tmp = Float64(Float64(Float64(l / t_m) * l) / Float64(t_m * Float64(Float64(k * 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 <= 7.5e+181)
		tmp = (l / k) * ((l / (t_m * t_m)) / (t_m * k));
	else
		tmp = ((l / t_m) * l) / (t_m * ((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, 7.5e+181], N[(N[(l / k), $MachinePrecision] * N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(t$95$m * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.5000000000000005e181

   1. Initial program 59.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 57.4

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

   5. Applied rewrites 57.4%

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

   7. Applied rewrites 57.4%

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

   9. Applied rewrites 56.4%

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

   11. Applied rewrites 65.0%

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

   if 7.5000000000000005e181 < k

   1. Initial program 41.7%

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

   3. 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 42.8

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

   5. Applied rewrites 42.8%

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

   7. Applied rewrites 42.8%

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

   9. Applied rewrites 49.2%

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

   11. Applied rewrites 59.6%

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

Alternative 16: 64.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.4 \cdot 10^{-166}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \ell}{\left(k \cdot \left(t\_m \cdot t\_m\right)\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \ell}{t\_m \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)}\\ \end{array} \end{array} \]

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 7.4e-166)
    (/ (* (/ l k) l) (* (* k (* t_m t_m)) t_m))
    (/ (* (/ l t_m) l) (* t_m (* (* 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 <= 7.4e-166) {
		tmp = ((l / k) * l) / ((k * (t_m * t_m)) * t_m);
	} else {
		tmp = ((l / t_m) * l) / (t_m * ((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 <= 7.4d-166) then
        tmp = ((l / k) * l) / ((k * (t_m * t_m)) * t_m)
    else
        tmp = ((l / t_m) * l) / (t_m * ((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 <= 7.4e-166) {
		tmp = ((l / k) * l) / ((k * (t_m * t_m)) * t_m);
	} else {
		tmp = ((l / t_m) * l) / (t_m * ((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 <= 7.4e-166:
		tmp = ((l / k) * l) / ((k * (t_m * t_m)) * t_m)
	else:
		tmp = ((l / t_m) * l) / (t_m * ((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 <= 7.4e-166)
		tmp = Float64(Float64(Float64(l / k) * l) / Float64(Float64(k * Float64(t_m * t_m)) * t_m));
	else
		tmp = Float64(Float64(Float64(l / t_m) * l) / Float64(t_m * Float64(Float64(k * 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 <= 7.4e-166)
		tmp = ((l / k) * l) / ((k * (t_m * t_m)) * t_m);
	else
		tmp = ((l / t_m) * l) / (t_m * ((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, 7.4e-166], N[(N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(t$95$m * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.4000000000000005e-166

   1. Initial program 60.8%

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

   3. 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 55.2

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

   5. Applied rewrites 55.2%

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

   7. Applied rewrites 62.0%

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

   9. Applied rewrites 63.9%

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

   if 7.4000000000000005e-166 < k

   1. Initial program 51.7%

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

   3. 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 56.6

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

   5. Applied rewrites 56.6%

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

   7. Applied rewrites 56.6%

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

   9. Applied rewrites 55.7%

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

   11. Applied rewrites 59.5%

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

Alternative 17: 62.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8.8 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \ell}{\left(k \cdot \left(t\_m \cdot t\_m\right)\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{t\_m \cdot t\_m}}{\left(k \cdot k\right) \cdot t\_m}\\ \end{array} \end{array} \]

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 8.8e-136)
    (/ (* (/ l k) l) (* (* k (* t_m t_m)) t_m))
    (* l (/ (/ l (* t_m t_m)) (* (* 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 <= 8.8e-136) {
		tmp = ((l / k) * l) / ((k * (t_m * t_m)) * t_m);
	} else {
		tmp = l * ((l / (t_m * t_m)) / ((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 <= 8.8d-136) then
        tmp = ((l / k) * l) / ((k * (t_m * t_m)) * t_m)
    else
        tmp = l * ((l / (t_m * t_m)) / ((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 <= 8.8e-136) {
		tmp = ((l / k) * l) / ((k * (t_m * t_m)) * t_m);
	} else {
		tmp = l * ((l / (t_m * t_m)) / ((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 <= 8.8e-136:
		tmp = ((l / k) * l) / ((k * (t_m * t_m)) * t_m)
	else:
		tmp = l * ((l / (t_m * t_m)) / ((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 <= 8.8e-136)
		tmp = Float64(Float64(Float64(l / k) * l) / Float64(Float64(k * Float64(t_m * t_m)) * t_m));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(t_m * t_m)) / Float64(Float64(k * 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 <= 8.8e-136)
		tmp = ((l / k) * l) / ((k * (t_m * t_m)) * t_m);
	else
		tmp = l * ((l / (t_m * t_m)) / ((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, 8.8e-136], N[(N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.8000000000000005e-136

   1. Initial program 61.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. 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 56.4

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

   5. Applied rewrites 56.4%

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

   7. Applied rewrites 63.0%

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

   9. Applied rewrites 64.8%

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

   if 8.8000000000000005e-136 < k

   1. Initial program 50.7%

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

   3. 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 54.8

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

   5. Applied rewrites 54.8%

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

   7. Applied rewrites 54.8%

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

   9. Applied rewrites 54.9%

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

   11. Applied rewrites 57.0%

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

Alternative 18: 59.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{\frac{\ell}{t\_m \cdot t\_m}}{\left(k \cdot k\right) \cdot t\_m}\right) \end{array} \]

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 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) {
	return t_s * (l * ((l / (t_m * t_m)) / ((k * k) * t_m)));
}

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 * k) * t_m)))
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 * k) * t_m)));
}

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 * k) * t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(Float64(l / Float64(t_m * t_m)) / Float64(Float64(k * k) * t_m))))
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 * k) * t_m)));
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[(l * N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.2%

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

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

5. Applied rewrites 55.8%

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

7. Applied rewrites 55.8%

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

9. Applied rewrites 55.6%

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

11. Applied rewrites 57.4%

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

Alternative 19: 54.1% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\ell \cdot \ell}{\left(t\_m \cdot t\_m\right) \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)} \end{array} \]

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 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) {
	return t_s * ((l * l) / ((t_m * t_m) * ((k * k) * t_m)));
}

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 * k) * t_m)))
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 * k) * t_m)));
}

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 * k) * t_m)))

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 * k) * t_m))))
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 * k) * t_m)));
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 * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.2%

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

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

5. Applied rewrites 55.8%

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

7. Applied rewrites 55.8%

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

9. Applied rewrites 55.6%

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

11. Applied rewrites 52.4%

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

Reproduce

herbie shell --seed 2025017 
(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))))