Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.9% → 82.4%

Time: 7.6s

Alternatives: 14

Speedup: 4.7×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.9% 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: 82.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-181}:\\ \;\;\;\;\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_m \cdot \frac{t\_m}{\ell}\right) \cdot \left(\frac{t\_m}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

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 3.5e-181)
    (/
     (* (* (* (cos k) l) l) 2.0)
     (* (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t_m) k) 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 <= 3.5e-181) {
		tmp = (((cos(k) * l) * l) * 2.0) / ((((0.5 - (cos((k + k)) * 0.5)) * t_m) * k) * 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 <= 3.5d-181) then
        tmp = (((cos(k) * l) * l) * 2.0d0) / ((((0.5d0 - (cos((k + k)) * 0.5d0)) * t_m) * k) * 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 <= 3.5e-181) {
		tmp = (((Math.cos(k) * l) * l) * 2.0) / ((((0.5 - (Math.cos((k + k)) * 0.5)) * t_m) * k) * 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 <= 3.5e-181:
		tmp = (((math.cos(k) * l) * l) * 2.0) / ((((0.5 - (math.cos((k + k)) * 0.5)) * t_m) * k) * 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 <= 3.5e-181)
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * l) * 2.0) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m) * k) * k));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * 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 <= 3.5e-181)
		tmp = (((cos(k) * l) * l) * 2.0) / ((((0.5 - (cos((k + k)) * 0.5)) * t_m) * k) * 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, 3.5e-181], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $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 < 3.49999999999999996e-181

   1. Initial program 27.5%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 69.3%

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

   6. Applied rewrites 78.2%

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

   if 3.49999999999999996e-181 < t

   1. Initial program 61.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 39.5

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

   3. Applied rewrites 39.5%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. exp-diff N/A

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

      8. pow-to-exp N/A

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

      9. pow-to-exp N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 73.9

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

   5. Applied rewrites 73.9%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   6. 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. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-sin.f64 83.4

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

   7. Applied rewrites 83.4%

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

Alternative 2: 68.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.4 \cdot 10^{-150}:\\ \;\;\;\;\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;t\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m \cdot t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log \ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9.4e-150)
    (/
     (* (* (* (cos k) l) l) 2.0)
     (* (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t_m) k) k))
    (if (<= t_m 1.35e+154)
      (/
       2.0
       (*
        (* (* (* (/ (* t_m t_m) l) (/ t_m l)) (sin k)) (tan k))
        (fma (/ k t_m) (/ k t_m) 2.0)))
      (/
       2.0
       (*
        (* (* (exp (fma (log t_m) 3.0 (* -2.0 (log l)))) (sin k)) (tan k))
        2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9.4e-150) {
		tmp = (((cos(k) * l) * l) * 2.0) / ((((0.5 - (cos((k + k)) * 0.5)) * t_m) * k) * k);
	} else if (t_m <= 1.35e+154) {
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * tan(k)) * fma((k / t_m), (k / t_m), 2.0));
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l)))) * sin(k)) * tan(k)) * 2.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 9.4e-150)
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * l) * 2.0) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m) * k) * k));
	elseif (t_m <= 1.35e+154)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l) * Float64(t_m / l)) * sin(k)) * tan(k)) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l)))) * sin(k)) * tan(k)) * 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9.4e-150], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.35e+154], 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 / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 9.3999999999999998e-150

   1. Initial program 28.3%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 69.3%

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

   6. Applied rewrites 77.5%

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

   if 9.3999999999999998e-150 < t < 1.35000000000000003e154

   1. Initial program 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 38.7

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

   3. Applied rewrites 38.7%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. exp-diff N/A

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

      8. pow-to-exp N/A

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

      9. pow-to-exp N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 78.0

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

   5. Applied rewrites 78.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 78.0

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

   7. Applied rewrites 78.0%

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

   if 1.35000000000000003e154 < t

   1. Initial program 65.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 42.7

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

   3. Applied rewrites 42.7%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. *-commutative N/A

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

      7. lower--.f64 N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 42.7

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

   5. Applied rewrites 42.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 41.2%

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

Alternative 3: 67.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.4 \cdot 10^{-150}:\\ \;\;\;\;\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m\right) \cdot k\right) \cdot k}\\ \mathbf{elif}\;t\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m \cdot t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log \ell\right)} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

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 9.4e-150)
    (/
     (* (* (* (cos k) l) l) 2.0)
     (* (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t_m) k) k))
    (if (<= t_m 1.35e+154)
      (/
       2.0
       (*
        (* (* (* (/ (* t_m t_m) l) (/ t_m l)) (sin k)) (tan k))
        (fma (/ k t_m) (/ k t_m) 2.0)))
      (/
       2.0
       (*
        (* (* (exp (fma (log t_m) 3.0 (* -2.0 (log l)))) 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 <= 9.4e-150) {
		tmp = (((cos(k) * l) * l) * 2.0) / ((((0.5 - (cos((k + k)) * 0.5)) * t_m) * k) * k);
	} else if (t_m <= 1.35e+154) {
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * tan(k)) * fma((k / t_m), (k / t_m), 2.0));
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l)))) * k) * tan(k)) * ((1.0 + 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 <= 9.4e-150)
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * l) * 2.0) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m) * k) * k));
	elseif (t_m <= 1.35e+154)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l) * Float64(t_m / l)) * sin(k)) * tan(k)) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l)))) * k) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 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, 9.4e-150], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.35e+154], 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 / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * k), $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 3 regimes

2. if t < 9.3999999999999998e-150

   1. Initial program 28.3%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 69.3%

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

   6. Applied rewrites 77.5%

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

   if 9.3999999999999998e-150 < t < 1.35000000000000003e154

   1. Initial program 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 38.7

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

   3. Applied rewrites 38.7%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. exp-diff N/A

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

      8. pow-to-exp N/A

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

      9. pow-to-exp N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 78.0

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

   5. Applied rewrites 78.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 78.0

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

   7. Applied rewrites 78.0%

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

   if 1.35000000000000003e154 < t

   1. Initial program 65.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 42.7

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

   3. Applied rewrites 42.7%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. *-commutative N/A

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

      7. lower--.f64 N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 42.7

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

   5. Applied rewrites 42.7%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 39.2%

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

Alternative 4: 67.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.75 \cdot 10^{-112}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log \ell\right)} \cdot k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(t\_m \cdot t\_m\right)\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m\right) \cdot k\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.75e-112)
    (/
     2.0
     (*
      (* (* (exp (fma (log t_m) 3.0 (* -2.0 (log l)))) k) (tan k))
      (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
    (if (<= k 0.075)
      (* l (/ l (* (* (* k k) (* t_m t_m)) t_m)))
      (/
       (* (* (* (cos k) l) l) 2.0)
       (* (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t_m) k) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.75e-112) {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l)))) * k) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else if (k <= 0.075) {
		tmp = l * (l / (((k * k) * (t_m * t_m)) * t_m));
	} else {
		tmp = (((cos(k) * l) * l) * 2.0) / ((((0.5 - (cos((k + k)) * 0.5)) * t_m) * k) * k);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.75e-112)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(fma(log(t_m), 3.0, Float64(-2.0 * log(l)))) * k) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	elseif (k <= 0.075)
		tmp = Float64(l * Float64(l / Float64(Float64(Float64(k * k) * Float64(t_m * t_m)) * t_m)));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * l) * 2.0) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m) * k) * k));
	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[k, 3.75e-112], N[(2.0 / N[(N[(N[(N[Exp[N[(N[Log[t$95$m], $MachinePrecision] * 3.0 + N[(-2.0 * N[Log[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * k), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.075], N[(l * N[(l / N[(N[(N[(k * k), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.7500000000000001e-112

   1. Initial program 57.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 37.0

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

   3. Applied rewrites 37.0%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. *-commutative N/A

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

      7. lower--.f64 N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 37.0

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

   5. Applied rewrites 37.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 34.6%

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

   if 3.7500000000000001e-112 < k < 0.0749999999999999972

   1. Initial program 59.9%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 67.2

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

   4. Applied rewrites 67.2%

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

   6. Applied rewrites 70.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow3 N/A

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

      7. unpow3 N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 73.1

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

   8. Applied rewrites 73.1%

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

   if 0.0749999999999999972 < k

   1. Initial program 47.8%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.1%

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

   6. Applied rewrites 73.1%

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

Alternative 5: 66.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.75 \cdot 10^{-112}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(t\_m \cdot t\_m\right)\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m\right) \cdot k\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.75e-112)
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l) 2.0))) (sin k)) k)
      (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
    (if (<= k 0.075)
      (* l (/ l (* (* (* k k) (* t_m t_m)) t_m)))
      (/
       (* (* (* (cos k) l) l) 2.0)
       (* (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t_m) k) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.75e-112) {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l) * 2.0))) * sin(k)) * k) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else if (k <= 0.075) {
		tmp = l * (l / (((k * k) * (t_m * t_m)) * t_m));
	} else {
		tmp = (((cos(k) * l) * l) * 2.0) / ((((0.5 - (cos((k + k)) * 0.5)) * t_m) * k) * k);
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.75d-112) then
        tmp = 2.0d0 / (((exp(((log(t_m) * 3.0d0) - (log(l) * 2.0d0))) * sin(k)) * k) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    else if (k <= 0.075d0) then
        tmp = l * (l / (((k * k) * (t_m * t_m)) * t_m))
    else
        tmp = (((cos(k) * l) * l) * 2.0d0) / ((((0.5d0 - (cos((k + k)) * 0.5d0)) * t_m) * k) * k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.75e-112) {
		tmp = 2.0 / (((Math.exp(((Math.log(t_m) * 3.0) - (Math.log(l) * 2.0))) * Math.sin(k)) * k) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	} else if (k <= 0.075) {
		tmp = l * (l / (((k * k) * (t_m * t_m)) * t_m));
	} else {
		tmp = (((Math.cos(k) * l) * l) * 2.0) / ((((0.5 - (Math.cos((k + k)) * 0.5)) * t_m) * k) * k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3.75e-112:
		tmp = 2.0 / (((math.exp(((math.log(t_m) * 3.0) - (math.log(l) * 2.0))) * math.sin(k)) * k) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	elif k <= 0.075:
		tmp = l * (l / (((k * k) * (t_m * t_m)) * t_m))
	else:
		tmp = (((math.cos(k) * l) * l) * 2.0) / ((((0.5 - (math.cos((k + k)) * 0.5)) * t_m) * k) * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.75e-112)
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l) * 2.0))) * sin(k)) * k) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	elseif (k <= 0.075)
		tmp = Float64(l * Float64(l / Float64(Float64(Float64(k * k) * Float64(t_m * t_m)) * t_m)));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * l) * 2.0) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m) * k) * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3.75e-112)
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l) * 2.0))) * sin(k)) * k) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	elseif (k <= 0.075)
		tmp = l * (l / (((k * k) * (t_m * t_m)) * t_m));
	else
		tmp = (((cos(k) * l) * l) * 2.0) / ((((0.5 - (cos((k + k)) * 0.5)) * t_m) * k) * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.75e-112], N[(2.0 / N[(N[(N[(N[Exp[N[(N[(N[Log[t$95$m], $MachinePrecision] * 3.0), $MachinePrecision] - N[(N[Log[l], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.075], N[(l * N[(l / N[(N[(N[(k * k), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.7500000000000001e-112

   1. Initial program 57.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 37.0

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

   3. Applied rewrites 37.0%

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

   4. Taylor expanded in k around 0

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

   6. Applied rewrites 34.6%

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

   if 3.7500000000000001e-112 < k < 0.0749999999999999972

   1. Initial program 59.9%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 67.2

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

   4. Applied rewrites 67.2%

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

   6. Applied rewrites 70.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow3 N/A

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

      7. unpow3 N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 73.1

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

   8. Applied rewrites 73.1%

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

   if 0.0749999999999999972 < k

   1. Initial program 47.8%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.1%

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

   6. Applied rewrites 73.1%

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

Alternative 6: 63.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m \cdot t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m\right) \cdot k\right) \cdot k}\\ \end{array} \end{array} \]

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 2.5e+14)
    (/ 2.0 (* (* (* (* (/ (* t_m t_m) l) (/ t_m l)) (sin k)) (tan k)) 2.0))
    (/
     (* (* (* (cos k) l) l) 2.0)
     (* (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t_m) k) k)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.5e+14) {
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * tan(k)) * 2.0);
	} else {
		tmp = (((cos(k) * l) * l) * 2.0) / ((((0.5 - (cos((k + k)) * 0.5)) * t_m) * k) * 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 <= 2.5d+14) then
        tmp = 2.0d0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * tan(k)) * 2.0d0)
    else
        tmp = (((cos(k) * l) * l) * 2.0d0) / ((((0.5d0 - (cos((k + k)) * 0.5d0)) * t_m) * k) * 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 <= 2.5e+14) {
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * Math.sin(k)) * Math.tan(k)) * 2.0);
	} else {
		tmp = (((Math.cos(k) * l) * l) * 2.0) / ((((0.5 - (Math.cos((k + k)) * 0.5)) * t_m) * k) * 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 <= 2.5e+14:
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * math.sin(k)) * math.tan(k)) * 2.0)
	else:
		tmp = (((math.cos(k) * l) * l) * 2.0) / ((((0.5 - (math.cos((k + k)) * 0.5)) * t_m) * k) * 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 <= 2.5e+14)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l) * Float64(t_m / l)) * sin(k)) * tan(k)) * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * l) * 2.0) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m) * k) * k));
	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 <= 2.5e+14)
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * tan(k)) * 2.0);
	else
		tmp = (((cos(k) * l) * l) * 2.0) / ((((0.5 - (cos((k + k)) * 0.5)) * t_m) * k) * 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, 2.5e+14], 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] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.5e14

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 36.8

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

   3. Applied rewrites 36.8%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. exp-diff N/A

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

      8. pow-to-exp N/A

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

      9. pow-to-exp N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 65.7%

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

   if 2.5e14 < k

   1. Initial program 48.1%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 67.9%

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

   6. Applied rewrites 73.2%

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

Alternative 7: 63.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m \cdot t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos k \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(0.5 - \cos \left(k + k\right) \cdot 0.5\right) \cdot t\_m\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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 2.5e+14)
    (/ 2.0 (* (* (* (* (/ (* t_m t_m) l) (/ t_m l)) (sin k)) (tan k)) 2.0))
    (/
     (* (* (* (cos k) l) l) 2.0)
     (* (* (- 0.5 (* (cos (+ k k)) 0.5)) t_m) (* k k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.5e+14) {
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * tan(k)) * 2.0);
	} else {
		tmp = (((cos(k) * l) * l) * 2.0) / (((0.5 - (cos((k + k)) * 0.5)) * t_m) * (k * 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 <= 2.5d+14) then
        tmp = 2.0d0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * tan(k)) * 2.0d0)
    else
        tmp = (((cos(k) * l) * l) * 2.0d0) / (((0.5d0 - (cos((k + k)) * 0.5d0)) * t_m) * (k * 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 <= 2.5e+14) {
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * Math.sin(k)) * Math.tan(k)) * 2.0);
	} else {
		tmp = (((Math.cos(k) * l) * l) * 2.0) / (((0.5 - (Math.cos((k + k)) * 0.5)) * t_m) * (k * 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 <= 2.5e+14:
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * math.sin(k)) * math.tan(k)) * 2.0)
	else:
		tmp = (((math.cos(k) * l) * l) * 2.0) / (((0.5 - (math.cos((k + k)) * 0.5)) * t_m) * (k * 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 <= 2.5e+14)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l) * Float64(t_m / l)) * sin(k)) * tan(k)) * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * l) * 2.0) / Float64(Float64(Float64(0.5 - Float64(cos(Float64(k + k)) * 0.5)) * t_m) * Float64(k * 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 <= 2.5e+14)
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * tan(k)) * 2.0);
	else
		tmp = (((cos(k) * l) * l) * 2.0) / (((0.5 - (cos((k + k)) * 0.5)) * t_m) * (k * 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, 2.5e+14], 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] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.5e14

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 36.8

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

   3. Applied rewrites 36.8%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. exp-diff N/A

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

      8. pow-to-exp N/A

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

      9. pow-to-exp N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 68.4

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 65.7%

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

   if 2.5e14 < k

   1. Initial program 48.1%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 67.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 67.9

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

   6. Applied rewrites 67.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-cos.f64 67.9

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

   8. Applied rewrites 67.9%

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

Alternative 8: 62.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m \cdot t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\_m\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3e+105)
    (/ 2.0 (* (* (* (* (/ (* t_m t_m) l) (/ t_m l)) (sin k)) (tan k)) 2.0))
    (/ (* -0.3333333333333333 (* l l)) (* k (* k t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3e+105) {
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * tan(k)) * 2.0);
	} else {
		tmp = (-0.3333333333333333 * (l * l)) / (k * (k * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3d+105) then
        tmp = 2.0d0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * tan(k)) * 2.0d0)
    else
        tmp = ((-0.3333333333333333d0) * (l * l)) / (k * (k * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3e+105) {
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * Math.sin(k)) * Math.tan(k)) * 2.0);
	} else {
		tmp = (-0.3333333333333333 * (l * l)) / (k * (k * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3e+105:
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * math.sin(k)) * math.tan(k)) * 2.0)
	else:
		tmp = (-0.3333333333333333 * (l * l)) / (k * (k * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3e+105)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l) * Float64(t_m / l)) * sin(k)) * tan(k)) * 2.0));
	else
		tmp = Float64(Float64(-0.3333333333333333 * Float64(l * l)) / Float64(k * Float64(k * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3e+105)
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * tan(k)) * 2.0);
	else
		tmp = (-0.3333333333333333 * (l * l)) / (k * (k * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3e+105], 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] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.0000000000000001e105

   1. Initial program 56.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 36.7

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

   3. Applied rewrites 36.7%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. exp-diff N/A

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

      8. pow-to-exp N/A

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

      9. pow-to-exp N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 64.7%

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

   if 3.0000000000000001e105 < k

   1. Initial program 47.1%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 64.3%

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

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{\color{blue}{4}}} \]

   7. Applied rewrites 12.7%

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

   8. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

      8. lift-*.f64 59.0

         \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

   10. Applied rewrites 59.0%

       \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]

       5. lower-*.f64 60.0

          \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]

   12. Applied rewrites 60.0%

       \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 61.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m \cdot t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \sin k\right) \cdot k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;k \leq 240000000000:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(t\_m \cdot t\_m\right)\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\_m\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.25e-94)
    (/
     2.0
     (*
      (* (* (* (/ (* t_m t_m) l) (/ t_m l)) (sin k)) k)
      (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
    (if (<= k 240000000000.0)
      (* l (/ l (* (* (* k k) (* t_m t_m)) t_m)))
      (/ (* -0.3333333333333333 (* l l)) (* k (* k t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.25e-94) {
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * k) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else if (k <= 240000000000.0) {
		tmp = l * (l / (((k * k) * (t_m * t_m)) * t_m));
	} else {
		tmp = (-0.3333333333333333 * (l * l)) / (k * (k * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.25d-94) then
        tmp = 2.0d0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * k) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    else if (k <= 240000000000.0d0) then
        tmp = l * (l / (((k * k) * (t_m * t_m)) * t_m))
    else
        tmp = ((-0.3333333333333333d0) * (l * l)) / (k * (k * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.25e-94) {
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * Math.sin(k)) * k) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	} else if (k <= 240000000000.0) {
		tmp = l * (l / (((k * k) * (t_m * t_m)) * t_m));
	} else {
		tmp = (-0.3333333333333333 * (l * l)) / (k * (k * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.25e-94:
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * math.sin(k)) * k) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	elif k <= 240000000000.0:
		tmp = l * (l / (((k * k) * (t_m * t_m)) * t_m))
	else:
		tmp = (-0.3333333333333333 * (l * l)) / (k * (k * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.25e-94)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l) * Float64(t_m / l)) * sin(k)) * k) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	elseif (k <= 240000000000.0)
		tmp = Float64(l * Float64(l / Float64(Float64(Float64(k * k) * Float64(t_m * t_m)) * t_m)));
	else
		tmp = Float64(Float64(-0.3333333333333333 * Float64(l * l)) / Float64(k * Float64(k * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.25e-94)
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * k) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	elseif (k <= 240000000000.0)
		tmp = l * (l / (((k * k) * (t_m * t_m)) * t_m));
	else
		tmp = (-0.3333333333333333 * (l * l)) / (k * (k * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.25e-94], 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] * k), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 240000000000.0], N[(l * N[(l / N[(N[(N[(k * k), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.2499999999999999e-94

   1. Initial program 57.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-log.f64 37.0

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

   3. Applied rewrites 37.0%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. exp-diff N/A

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

      8. pow-to-exp N/A

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

      9. pow-to-exp N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 64.7%

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

   if 1.2499999999999999e-94 < k < 2.4e11

   1. Initial program 58.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 65.9

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

   4. Applied rewrites 65.9%

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

   6. Applied rewrites 68.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow3 N/A

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

      7. unpow3 N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 71.0

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

   8. Applied rewrites 71.0%

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

   if 2.4e11 < k

   1. Initial program 48.0%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.1%

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

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{\color{blue}{4}}} \]

   7. Applied rewrites 20.2%

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

   8. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

      8. lift-*.f64 55.1

         \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

   10. Applied rewrites 55.1%

       \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]

       5. lower-*.f64 55.8

          \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]

   12. Applied rewrites 55.8%

       \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 57.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-163}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)\right)}\\ \mathbf{elif}\;k \leq 240000000000:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(\left(k \cdot k\right) \cdot \left(t\_m \cdot t\_m\right)\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\_m\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 5.8e-163)
    (* l (/ l (* k (* k (* (* t_m t_m) t_m)))))
    (if (<= k 240000000000.0)
      (* l (/ l (* (* (* k k) (* t_m t_m)) t_m)))
      (/ (* -0.3333333333333333 (* l l)) (* k (* k t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5.8e-163) {
		tmp = l * (l / (k * (k * ((t_m * t_m) * t_m))));
	} else if (k <= 240000000000.0) {
		tmp = l * (l / (((k * k) * (t_m * t_m)) * t_m));
	} else {
		tmp = (-0.3333333333333333 * (l * l)) / (k * (k * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.8d-163) then
        tmp = l * (l / (k * (k * ((t_m * t_m) * t_m))))
    else if (k <= 240000000000.0d0) then
        tmp = l * (l / (((k * k) * (t_m * t_m)) * t_m))
    else
        tmp = ((-0.3333333333333333d0) * (l * l)) / (k * (k * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5.8e-163) {
		tmp = l * (l / (k * (k * ((t_m * t_m) * t_m))));
	} else if (k <= 240000000000.0) {
		tmp = l * (l / (((k * k) * (t_m * t_m)) * t_m));
	} else {
		tmp = (-0.3333333333333333 * (l * l)) / (k * (k * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 5.8e-163:
		tmp = l * (l / (k * (k * ((t_m * t_m) * t_m))))
	elif k <= 240000000000.0:
		tmp = l * (l / (((k * k) * (t_m * t_m)) * t_m))
	else:
		tmp = (-0.3333333333333333 * (l * l)) / (k * (k * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 5.8e-163)
		tmp = Float64(l * Float64(l / Float64(k * Float64(k * Float64(Float64(t_m * t_m) * t_m)))));
	elseif (k <= 240000000000.0)
		tmp = Float64(l * Float64(l / Float64(Float64(Float64(k * k) * Float64(t_m * t_m)) * t_m)));
	else
		tmp = Float64(Float64(-0.3333333333333333 * Float64(l * l)) / Float64(k * Float64(k * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 5.8e-163)
		tmp = l * (l / (k * (k * ((t_m * t_m) * t_m))));
	elseif (k <= 240000000000.0)
		tmp = l * (l / (((k * k) * (t_m * t_m)) * t_m));
	else
		tmp = (-0.3333333333333333 * (l * l)) / (k * (k * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 5.8e-163], N[(l * N[(l / N[(k * N[(k * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 240000000000.0], N[(l * N[(l / N[(N[(N[(k * k), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 5.8000000000000002e-163

   1. Initial program 56.2%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 50.1

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

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow3 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 62.2

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

   8. Applied rewrites 62.2%

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

   if 5.8000000000000002e-163 < k < 2.4e11

   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. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 66.6

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

   4. Applied rewrites 66.6%

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

   6. Applied rewrites 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow3 N/A

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

      7. unpow3 N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 74.5

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

   8. Applied rewrites 74.5%

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

   if 2.4e11 < k

   1. Initial program 48.0%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.1%

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

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{\color{blue}{4}}} \]

   7. Applied rewrites 20.2%

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

   8. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

      8. lift-*.f64 55.1

         \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

   10. Applied rewrites 55.1%

       \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]

       5. lower-*.f64 55.8

          \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]

   12. Applied rewrites 55.8%

       \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 47.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 240000000000:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left(k \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\_m\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 240000000000.0)
    (* l (/ l (* k (* k (* (* t_m t_m) t_m)))))
    (/ (* -0.3333333333333333 (* l l)) (* k (* k t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 240000000000.0) {
		tmp = l * (l / (k * (k * ((t_m * t_m) * t_m))));
	} else {
		tmp = (-0.3333333333333333 * (l * l)) / (k * (k * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 240000000000.0d0) then
        tmp = l * (l / (k * (k * ((t_m * t_m) * t_m))))
    else
        tmp = ((-0.3333333333333333d0) * (l * l)) / (k * (k * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 240000000000.0) {
		tmp = l * (l / (k * (k * ((t_m * t_m) * t_m))));
	} else {
		tmp = (-0.3333333333333333 * (l * l)) / (k * (k * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 240000000000.0:
		tmp = l * (l / (k * (k * ((t_m * t_m) * t_m))))
	else:
		tmp = (-0.3333333333333333 * (l * l)) / (k * (k * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 240000000000.0)
		tmp = Float64(l * Float64(l / Float64(k * Float64(k * Float64(Float64(t_m * t_m) * t_m)))));
	else
		tmp = Float64(Float64(-0.3333333333333333 * Float64(l * l)) / Float64(k * Float64(k * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 240000000000.0)
		tmp = l * (l / (k * (k * ((t_m * t_m) * t_m))));
	else
		tmp = (-0.3333333333333333 * (l * l)) / (k * (k * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 240000000000.0], N[(l * N[(l / N[(k * N[(k * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.4e11

   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. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 53.0

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

   4. Applied rewrites 53.0%

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

   6. Applied rewrites 58.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow3 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow3 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 63.8

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

   8. Applied rewrites 63.8%

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

   if 2.4e11 < k

   1. Initial program 48.0%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.1%

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

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{\color{blue}{4}}} \]

   7. Applied rewrites 20.2%

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

   8. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

      8. lift-*.f64 55.1

         \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

   10. Applied rewrites 55.1%

       \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]

       5. lower-*.f64 55.8

          \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]

   12. Applied rewrites 55.8%

       \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 47.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 240000000000:\\ \;\;\;\;\frac{\ell}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\_m\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 240000000000.0)
    (* (/ l (* (* k k) (* (* t_m t_m) t_m))) l)
    (/ (* -0.3333333333333333 (* l l)) (* k (* k t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 240000000000.0) {
		tmp = (l / ((k * k) * ((t_m * t_m) * t_m))) * l;
	} else {
		tmp = (-0.3333333333333333 * (l * l)) / (k * (k * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 240000000000.0d0) then
        tmp = (l / ((k * k) * ((t_m * t_m) * t_m))) * l
    else
        tmp = ((-0.3333333333333333d0) * (l * l)) / (k * (k * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 240000000000.0) {
		tmp = (l / ((k * k) * ((t_m * t_m) * t_m))) * l;
	} else {
		tmp = (-0.3333333333333333 * (l * l)) / (k * (k * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 240000000000.0:
		tmp = (l / ((k * k) * ((t_m * t_m) * t_m))) * l
	else:
		tmp = (-0.3333333333333333 * (l * l)) / (k * (k * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 240000000000.0)
		tmp = Float64(Float64(l / Float64(Float64(k * k) * Float64(Float64(t_m * t_m) * t_m))) * l);
	else
		tmp = Float64(Float64(-0.3333333333333333 * Float64(l * l)) / Float64(k * Float64(k * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 240000000000.0)
		tmp = (l / ((k * k) * ((t_m * t_m) * t_m))) * l;
	else
		tmp = (-0.3333333333333333 * (l * l)) / (k * (k * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 240000000000.0], N[(N[(l / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(-0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.4e11

   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. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 53.0

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

   4. Applied rewrites 53.0%

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

   6. Applied rewrites 58.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 58.2

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

   8. Applied rewrites 58.2%

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

   if 2.4e11 < k

   1. Initial program 48.0%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   4. Applied rewrites 68.1%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{\color{blue}{4}}} \]

   7. Applied rewrites 20.2%

      \[\leadsto \frac{2 \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\color{blue}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   8. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

      4. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot t} \]

      7. pow2 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

      8. lift-*.f64 55.1

         \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

   10. Applied rewrites 55.1%

       \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]

       5. lower-*.f64 55.8

          \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]

   12. Applied rewrites 55.8%

       \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 33.1% accurate, 7.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\_m\right)} \end{array} \]

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 (/ (* -0.3333333333333333 (* l l)) (* k (* k t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((-0.3333333333333333 * (l * l)) / (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 * (((-0.3333333333333333d0) * (l * l)) / (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 * ((-0.3333333333333333 * (l * l)) / (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 * ((-0.3333333333333333 * (l * l)) / (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(-0.3333333333333333 * Float64(l * l)) / Float64(k * Float64(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 * ((-0.3333333333333333 * (l * l)) / (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[(-0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
3. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   6. lower-cos.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   7. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

4. Applied rewrites 55.5%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]

5. Taylor expanded in k around 0

   \[\leadsto \frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{\color{blue}{4}}} \]

7. Applied rewrites 22.7%

   \[\leadsto \frac{2 \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\color{blue}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

8. Taylor expanded in k around inf

   \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

   4. pow2 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot t} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot t} \]

   7. pow2 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

   8. lift-*.f64 31.3

      \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

10. Applied rewrites 31.3%

    \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

    3. associate-*l* N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]

    5. lower-*.f64 33.1

       \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]

12. Applied rewrites 33.1%

    \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{k \cdot \left(k \cdot t\right)} \]
13. Add Preprocessing

Alternative 14: 31.2% accurate, 7.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t\_m} \cdot -0.3333333333333333\right) \end{array} \]

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) (* (* k k) t_m)) -0.3333333333333333)))

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) / ((k * k) * t_m)) * -0.3333333333333333);
}

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) / ((k * k) * t_m)) * (-0.3333333333333333d0))
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) / ((k * k) * t_m)) * -0.3333333333333333);
}

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) / ((k * k) * t_m)) * -0.3333333333333333)

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(Float64(l * l) / Float64(Float64(k * k) * t_m)) * -0.3333333333333333))
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) / ((k * k) * t_m)) * -0.3333333333333333);
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[(N[(l * l), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
3. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{{k}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{\color{blue}{2}} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   6. lower-cos.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   7. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(t \cdot {\sin k}^{2}\right) \cdot \color{blue}{{k}^{2}}} \]

4. Applied rewrites 55.5%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]

5. Taylor expanded in k around 0

   \[\leadsto \frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{\color{blue}{4}}} \]

7. Applied rewrites 22.7%

   \[\leadsto \frac{2 \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\color{blue}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

8. Taylor expanded in k around inf

   \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

   4. pow2 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot t} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot t} \]

   7. pow2 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

   8. lift-*.f64 31.3

      \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

10. Applied rewrites 31.3%

    \[\leadsto \frac{-0.3333333333333333 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]
11. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \]

    4. pow2 N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

    5. associate-/l* N/A

       \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

    6. lift-*.f64 N/A

       \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

    7. lift-*.f64 N/A

       \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot t} \]

    8. pow2 N/A

       \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} \]

    9. *-commutative N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

    10. lower-*.f64 N/A

        \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

    11. lower-/.f64 N/A

        \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

    12. pow2 N/A

        \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

    13. lift-*.f64 N/A

        \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

    14. pow2 N/A

        \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

    15. lift-*.f64 N/A

        \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{-1}{3} \]

    16. lift-*.f64 31.2

        \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]

12. Applied rewrites 31.2%

    \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025123 
(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))))