Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.4% → 85.1%

Time: 7.7s

Alternatives: 20

Speedup: 12.5×

• 27.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-72}:\\ \;\;\;\;\left(\frac{\frac{\cos k}{t\_m}}{{\sin k}^{2}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot 2\\ \mathbf{elif}\;t\_m \leq 2.35 \cdot 10^{+153}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m \cdot t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1e-72)
    (* (* (/ (/ (cos k) t_m) (pow (sin k) 2.0)) (* (/ l k) (/ l k))) 2.0)
    (if (<= t_m 2.35e+153)
      (/
       2.0
       (*
        (* (* (* (/ (* t_m t_m) l) (/ t_m l)) (sin k)) (tan k))
        (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
      (* (/ l (pow (* k t_m) 2.0)) (/ l t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e-72) {
		tmp = (((cos(k) / t_m) / pow(sin(k), 2.0)) * ((l / k) * (l / k))) * 2.0;
	} else if (t_m <= 2.35e+153) {
		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));
	} else {
		tmp = (l / pow((k * t_m), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1d-72) then
        tmp = (((cos(k) / t_m) / (sin(k) ** 2.0d0)) * ((l / k) * (l / k))) * 2.0d0
    else if (t_m <= 2.35d+153) then
        tmp = 2.0d0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    else
        tmp = (l / ((k * t_m) ** 2.0d0)) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e-72) {
		tmp = (((Math.cos(k) / t_m) / Math.pow(Math.sin(k), 2.0)) * ((l / k) * (l / k))) * 2.0;
	} else if (t_m <= 2.35e+153) {
		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));
	} else {
		tmp = (l / Math.pow((k * t_m), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1e-72:
		tmp = (((math.cos(k) / t_m) / math.pow(math.sin(k), 2.0)) * ((l / k) * (l / k))) * 2.0
	elif t_m <= 2.35e+153:
		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))
	else:
		tmp = (l / math.pow((k * t_m), 2.0)) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1e-72)
		tmp = Float64(Float64(Float64(Float64(cos(k) / t_m) / (sin(k) ^ 2.0)) * Float64(Float64(l / k) * Float64(l / k))) * 2.0);
	elseif (t_m <= 2.35e+153)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l) * Float64(t_m / l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1e-72)
		tmp = (((cos(k) / t_m) / (sin(k) ^ 2.0)) * ((l / k) * (l / k))) * 2.0;
	elseif (t_m <= 2.35e+153)
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	else
		tmp = (l / ((k * t_m) ^ 2.0)) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1e-72], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$m, 2.35e+153], 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[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 9.9999999999999997e-73

   1. Initial program 50.8%

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

   3. Taylor expanded in k around 0

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

      1. 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. lift-pow.f64 50.2

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

   5. Applied rewrites 50.2%

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

   6. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. associate-*r* N/A

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

   8. Applied rewrites 59.7%

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

   10. Applied rewrites 71.9%

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

   if 9.9999999999999997e-73 < t < 2.34999999999999984e153

   1. Initial program 64.8%

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

      1. lift-*.f64 N/A

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

      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 46.7

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

   4. Applied rewrites 46.7%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{e^{\log t \cdot 3 - \log \ell \cdot 2}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. 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. pow3 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 87.2

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

   6. Applied rewrites 87.2%

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

   if 2.34999999999999984e153 < t

   1. Initial program 74.1%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 55.6

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

   5. Applied rewrites 55.6%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 55.6

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

   7. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 55.6

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

   9. Applied rewrites 55.6%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. pow2 N/A

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

       11. lower-/.f64 N/A

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

       12. pow-prod-down N/A

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

       13. lower-pow.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lower-/.f64 89.2

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

   11. Applied rewrites 89.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-72}:\\ \;\;\;\;\left(\frac{\frac{\cos k}{t}}{{\sin k}^{2}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot 2\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{+153}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-72}:\\ \;\;\;\;\left(\frac{\frac{\cos k}{t\_m}}{{\sin k}^{2}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log \ell \cdot 2} \cdot \frac{1}{{\sin k}^{-1}}\right) \cdot \frac{\sin k}{\cos k}\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

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 1e-72)
    (* (* (/ (/ (cos k) t_m) (pow (sin k) 2.0)) (* (/ l k) (/ l k))) 2.0)
    (/
     2.0
     (*
      (*
       (*
        (exp (- (* (log t_m) 3.0) (* (log l) 2.0)))
        (/ 1.0 (pow (sin k) -1.0)))
       (/ (sin k) (cos 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 <= 1e-72) {
		tmp = (((cos(k) / t_m) / pow(sin(k), 2.0)) * ((l / k) * (l / k))) * 2.0;
	} else {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l) * 2.0))) * (1.0 / pow(sin(k), -1.0))) * (sin(k) / cos(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 <= 1d-72) then
        tmp = (((cos(k) / t_m) / (sin(k) ** 2.0d0)) * ((l / k) * (l / k))) * 2.0d0
    else
        tmp = 2.0d0 / (((exp(((log(t_m) * 3.0d0) - (log(l) * 2.0d0))) * (1.0d0 / (sin(k) ** (-1.0d0)))) * (sin(k) / cos(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 <= 1e-72) {
		tmp = (((Math.cos(k) / t_m) / Math.pow(Math.sin(k), 2.0)) * ((l / k) * (l / k))) * 2.0;
	} else {
		tmp = 2.0 / (((Math.exp(((Math.log(t_m) * 3.0) - (Math.log(l) * 2.0))) * (1.0 / Math.pow(Math.sin(k), -1.0))) * (Math.sin(k) / Math.cos(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 <= 1e-72:
		tmp = (((math.cos(k) / t_m) / math.pow(math.sin(k), 2.0)) * ((l / k) * (l / k))) * 2.0
	else:
		tmp = 2.0 / (((math.exp(((math.log(t_m) * 3.0) - (math.log(l) * 2.0))) * (1.0 / math.pow(math.sin(k), -1.0))) * (math.sin(k) / math.cos(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 <= 1e-72)
		tmp = Float64(Float64(Float64(Float64(cos(k) / t_m) / (sin(k) ^ 2.0)) * Float64(Float64(l / k) * Float64(l / k))) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l) * 2.0))) * Float64(1.0 / (sin(k) ^ -1.0))) * Float64(sin(k) / cos(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 <= 1e-72)
		tmp = (((cos(k) / t_m) / (sin(k) ^ 2.0)) * ((l / k) * (l / k))) * 2.0;
	else
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l) * 2.0))) * (1.0 / (sin(k) ^ -1.0))) * (sin(k) / cos(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, 1e-72], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], 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[(1.0 / N[Power[N[Sin[k], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $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 < 9.9999999999999997e-73

   1. Initial program 50.8%

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

   3. Taylor expanded in k around 0

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

      1. 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. lift-pow.f64 50.2

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

   5. Applied rewrites 50.2%

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

   6. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. associate-*r* N/A

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

   8. Applied rewrites 59.7%

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

   10. Applied rewrites 71.9%

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

   if 9.9999999999999997e-73 < t

   1. Initial program 68.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 44.4

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

   4. Applied rewrites 44.4%

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

      1. lift-sin.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. pow-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-sin.f64 44.4

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

   6. Applied rewrites 44.4%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 44.4

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

   8. Applied rewrites 44.4%

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \frac{1}{{\sin k}^{-1}}\right) \cdot \color{blue}{\frac{\sin k}{\cos k}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-72}:\\ \;\;\;\;\left(\frac{\frac{\cos k}{t}}{{\sin k}^{2}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \frac{1}{{\sin k}^{-1}}\right) \cdot \frac{\sin k}{\cos k}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-72}:\\ \;\;\;\;\left(\frac{\frac{\cos k}{t\_m}}{{\sin k}^{2}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \frac{\sin k}{\cos k}\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

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 1e-72)
    (* (* (/ (/ (cos k) t_m) (pow (sin k) 2.0)) (* (/ l k) (/ l k))) 2.0)
    (/
     2.0
     (*
      (*
       (* (exp (- (* (log t_m) 3.0) (* (log l) 2.0))) (sin k))
       (/ (sin k) (cos 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 <= 1e-72) {
		tmp = (((cos(k) / t_m) / pow(sin(k), 2.0)) * ((l / k) * (l / k))) * 2.0;
	} else {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l) * 2.0))) * sin(k)) * (sin(k) / cos(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 <= 1d-72) then
        tmp = (((cos(k) / t_m) / (sin(k) ** 2.0d0)) * ((l / k) * (l / k))) * 2.0d0
    else
        tmp = 2.0d0 / (((exp(((log(t_m) * 3.0d0) - (log(l) * 2.0d0))) * sin(k)) * (sin(k) / cos(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 <= 1e-72) {
		tmp = (((Math.cos(k) / t_m) / Math.pow(Math.sin(k), 2.0)) * ((l / k) * (l / k))) * 2.0;
	} else {
		tmp = 2.0 / (((Math.exp(((Math.log(t_m) * 3.0) - (Math.log(l) * 2.0))) * Math.sin(k)) * (Math.sin(k) / Math.cos(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 <= 1e-72:
		tmp = (((math.cos(k) / t_m) / math.pow(math.sin(k), 2.0)) * ((l / k) * (l / k))) * 2.0
	else:
		tmp = 2.0 / (((math.exp(((math.log(t_m) * 3.0) - (math.log(l) * 2.0))) * math.sin(k)) * (math.sin(k) / math.cos(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 <= 1e-72)
		tmp = Float64(Float64(Float64(Float64(cos(k) / t_m) / (sin(k) ^ 2.0)) * Float64(Float64(l / k) * Float64(l / k))) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l) * 2.0))) * sin(k)) * Float64(sin(k) / cos(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 <= 1e-72)
		tmp = (((cos(k) / t_m) / (sin(k) ^ 2.0)) * ((l / k) * (l / k))) * 2.0;
	else
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l) * 2.0))) * sin(k)) * (sin(k) / cos(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, 1e-72], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], 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] * N[(N[Sin[k], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $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 < 9.9999999999999997e-73

   1. Initial program 50.8%

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

   3. Taylor expanded in k around 0

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

      1. 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. lift-pow.f64 50.2

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

   5. Applied rewrites 50.2%

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

   6. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. associate-*r* N/A

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

   8. Applied rewrites 59.7%

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

   10. Applied rewrites 71.9%

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

   if 9.9999999999999997e-73 < t

   1. Initial program 68.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 44.4

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

   4. Applied rewrites 44.4%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lower-cos.f64 44.4

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

   6. Applied rewrites 44.4%

      \[\leadsto \frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \color{blue}{\frac{\sin k}{\cos k}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-72}:\\ \;\;\;\;\left(\frac{\frac{\cos k}{t}}{{\sin k}^{2}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \frac{\sin k}{\cos k}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-72}:\\ \;\;\;\;\left(\frac{\frac{\cos k}{t\_m}}{{\sin k}^{2}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t\_m \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1e-72)
    (* (* (/ (/ (cos k) t_m) (pow (sin k) 2.0)) (* (/ l k) (/ l k))) 2.0)
    (/
     2.0
     (*
      (* (* (exp (- (* (log t_m) 3.0) (* (log l) 2.0))) (sin k)) (tan k))
      (fma (/ k t_m) (/ k t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e-72) {
		tmp = (((cos(k) / t_m) / pow(sin(k), 2.0)) * ((l / k) * (l / k))) * 2.0;
	} else {
		tmp = 2.0 / (((exp(((log(t_m) * 3.0) - (log(l) * 2.0))) * sin(k)) * tan(k)) * fma((k / t_m), (k / t_m), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1e-72)
		tmp = Float64(Float64(Float64(Float64(cos(k) / t_m) / (sin(k) ^ 2.0)) * Float64(Float64(l / k) * Float64(l / k))) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(exp(Float64(Float64(log(t_m) * 3.0) - Float64(log(l) * 2.0))) * sin(k)) * tan(k)) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < 9.9999999999999997e-73

   1. Initial program 50.8%

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

   3. Taylor expanded in k around 0

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

      1. 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. lift-pow.f64 50.2

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

   5. Applied rewrites 50.2%

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

   6. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. associate-*r* N/A

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

   8. Applied rewrites 59.7%

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

   10. Applied rewrites 71.9%

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

   if 9.9999999999999997e-73 < t

   1. Initial program 68.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 44.4

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

   4. Applied rewrites 44.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 44.4

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

   6. Applied rewrites 44.4%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-72}:\\ \;\;\;\;\left(\frac{\frac{\cos k}{t}}{{\sin k}^{2}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\log t \cdot 3 - \log \ell \cdot 2} \cdot \sin k\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-72}:\\ \;\;\;\;\left(\frac{\frac{\cos k}{t\_m}}{{\sin k}^{2}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t\_m, 3, -2 \cdot \log \ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1e-72)
    (* (* (/ (/ (cos k) t_m) (pow (sin k) 2.0)) (* (/ l k) (/ l k))) 2.0)
    (/
     2.0
     (*
      (* (* (exp (fma (log t_m) 3.0 (* -2.0 (log l)))) (sin k)) (tan k))
      (fma (/ k t_m) (/ k t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e-72) {
		tmp = (((cos(k) / t_m) / pow(sin(k), 2.0)) * ((l / k) * (l / k))) * 2.0;
	} else {
		tmp = 2.0 / (((exp(fma(log(t_m), 3.0, (-2.0 * log(l)))) * sin(k)) * tan(k)) * fma((k / t_m), (k / t_m), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1e-72)
		tmp = Float64(Float64(Float64(Float64(cos(k) / t_m) / (sin(k) ^ 2.0)) * Float64(Float64(l / k) * Float64(l / k))) * 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)) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if t < 9.9999999999999997e-73

   1. Initial program 50.8%

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

   3. Taylor expanded in k around 0

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

      1. 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. lift-pow.f64 50.2

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

   5. Applied rewrites 50.2%

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

   6. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. associate-*r* N/A

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

   8. Applied rewrites 59.7%

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

   10. Applied rewrites 71.9%

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

   if 9.9999999999999997e-73 < t

   1. Initial program 68.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 44.4

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

   4. Applied rewrites 44.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 44.4

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

   6. Applied rewrites 44.4%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. *-commutative N/A

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

      7. lower--.f64 N/A

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

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

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 44.3

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

   8. Applied rewrites 44.3%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-72}:\\ \;\;\;\;\left(\frac{\frac{\cos k}{t}}{{\sin k}^{2}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(e^{\mathsf{fma}\left(\log t, 3, -2 \cdot \log \ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+179}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)\right) \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_m\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       2e+179)
    (/ (* l l) (* (* (* k t_m) (* k t_m)) t_m))
    (/ (* l l) (* (* (* (* k k) t_m) t_m) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / ((((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 2e+179) {
		tmp = (l * l) / (((k * t_m) * (k * t_m)) * t_m);
	} else {
		tmp = (l * l) / ((((k * k) * t_m) * t_m) * t_m);
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 2d+179) then
        tmp = (l * l) / (((k * t_m) * (k * t_m)) * t_m)
    else
        tmp = (l * l) / ((((k * k) * t_m) * t_m) * t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((2.0 / ((((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 2e+179) {
		tmp = (l * l) / (((k * t_m) * (k * t_m)) * t_m);
	} else {
		tmp = (l * l) / ((((k * k) * t_m) * t_m) * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (2.0 / ((((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 2e+179:
		tmp = (l * l) / (((k * t_m) * (k * t_m)) * t_m)
	else:
		tmp = (l * l) / ((((k * k) * t_m) * t_m) * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t_m ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0))) <= 2e+179)
		tmp = Float64(Float64(l * l) / Float64(Float64(Float64(k * t_m) * Float64(k * t_m)) * t_m));
	else
		tmp = Float64(Float64(l * l) / Float64(Float64(Float64(Float64(k * k) * t_m) * t_m) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((2.0 / (((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 2e+179)
		tmp = (l * l) / (((k * t_m) * (k * t_m)) * t_m);
	else
		tmp = (l * l) / ((((k * k) * t_m) * t_m) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+179], N[(N[(l * l), $MachinePrecision] / N[(N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999996e179

   1. Initial program 83.1%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 68.5

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

   5. Applied rewrites 68.5%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 68.5

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

   7. Applied rewrites 68.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 69.3

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

   9. Applied rewrites 69.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. unswap-sqr N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 82.6

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

   11. Applied rewrites 82.6%

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

   if 1.99999999999999996e179 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 24.0%

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

   3. Taylor expanded in k around 0

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

      1. 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. lift-pow.f64 27.9

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

   5. Applied rewrites 27.9%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 27.9

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

   7. Applied rewrites 27.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 31.5

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

   9. Applied rewrites 31.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 40.4

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

   11. Applied rewrites 40.4%

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

Alternative 7: 76.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 980000000:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_2} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\cos k}{t\_m}}{t\_2} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot 2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (sin k) 2.0)))
   (*
    t_s
    (if (<= k 980000000.0)
      (* (/ l (pow (* k t_m) 2.0)) (/ l t_m))
      (if (<= k 2.5e+106)
        (* (/ (* (* (cos k) l) l) (* (* (* k k) t_m) t_2)) 2.0)
        (* (* (/ (/ (cos k) t_m) t_2) (* (/ l k) (/ l 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 t_2 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 980000000.0) {
		tmp = (l / pow((k * t_m), 2.0)) * (l / t_m);
	} else if (k <= 2.5e+106) {
		tmp = (((cos(k) * l) * l) / (((k * k) * t_m) * t_2)) * 2.0;
	} else {
		tmp = (((cos(k) / t_m) / t_2) * ((l / k) * (l / k))) * 2.0;
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sin(k) ** 2.0d0
    if (k <= 980000000.0d0) then
        tmp = (l / ((k * t_m) ** 2.0d0)) * (l / t_m)
    else if (k <= 2.5d+106) then
        tmp = (((cos(k) * l) * l) / (((k * k) * t_m) * t_2)) * 2.0d0
    else
        tmp = (((cos(k) / t_m) / t_2) * ((l / k) * (l / k))) * 2.0d0
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 980000000.0) {
		tmp = (l / Math.pow((k * t_m), 2.0)) * (l / t_m);
	} else if (k <= 2.5e+106) {
		tmp = (((Math.cos(k) * l) * l) / (((k * k) * t_m) * t_2)) * 2.0;
	} else {
		tmp = (((Math.cos(k) / t_m) / t_2) * ((l / k) * (l / k))) * 2.0;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 980000000.0:
		tmp = (l / math.pow((k * t_m), 2.0)) * (l / t_m)
	elif k <= 2.5e+106:
		tmp = (((math.cos(k) * l) * l) / (((k * k) * t_m) * t_2)) * 2.0
	else:
		tmp = (((math.cos(k) / t_m) / t_2) * ((l / k) * (l / 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)
	t_2 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 980000000.0)
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	elseif (k <= 2.5e+106)
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * l) / Float64(Float64(Float64(k * k) * t_m) * t_2)) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(cos(k) / t_m) / t_2) * Float64(Float64(l / k) * Float64(l / k))) * 2.0);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 980000000.0)
		tmp = (l / ((k * t_m) ^ 2.0)) * (l / t_m);
	elseif (k <= 2.5e+106)
		tmp = (((cos(k) * l) * l) / (((k * k) * t_m) * t_2)) * 2.0;
	else
		tmp = (((cos(k) / t_m) / t_2) * ((l / k) * (l / k))) * 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 980000000.0], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.5e+106], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 9.8e8

   1. Initial program 59.3%

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

   3. Taylor expanded in k around 0

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

      1. 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. lift-pow.f64 52.2

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

   5. Applied rewrites 52.2%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 52.2

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

   7. Applied rewrites 52.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 54.3

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

   9. Applied rewrites 54.3%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. pow2 N/A

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

       11. lower-/.f64 N/A

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

       12. pow-prod-down N/A

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

       13. lower-pow.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lower-/.f64 73.2

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

   11. Applied rewrites 73.2%

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

   if 9.8e8 < k < 2.4999999999999999e106

   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. Add Preprocessing

   3. Taylor expanded in k around 0

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

      1. 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. lift-pow.f64 42.5

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

   5. Applied rewrites 42.5%

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

   6. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. associate-*r* N/A

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

   8. Applied rewrites 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   10. Applied rewrites 81.7%

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

   if 2.4999999999999999e106 < k

   1. Initial program 44.4%

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

   3. Taylor expanded in k around 0

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

      1. 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. lift-pow.f64 42.3

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

   5. Applied rewrites 42.3%

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

   6. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. associate-*r* N/A

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

   8. Applied rewrites 52.7%

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

   10. Applied rewrites 82.3%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 980000000:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\cos k}{t}}{{\sin k}^{2}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 980000000:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_2} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{t\_2 \cdot t\_m} \cdot 2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (sin k) 2.0)))
   (*
    t_s
    (if (<= k 980000000.0)
      (* (/ l (pow (* k t_m) 2.0)) (/ l t_m))
      (if (<= k 2.5e+106)
        (* (/ (* (* (cos k) l) l) (* (* (* k k) t_m) t_2)) 2.0)
        (* (/ (* (* (/ l k) (/ l k)) (cos k)) (* t_2 t_m)) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 980000000.0) {
		tmp = (l / pow((k * t_m), 2.0)) * (l / t_m);
	} else if (k <= 2.5e+106) {
		tmp = (((cos(k) * l) * l) / (((k * k) * t_m) * t_2)) * 2.0;
	} else {
		tmp = ((((l / k) * (l / k)) * cos(k)) / (t_2 * t_m)) * 2.0;
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sin(k) ** 2.0d0
    if (k <= 980000000.0d0) then
        tmp = (l / ((k * t_m) ** 2.0d0)) * (l / t_m)
    else if (k <= 2.5d+106) then
        tmp = (((cos(k) * l) * l) / (((k * k) * t_m) * t_2)) * 2.0d0
    else
        tmp = ((((l / k) * (l / k)) * cos(k)) / (t_2 * t_m)) * 2.0d0
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 980000000.0) {
		tmp = (l / Math.pow((k * t_m), 2.0)) * (l / t_m);
	} else if (k <= 2.5e+106) {
		tmp = (((Math.cos(k) * l) * l) / (((k * k) * t_m) * t_2)) * 2.0;
	} else {
		tmp = ((((l / k) * (l / k)) * Math.cos(k)) / (t_2 * t_m)) * 2.0;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 980000000.0:
		tmp = (l / math.pow((k * t_m), 2.0)) * (l / t_m)
	elif k <= 2.5e+106:
		tmp = (((math.cos(k) * l) * l) / (((k * k) * t_m) * t_2)) * 2.0
	else:
		tmp = ((((l / k) * (l / k)) * math.cos(k)) / (t_2 * t_m)) * 2.0
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 980000000.0)
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	elseif (k <= 2.5e+106)
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * l) / Float64(Float64(Float64(k * k) * t_m) * t_2)) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) * cos(k)) / Float64(t_2 * t_m)) * 2.0);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 980000000.0)
		tmp = (l / ((k * t_m) ^ 2.0)) * (l / t_m);
	elseif (k <= 2.5e+106)
		tmp = (((cos(k) * l) * l) / (((k * k) * t_m) * t_2)) * 2.0;
	else
		tmp = ((((l / k) * (l / k)) * cos(k)) / (t_2 * t_m)) * 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 980000000.0], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.5e+106], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 9.8e8

   1. Initial program 59.3%

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

   3. Taylor expanded in k around 0

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

      1. 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. lift-pow.f64 52.2

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

   5. Applied rewrites 52.2%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 52.2

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

   7. Applied rewrites 52.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 54.3

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

   9. Applied rewrites 54.3%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. pow2 N/A

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

       11. lower-/.f64 N/A

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

       12. pow-prod-down N/A

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

       13. lower-pow.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lower-/.f64 73.2

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

   11. Applied rewrites 73.2%

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

   if 9.8e8 < k < 2.4999999999999999e106

   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. Add Preprocessing

   3. Taylor expanded in k around 0

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

      1. 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. lift-pow.f64 42.5

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

   5. Applied rewrites 42.5%

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

   6. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. associate-*r* N/A

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

   8. Applied rewrites 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   10. Applied rewrites 81.7%

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

   if 2.4999999999999999e106 < k

   1. Initial program 44.4%

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

   3. Taylor expanded in k around 0

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

      1. 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. lift-pow.f64 42.3

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

   5. Applied rewrites 42.3%

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

   6. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. associate-*r* N/A

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

   8. Applied rewrites 52.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-cos.f64 N/A

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

      18. lift-sin.f64 N/A

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

      19. lift-pow.f64 N/A

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

      20. lift-*.f64 82.2

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

   10. Applied rewrites 82.2%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 980000000:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \cos k}{{\sin k}^{2} \cdot t} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 980000000:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot t\_2} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t\_2 \cdot t\_m}\right) \cdot 2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (sin k) 2.0)))
   (*
    t_s
    (if (<= k 980000000.0)
      (* (/ l (pow (* k t_m) 2.0)) (/ l t_m))
      (if (<= k 2.5e+106)
        (* (/ (* (* (cos k) l) l) (* (* (* k k) t_m) t_2)) 2.0)
        (* (* (* (/ l k) (/ l k)) (/ (cos k) (* t_2 t_m))) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 980000000.0) {
		tmp = (l / pow((k * t_m), 2.0)) * (l / t_m);
	} else if (k <= 2.5e+106) {
		tmp = (((cos(k) * l) * l) / (((k * k) * t_m) * t_2)) * 2.0;
	} else {
		tmp = (((l / k) * (l / k)) * (cos(k) / (t_2 * t_m))) * 2.0;
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sin(k) ** 2.0d0
    if (k <= 980000000.0d0) then
        tmp = (l / ((k * t_m) ** 2.0d0)) * (l / t_m)
    else if (k <= 2.5d+106) then
        tmp = (((cos(k) * l) * l) / (((k * k) * t_m) * t_2)) * 2.0d0
    else
        tmp = (((l / k) * (l / k)) * (cos(k) / (t_2 * t_m))) * 2.0d0
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 980000000.0) {
		tmp = (l / Math.pow((k * t_m), 2.0)) * (l / t_m);
	} else if (k <= 2.5e+106) {
		tmp = (((Math.cos(k) * l) * l) / (((k * k) * t_m) * t_2)) * 2.0;
	} else {
		tmp = (((l / k) * (l / k)) * (Math.cos(k) / (t_2 * t_m))) * 2.0;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 980000000.0:
		tmp = (l / math.pow((k * t_m), 2.0)) * (l / t_m)
	elif k <= 2.5e+106:
		tmp = (((math.cos(k) * l) * l) / (((k * k) * t_m) * t_2)) * 2.0
	else:
		tmp = (((l / k) * (l / k)) * (math.cos(k) / (t_2 * t_m))) * 2.0
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 980000000.0)
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	elseif (k <= 2.5e+106)
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * l) / Float64(Float64(Float64(k * k) * t_m) * t_2)) * 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t_2 * t_m))) * 2.0);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 980000000.0)
		tmp = (l / ((k * t_m) ^ 2.0)) * (l / t_m);
	elseif (k <= 2.5e+106)
		tmp = (((cos(k) * l) * l) / (((k * k) * t_m) * t_2)) * 2.0;
	else
		tmp = (((l / k) * (l / k)) * (cos(k) / (t_2 * t_m))) * 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 980000000.0], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.5e+106], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$2 * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 9.8e8

   1. Initial program 59.3%

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

   3. Taylor expanded in k around 0

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

      1. 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. lift-pow.f64 52.2

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

   5. Applied rewrites 52.2%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 52.2

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

   7. Applied rewrites 52.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 54.3

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

   9. Applied rewrites 54.3%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. pow2 N/A

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

       11. lower-/.f64 N/A

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

       12. pow-prod-down N/A

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

       13. lower-pow.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lower-/.f64 73.2

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

   11. Applied rewrites 73.2%

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

   if 9.8e8 < k < 2.4999999999999999e106

   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. Add Preprocessing

   3. Taylor expanded in k around 0

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

      1. 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. lift-pow.f64 42.5

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

   5. Applied rewrites 42.5%

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

   6. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. associate-*r* N/A

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

   8. Applied rewrites 65.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   10. Applied rewrites 81.7%

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

   if 2.4999999999999999e106 < k

   1. Initial program 44.4%

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

   3. Taylor expanded in k around 0

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

      1. 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. lift-pow.f64 42.3

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

   5. Applied rewrites 42.3%

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

   6. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. associate-*r* N/A

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

   8. Applied rewrites 52.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 82.2

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

   10. Applied rewrites 82.2%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 980000000:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{-73}:\\ \;\;\;\;\left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t\_m}\right) \cdot 2\\ \mathbf{elif}\;t\_m \leq 7.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t\_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9e-73)
    (* (* (* l (/ l (* k k))) (/ (cos k) (* (pow (sin k) 2.0) t_m))) 2.0)
    (if (<= t_m 7.4e+49)
      (/ (* (/ l k) (/ l k)) (pow t_m 3.0))
      (* (/ l (pow (* k t_m) 2.0)) (/ l t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9e-73) {
		tmp = ((l * (l / (k * k))) * (cos(k) / (pow(sin(k), 2.0) * t_m))) * 2.0;
	} else if (t_m <= 7.4e+49) {
		tmp = ((l / k) * (l / k)) / pow(t_m, 3.0);
	} else {
		tmp = (l / pow((k * t_m), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 9d-73) then
        tmp = ((l * (l / (k * k))) * (cos(k) / ((sin(k) ** 2.0d0) * t_m))) * 2.0d0
    else if (t_m <= 7.4d+49) then
        tmp = ((l / k) * (l / k)) / (t_m ** 3.0d0)
    else
        tmp = (l / ((k * t_m) ** 2.0d0)) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9e-73) {
		tmp = ((l * (l / (k * k))) * (Math.cos(k) / (Math.pow(Math.sin(k), 2.0) * t_m))) * 2.0;
	} else if (t_m <= 7.4e+49) {
		tmp = ((l / k) * (l / k)) / Math.pow(t_m, 3.0);
	} else {
		tmp = (l / Math.pow((k * t_m), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 9e-73:
		tmp = ((l * (l / (k * k))) * (math.cos(k) / (math.pow(math.sin(k), 2.0) * t_m))) * 2.0
	elif t_m <= 7.4e+49:
		tmp = ((l / k) * (l / k)) / math.pow(t_m, 3.0)
	else:
		tmp = (l / math.pow((k * t_m), 2.0)) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 9e-73)
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(k * k))) * Float64(cos(k) / Float64((sin(k) ^ 2.0) * t_m))) * 2.0);
	elseif (t_m <= 7.4e+49)
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) / (t_m ^ 3.0));
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 9e-73)
		tmp = ((l * (l / (k * k))) * (cos(k) / ((sin(k) ^ 2.0) * t_m))) * 2.0;
	elseif (t_m <= 7.4e+49)
		tmp = ((l / k) * (l / k)) / (t_m ^ 3.0);
	else
		tmp = (l / ((k * t_m) ^ 2.0)) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9e-73], N[(N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$m, 7.4e+49], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 9e-73

   1. Initial program 50.8%

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

   3. Taylor expanded in k around 0

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

      1. 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. lift-pow.f64 50.2

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

   5. Applied rewrites 50.2%

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

   6. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. associate-*r* N/A

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

   8. Applied rewrites 59.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 64.3

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

   10. Applied rewrites 64.3%

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

   if 9e-73 < t < 7.40000000000000036e49

   1. Initial program 76.7%

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

   3. Taylor expanded in k around 0

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

      1. 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. lift-pow.f64 55.9

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

   5. Applied rewrites 55.9%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 55.9

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

   7. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 55.9

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

   9. Applied rewrites 55.9%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. pow2 N/A

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

       9. pow2 N/A

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

       10. associate-*l* N/A

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

       11. pow2 N/A

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

       12. unpow3 N/A

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

       13. associate-/r* N/A

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

       14. pow2 N/A

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

       15. pow2 N/A

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

       16. lower-/.f64 N/A

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

       17. times-frac N/A

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

       18. lower-*.f64 N/A

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

       19. lower-/.f64 N/A

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

       20. lower-/.f64 N/A

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

       21. lift-pow.f64 77.6

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

   11. Applied rewrites 77.6%

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

   if 7.40000000000000036e49 < t

   1. Initial program 63.1%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 45.8

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

   5. Applied rewrites 45.8%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 45.8

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

   7. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 49.9

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

   9. Applied rewrites 49.9%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. pow2 N/A

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

       11. lower-/.f64 N/A

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

       12. pow-prod-down N/A

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

       13. lower-pow.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lower-/.f64 79.1

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

   11. Applied rewrites 79.1%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-73}:\\ \;\;\;\;\left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-108}:\\ \;\;\;\;\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot {\sin k}^{2}} \cdot 2\\ \mathbf{elif}\;t\_m \leq 7.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t\_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.6e-108)
    (* (/ (* (* (cos k) l) l) (* (* (* k k) t_m) (pow (sin k) 2.0))) 2.0)
    (if (<= t_m 7.4e+49)
      (/ (* (/ l k) (/ l k)) (pow t_m 3.0))
      (* (/ l (pow (* k t_m) 2.0)) (/ l t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.6e-108) {
		tmp = (((cos(k) * l) * l) / (((k * k) * t_m) * pow(sin(k), 2.0))) * 2.0;
	} else if (t_m <= 7.4e+49) {
		tmp = ((l / k) * (l / k)) / pow(t_m, 3.0);
	} else {
		tmp = (l / pow((k * t_m), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.6d-108) then
        tmp = (((cos(k) * l) * l) / (((k * k) * t_m) * (sin(k) ** 2.0d0))) * 2.0d0
    else if (t_m <= 7.4d+49) then
        tmp = ((l / k) * (l / k)) / (t_m ** 3.0d0)
    else
        tmp = (l / ((k * t_m) ** 2.0d0)) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.6e-108) {
		tmp = (((Math.cos(k) * l) * l) / (((k * k) * t_m) * Math.pow(Math.sin(k), 2.0))) * 2.0;
	} else if (t_m <= 7.4e+49) {
		tmp = ((l / k) * (l / k)) / Math.pow(t_m, 3.0);
	} else {
		tmp = (l / Math.pow((k * t_m), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.6e-108:
		tmp = (((math.cos(k) * l) * l) / (((k * k) * t_m) * math.pow(math.sin(k), 2.0))) * 2.0
	elif t_m <= 7.4e+49:
		tmp = ((l / k) * (l / k)) / math.pow(t_m, 3.0)
	else:
		tmp = (l / math.pow((k * t_m), 2.0)) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.6e-108)
		tmp = Float64(Float64(Float64(Float64(cos(k) * l) * l) / Float64(Float64(Float64(k * k) * t_m) * (sin(k) ^ 2.0))) * 2.0);
	elseif (t_m <= 7.4e+49)
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) / (t_m ^ 3.0));
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.6e-108)
		tmp = (((cos(k) * l) * l) / (((k * k) * t_m) * (sin(k) ^ 2.0))) * 2.0;
	elseif (t_m <= 7.4e+49)
		tmp = ((l / k) * (l / k)) / (t_m ^ 3.0);
	else
		tmp = (l / ((k * t_m) ^ 2.0)) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.6e-108], N[(N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$m, 7.4e+49], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.6e-108

   1. Initial program 49.0%

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

   3. Taylor expanded in k around 0

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

      1. 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. lift-pow.f64 48.5

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

   5. Applied rewrites 48.5%

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

   6. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. associate-*r* N/A

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

   8. Applied rewrites 58.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

   10. Applied rewrites 58.8%

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

   if 1.6e-108 < t < 7.40000000000000036e49

   1. Initial program 77.6%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 62.3

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

   5. Applied rewrites 62.3%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 62.2

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

   7. Applied rewrites 62.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 62.3

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

   9. Applied rewrites 62.3%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. pow2 N/A

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

       9. pow2 N/A

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

       10. associate-*l* N/A

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

       11. pow2 N/A

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

       12. unpow3 N/A

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

       13. associate-/r* N/A

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

       14. pow2 N/A

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

       15. pow2 N/A

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

       16. lower-/.f64 N/A

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

       17. times-frac N/A

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

       18. lower-*.f64 N/A

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

       19. lower-/.f64 N/A

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

       20. lower-/.f64 N/A

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

       21. lift-pow.f64 80.8

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

   11. Applied rewrites 80.8%

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

   if 7.40000000000000036e49 < t

   1. Initial program 63.1%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 45.8

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

   5. Applied rewrites 45.8%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 45.8

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

   7. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 49.9

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

   9. Applied rewrites 49.9%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. pow2 N/A

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

       11. lower-/.f64 N/A

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

       12. pow-prod-down N/A

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

       13. lower-pow.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lower-/.f64 79.1

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

   11. Applied rewrites 79.1%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-108}:\\ \;\;\;\;\frac{\left(\cos k \cdot \ell\right) \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}} \cdot 2\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.7 \cdot 10^{-145}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\_m}\right) \cdot 2\\ \mathbf{elif}\;t\_m \leq 7.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t\_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.7e-145)
    (*
     (*
      (/ (* l l) (* k k))
      (/ (cos k) (* (- 0.5 (* 0.5 (cos (* 2.0 k)))) t_m)))
     2.0)
    (if (<= t_m 7.4e+49)
      (/ (* (/ l k) (/ l k)) (pow t_m 3.0))
      (* (/ l (pow (* k t_m) 2.0)) (/ l t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.7e-145) {
		tmp = (((l * l) / (k * k)) * (cos(k) / ((0.5 - (0.5 * cos((2.0 * k)))) * t_m))) * 2.0;
	} else if (t_m <= 7.4e+49) {
		tmp = ((l / k) * (l / k)) / pow(t_m, 3.0);
	} else {
		tmp = (l / pow((k * t_m), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.7d-145) then
        tmp = (((l * l) / (k * k)) * (cos(k) / ((0.5d0 - (0.5d0 * cos((2.0d0 * k)))) * t_m))) * 2.0d0
    else if (t_m <= 7.4d+49) then
        tmp = ((l / k) * (l / k)) / (t_m ** 3.0d0)
    else
        tmp = (l / ((k * t_m) ** 2.0d0)) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.7e-145) {
		tmp = (((l * l) / (k * k)) * (Math.cos(k) / ((0.5 - (0.5 * Math.cos((2.0 * k)))) * t_m))) * 2.0;
	} else if (t_m <= 7.4e+49) {
		tmp = ((l / k) * (l / k)) / Math.pow(t_m, 3.0);
	} else {
		tmp = (l / Math.pow((k * t_m), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.7e-145:
		tmp = (((l * l) / (k * k)) * (math.cos(k) / ((0.5 - (0.5 * math.cos((2.0 * k)))) * t_m))) * 2.0
	elif t_m <= 7.4e+49:
		tmp = ((l / k) * (l / k)) / math.pow(t_m, 3.0)
	else:
		tmp = (l / math.pow((k * t_m), 2.0)) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.7e-145)
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k * k)) * Float64(cos(k) / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k)))) * t_m))) * 2.0);
	elseif (t_m <= 7.4e+49)
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) / (t_m ^ 3.0));
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.7e-145)
		tmp = (((l * l) / (k * k)) * (cos(k) / ((0.5 - (0.5 * cos((2.0 * k)))) * t_m))) * 2.0;
	elseif (t_m <= 7.4e+49)
		tmp = ((l / k) * (l / k)) / (t_m ^ 3.0);
	else
		tmp = (l / ((k * t_m) ^ 2.0)) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.7e-145], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$m, 7.4e+49], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.7e-145

   1. Initial program 50.5%

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

   3. Taylor expanded in k around 0

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

      1. 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. lift-pow.f64 49.9

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. associate-*r* N/A

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

   8. Applied rewrites 58.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 56.0

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

   10. Applied rewrites 56.0%

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

   if 2.7e-145 < t < 7.40000000000000036e49

   1. Initial program 67.1%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 54.7

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

   5. Applied rewrites 54.7%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 54.7

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

   7. Applied rewrites 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \]

      10. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

      12. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

      13. lift-*.f64 56.9

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

   9. Applied rewrites 56.9%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right)} \cdot t} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t}} \]

       3. pow2 N/A

          \[\leadsto \frac{{\ell}^{2}}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right)} \cdot t} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       8. pow2 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left({k}^{2} \cdot \left(t \cdot t\right)\right) \cdot t} \]

       9. pow2 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \]

       10. associate-*l* N/A

           \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{\left({t}^{2} \cdot t\right)}} \]

       11. pow2 N/A

           \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

       12. unpow3 N/A

           \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

       13. associate-/r* N/A

           \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{\color{blue}{{t}^{3}}} \]

       14. pow2 N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{{t}^{3}} \]

       15. pow2 N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{{t}^{3}} \]

       16. lower-/.f64 N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{\color{blue}{{t}^{3}}} \]

       17. times-frac N/A

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\color{blue}{t}}^{3}} \]

       18. lower-*.f64 N/A

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\color{blue}{t}}^{3}} \]

       19. lower-/.f64 N/A

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}} \]

       20. lower-/.f64 N/A

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}} \]

       21. lift-pow.f64 72.0

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{\color{blue}{3}}} \]

   11. Applied rewrites 72.0%

       \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{{t}^{3}}} \]

   if 7.40000000000000036e49 < t

   1. Initial program 63.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 45.8

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 45.8

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   7. Applied rewrites 45.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \]

      10. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

      12. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

      13. lift-*.f64 49.9

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

   9. Applied rewrites 49.9%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right)} \cdot t} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t}} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       7. times-frac N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

       9. pow2 N/A

          \[\leadsto \frac{\ell}{{k}^{2} \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{t} \]

       10. pow2 N/A

           \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \frac{\ell}{t} \]

       11. lower-/.f64 N/A

           \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \frac{\color{blue}{\ell}}{t} \]

       12. pow-prod-down N/A

           \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t} \]

       13. lower-pow.f64 N/A

           \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t} \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t} \]

       15. lower-/.f64 79.1

           \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{\color{blue}{t}} \]

   11. Applied rewrites 79.1%

       \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \color{blue}{\frac{\ell}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{-145}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t}\right) \cdot 2\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{-108}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{\left(k \cdot k\right) \cdot t\_m}\right) \cdot 2\\ \mathbf{elif}\;t\_m \leq 7.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t\_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.35e-108)
    (* (* (/ (* l l) (* k k)) (/ (cos k) (* (* k k) t_m))) 2.0)
    (if (<= t_m 7.4e+49)
      (/ (* (/ l k) (/ l k)) (pow t_m 3.0))
      (* (/ l (pow (* k t_m) 2.0)) (/ l t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.35e-108) {
		tmp = (((l * l) / (k * k)) * (cos(k) / ((k * k) * t_m))) * 2.0;
	} else if (t_m <= 7.4e+49) {
		tmp = ((l / k) * (l / k)) / pow(t_m, 3.0);
	} else {
		tmp = (l / pow((k * t_m), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.35d-108) then
        tmp = (((l * l) / (k * k)) * (cos(k) / ((k * k) * t_m))) * 2.0d0
    else if (t_m <= 7.4d+49) then
        tmp = ((l / k) * (l / k)) / (t_m ** 3.0d0)
    else
        tmp = (l / ((k * t_m) ** 2.0d0)) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.35e-108) {
		tmp = (((l * l) / (k * k)) * (Math.cos(k) / ((k * k) * t_m))) * 2.0;
	} else if (t_m <= 7.4e+49) {
		tmp = ((l / k) * (l / k)) / Math.pow(t_m, 3.0);
	} else {
		tmp = (l / Math.pow((k * t_m), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.35e-108:
		tmp = (((l * l) / (k * k)) * (math.cos(k) / ((k * k) * t_m))) * 2.0
	elif t_m <= 7.4e+49:
		tmp = ((l / k) * (l / k)) / math.pow(t_m, 3.0)
	else:
		tmp = (l / math.pow((k * t_m), 2.0)) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.35e-108)
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k * k)) * Float64(cos(k) / Float64(Float64(k * k) * t_m))) * 2.0);
	elseif (t_m <= 7.4e+49)
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) / (t_m ^ 3.0));
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.35e-108)
		tmp = (((l * l) / (k * k)) * (cos(k) / ((k * k) * t_m))) * 2.0;
	elseif (t_m <= 7.4e+49)
		tmp = ((l / k) * (l / k)) / (t_m ^ 3.0);
	else
		tmp = (l / ((k * t_m) ^ 2.0)) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.35e-108], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$m, 7.4e+49], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.35000000000000002e-108

   1. Initial program 49.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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. lift-pow.f64 48.5

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 48.5%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   7. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. +-commutative N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. associate-*r* N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   8. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]

   9. Taylor expanded in k around 0

      \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{k}^{2} \cdot t}\right) \cdot 2 \]
   10. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{\left(k \cdot k\right) \cdot t}\right) \cdot 2 \]

       2. lift-*.f64 54.5

          \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{\left(k \cdot k\right) \cdot t}\right) \cdot 2 \]

   11. Applied rewrites 54.5%

       \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{\left(k \cdot k\right) \cdot t}\right) \cdot 2 \]

   if 1.35000000000000002e-108 < t < 7.40000000000000036e49

   1. Initial program 77.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 62.3

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 62.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 62.2

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   7. Applied rewrites 62.2%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \]

      10. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

      12. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

      13. lift-*.f64 62.3

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

   9. Applied rewrites 62.3%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right)} \cdot t} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t}} \]

       3. pow2 N/A

          \[\leadsto \frac{{\ell}^{2}}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right)} \cdot t} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       8. pow2 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left({k}^{2} \cdot \left(t \cdot t\right)\right) \cdot t} \]

       9. pow2 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \]

       10. associate-*l* N/A

           \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{\left({t}^{2} \cdot t\right)}} \]

       11. pow2 N/A

           \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

       12. unpow3 N/A

           \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

       13. associate-/r* N/A

           \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{\color{blue}{{t}^{3}}} \]

       14. pow2 N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{{t}^{3}} \]

       15. pow2 N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{{t}^{3}} \]

       16. lower-/.f64 N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{\color{blue}{{t}^{3}}} \]

       17. times-frac N/A

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\color{blue}{t}}^{3}} \]

       18. lower-*.f64 N/A

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\color{blue}{t}}^{3}} \]

       19. lower-/.f64 N/A

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}} \]

       20. lower-/.f64 N/A

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}} \]

       21. lift-pow.f64 80.8

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{\color{blue}{3}}} \]

   11. Applied rewrites 80.8%

       \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{{t}^{3}}} \]

   if 7.40000000000000036e49 < t

   1. Initial program 63.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 45.8

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 45.8

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   7. Applied rewrites 45.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \]

      10. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

      12. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

      13. lift-*.f64 49.9

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

   9. Applied rewrites 49.9%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right)} \cdot t} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t}} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       7. times-frac N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

       9. pow2 N/A

          \[\leadsto \frac{\ell}{{k}^{2} \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{t} \]

       10. pow2 N/A

           \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \frac{\ell}{t} \]

       11. lower-/.f64 N/A

           \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \frac{\color{blue}{\ell}}{t} \]

       12. pow-prod-down N/A

           \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t} \]

       13. lower-pow.f64 N/A

           \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t} \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t} \]

       15. lower-/.f64 79.1

           \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{\color{blue}{t}} \]

   11. Applied rewrites 79.1%

       \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \color{blue}{\frac{\ell}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-108}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{\left(k \cdot k\right) \cdot t}\right) \cdot 2\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{-146}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{{k}^{-2}}{t\_m}\right) \cdot 2\\ \mathbf{elif}\;t\_m \leq 7.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t\_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.3e-146)
    (* (* (/ (* l l) (* k k)) (/ (pow k -2.0) t_m)) 2.0)
    (if (<= t_m 7.4e+49)
      (/ (* (/ l k) (/ l k)) (pow t_m 3.0))
      (* (/ l (pow (* k t_m) 2.0)) (/ l t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.3e-146) {
		tmp = (((l * l) / (k * k)) * (pow(k, -2.0) / t_m)) * 2.0;
	} else if (t_m <= 7.4e+49) {
		tmp = ((l / k) * (l / k)) / pow(t_m, 3.0);
	} else {
		tmp = (l / pow((k * t_m), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.3d-146) then
        tmp = (((l * l) / (k * k)) * ((k ** (-2.0d0)) / t_m)) * 2.0d0
    else if (t_m <= 7.4d+49) then
        tmp = ((l / k) * (l / k)) / (t_m ** 3.0d0)
    else
        tmp = (l / ((k * t_m) ** 2.0d0)) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.3e-146) {
		tmp = (((l * l) / (k * k)) * (Math.pow(k, -2.0) / t_m)) * 2.0;
	} else if (t_m <= 7.4e+49) {
		tmp = ((l / k) * (l / k)) / Math.pow(t_m, 3.0);
	} else {
		tmp = (l / Math.pow((k * t_m), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3.3e-146:
		tmp = (((l * l) / (k * k)) * (math.pow(k, -2.0) / t_m)) * 2.0
	elif t_m <= 7.4e+49:
		tmp = ((l / k) * (l / k)) / math.pow(t_m, 3.0)
	else:
		tmp = (l / math.pow((k * t_m), 2.0)) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.3e-146)
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k * k)) * Float64((k ^ -2.0) / t_m)) * 2.0);
	elseif (t_m <= 7.4e+49)
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) / (t_m ^ 3.0));
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3.3e-146)
		tmp = (((l * l) / (k * k)) * ((k ^ -2.0) / t_m)) * 2.0;
	elseif (t_m <= 7.4e+49)
		tmp = ((l / k) * (l / k)) / (t_m ^ 3.0);
	else
		tmp = (l / ((k * t_m) ^ 2.0)) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.3e-146], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, -2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$m, 7.4e+49], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.3e-146

   1. Initial program 50.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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. lift-pow.f64 50.2

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   7. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. +-commutative N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. associate-*r* N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   8. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]

   9. Taylor expanded in k around 0

      \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{1}{{k}^{2} \cdot t}\right) \cdot 2 \]
   10. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\frac{1}{{k}^{2}}}{t}\right) \cdot 2 \]

       2. lower-/.f64 N/A

          \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\frac{1}{{k}^{2}}}{t}\right) \cdot 2 \]

       3. pow-flip N/A

          \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{{k}^{\left(\mathsf{neg}\left(2\right)\right)}}{t}\right) \cdot 2 \]

       4. metadata-eval N/A

          \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{{k}^{-2}}{t}\right) \cdot 2 \]

       5. lower-pow.f64 54.6

          \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{{k}^{-2}}{t}\right) \cdot 2 \]

   11. Applied rewrites 54.6%

       \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{{k}^{-2}}{t}\right) \cdot 2 \]

   if 3.3e-146 < t < 7.40000000000000036e49

   1. Initial program 65.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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. lift-pow.f64 53.6

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 53.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 53.6

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   7. Applied rewrites 53.6%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \]

      10. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

      12. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

      13. lift-*.f64 55.8

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

   9. Applied rewrites 55.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right)} \cdot t} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t}} \]

       3. pow2 N/A

          \[\leadsto \frac{{\ell}^{2}}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right)} \cdot t} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       8. pow2 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left({k}^{2} \cdot \left(t \cdot t\right)\right) \cdot t} \]

       9. pow2 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \]

       10. associate-*l* N/A

           \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{\left({t}^{2} \cdot t\right)}} \]

       11. pow2 N/A

           \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

       12. unpow3 N/A

           \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

       13. associate-/r* N/A

           \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{\color{blue}{{t}^{3}}} \]

       14. pow2 N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{{t}^{3}} \]

       15. pow2 N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{{t}^{3}} \]

       16. lower-/.f64 N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{\color{blue}{{t}^{3}}} \]

       17. times-frac N/A

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\color{blue}{t}}^{3}} \]

       18. lower-*.f64 N/A

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\color{blue}{t}}^{3}} \]

       19. lower-/.f64 N/A

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}} \]

       20. lower-/.f64 N/A

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}} \]

       21. lift-pow.f64 70.5

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{\color{blue}{3}}} \]

   11. Applied rewrites 70.5%

       \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{{t}^{3}}} \]

   if 7.40000000000000036e49 < t

   1. Initial program 63.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 45.8

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 45.8

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   7. Applied rewrites 45.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \]

      10. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

      12. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

      13. lift-*.f64 49.9

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

   9. Applied rewrites 49.9%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right)} \cdot t} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t}} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       7. times-frac N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

       9. pow2 N/A

          \[\leadsto \frac{\ell}{{k}^{2} \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{t} \]

       10. pow2 N/A

           \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \frac{\ell}{t} \]

       11. lower-/.f64 N/A

           \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \frac{\color{blue}{\ell}}{t} \]

       12. pow-prod-down N/A

           \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t} \]

       13. lower-pow.f64 N/A

           \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t} \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t} \]

       15. lower-/.f64 79.1

           \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{\color{blue}{t}} \]

   11. Applied rewrites 79.1%

       \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \color{blue}{\frac{\ell}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-146}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{{k}^{-2}}{t}\right) \cdot 2\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 70.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.3 \cdot 10^{-146}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{k}^{4} \cdot t\_m} \cdot 2\\ \mathbf{elif}\;t\_m \leq 7.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t\_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.3e-146)
    (* (/ (* l l) (* (pow k 4.0) t_m)) 2.0)
    (if (<= t_m 7.4e+49)
      (/ (* (/ l k) (/ l k)) (pow t_m 3.0))
      (* (/ l (pow (* k t_m) 2.0)) (/ l t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.3e-146) {
		tmp = ((l * l) / (pow(k, 4.0) * t_m)) * 2.0;
	} else if (t_m <= 7.4e+49) {
		tmp = ((l / k) * (l / k)) / pow(t_m, 3.0);
	} else {
		tmp = (l / pow((k * t_m), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.3d-146) then
        tmp = ((l * l) / ((k ** 4.0d0) * t_m)) * 2.0d0
    else if (t_m <= 7.4d+49) then
        tmp = ((l / k) * (l / k)) / (t_m ** 3.0d0)
    else
        tmp = (l / ((k * t_m) ** 2.0d0)) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.3e-146) {
		tmp = ((l * l) / (Math.pow(k, 4.0) * t_m)) * 2.0;
	} else if (t_m <= 7.4e+49) {
		tmp = ((l / k) * (l / k)) / Math.pow(t_m, 3.0);
	} else {
		tmp = (l / Math.pow((k * t_m), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3.3e-146:
		tmp = ((l * l) / (math.pow(k, 4.0) * t_m)) * 2.0
	elif t_m <= 7.4e+49:
		tmp = ((l / k) * (l / k)) / math.pow(t_m, 3.0)
	else:
		tmp = (l / math.pow((k * t_m), 2.0)) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.3e-146)
		tmp = Float64(Float64(Float64(l * l) / Float64((k ^ 4.0) * t_m)) * 2.0);
	elseif (t_m <= 7.4e+49)
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) / (t_m ^ 3.0));
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3.3e-146)
		tmp = ((l * l) / ((k ^ 4.0) * t_m)) * 2.0;
	elseif (t_m <= 7.4e+49)
		tmp = ((l / k) * (l / k)) / (t_m ^ 3.0);
	else
		tmp = (l / ((k * t_m) ^ 2.0)) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.3e-146], N[(N[(N[(l * l), $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$m, 7.4e+49], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.3e-146

   1. Initial program 50.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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. lift-pow.f64 50.2

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   7. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. +-commutative N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. associate-*r* N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   8. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

       2. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

       5. lower-pow.f64 53.3

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

   11. Applied rewrites 53.3%

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

   if 3.3e-146 < t < 7.40000000000000036e49

   1. Initial program 65.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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. lift-pow.f64 53.6

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 53.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 53.6

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   7. Applied rewrites 53.6%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \]

      10. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

      12. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

      13. lift-*.f64 55.8

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

   9. Applied rewrites 55.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right)} \cdot t} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t}} \]

       3. pow2 N/A

          \[\leadsto \frac{{\ell}^{2}}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right)} \cdot t} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       8. pow2 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left({k}^{2} \cdot \left(t \cdot t\right)\right) \cdot t} \]

       9. pow2 N/A

          \[\leadsto \frac{{\ell}^{2}}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \]

       10. associate-*l* N/A

           \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \color{blue}{\left({t}^{2} \cdot t\right)}} \]

       11. pow2 N/A

           \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

       12. unpow3 N/A

           \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

       13. associate-/r* N/A

           \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{\color{blue}{{t}^{3}}} \]

       14. pow2 N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{{t}^{3}} \]

       15. pow2 N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{{t}^{3}} \]

       16. lower-/.f64 N/A

           \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{\color{blue}{{t}^{3}}} \]

       17. times-frac N/A

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\color{blue}{t}}^{3}} \]

       18. lower-*.f64 N/A

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\color{blue}{t}}^{3}} \]

       19. lower-/.f64 N/A

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}} \]

       20. lower-/.f64 N/A

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}} \]

       21. lift-pow.f64 70.5

           \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{\color{blue}{3}}} \]

   11. Applied rewrites 70.5%

       \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{{t}^{3}}} \]

   if 7.40000000000000036e49 < t

   1. Initial program 63.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 45.8

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 45.8

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   7. Applied rewrites 45.8%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \]

      10. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

      12. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

      13. lift-*.f64 49.9

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

   9. Applied rewrites 49.9%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right)} \cdot t} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t}} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       7. times-frac N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

       9. pow2 N/A

          \[\leadsto \frac{\ell}{{k}^{2} \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{t} \]

       10. pow2 N/A

           \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \frac{\ell}{t} \]

       11. lower-/.f64 N/A

           \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \frac{\color{blue}{\ell}}{t} \]

       12. pow-prod-down N/A

           \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t} \]

       13. lower-pow.f64 N/A

           \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t} \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t} \]

       15. lower-/.f64 79.1

           \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{\color{blue}{t}} \]

   11. Applied rewrites 79.1%

       \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \color{blue}{\frac{\ell}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-146}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10^{-86}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{k}^{4} \cdot t\_m} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\_m\right)}^{2}} \cdot \frac{\ell}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1e-86)
    (* (/ (* l l) (* (pow k 4.0) t_m)) 2.0)
    (* (/ l (pow (* k t_m) 2.0)) (/ l t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e-86) {
		tmp = ((l * l) / (pow(k, 4.0) * t_m)) * 2.0;
	} else {
		tmp = (l / pow((k * t_m), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1d-86) then
        tmp = ((l * l) / ((k ** 4.0d0) * t_m)) * 2.0d0
    else
        tmp = (l / ((k * t_m) ** 2.0d0)) * (l / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e-86) {
		tmp = ((l * l) / (Math.pow(k, 4.0) * t_m)) * 2.0;
	} else {
		tmp = (l / Math.pow((k * t_m), 2.0)) * (l / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1e-86:
		tmp = ((l * l) / (math.pow(k, 4.0) * t_m)) * 2.0
	else:
		tmp = (l / math.pow((k * t_m), 2.0)) * (l / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1e-86)
		tmp = Float64(Float64(Float64(l * l) / Float64((k ^ 4.0) * t_m)) * 2.0);
	else
		tmp = Float64(Float64(l / (Float64(k * t_m) ^ 2.0)) * Float64(l / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1e-86)
		tmp = ((l * l) / ((k ^ 4.0) * t_m)) * 2.0;
	else
		tmp = (l / ((k * t_m) ^ 2.0)) * (l / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1e-86], N[(N[(N[(l * l), $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(l / N[Power[N[(k * t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.00000000000000008e-86

   1. Initial program 50.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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. lift-pow.f64 50.0

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   7. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. +-commutative N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. associate-*r* N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   8. Applied rewrites 59.6%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

       2. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

       5. lower-pow.f64 53.9

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

   11. Applied rewrites 53.9%

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

   if 1.00000000000000008e-86 < t

   1. Initial program 67.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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. lift-pow.f64 50.0

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 50.0

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   7. Applied rewrites 50.0%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \]

      10. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

      12. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

      13. lift-*.f64 52.5

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

   9. Applied rewrites 52.5%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right)} \cdot t} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t}} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       7. times-frac N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \color{blue}{\frac{\ell}{t}} \]

       9. pow2 N/A

          \[\leadsto \frac{\ell}{{k}^{2} \cdot \left(t \cdot t\right)} \cdot \frac{\ell}{t} \]

       10. pow2 N/A

           \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \frac{\ell}{t} \]

       11. lower-/.f64 N/A

           \[\leadsto \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \frac{\color{blue}{\ell}}{t} \]

       12. pow-prod-down N/A

           \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t} \]

       13. lower-pow.f64 N/A

           \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t} \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t} \]

       15. lower-/.f64 72.0

           \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{\color{blue}{t}} \]

   11. Applied rewrites 72.0%

       \[\leadsto \frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \color{blue}{\frac{\ell}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{-86}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(k \cdot t\right)}^{2}} \cdot \frac{\ell}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 64.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{k}^{4} \cdot t\_m} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)\right) \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.5e-23)
    (* (/ (* l l) (* (pow k 4.0) t_m)) 2.0)
    (/ (* l l) (* (* (* k t_m) (* k t_m)) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.5e-23) {
		tmp = ((l * l) / (pow(k, 4.0) * t_m)) * 2.0;
	} else {
		tmp = (l * l) / (((k * t_m) * (k * t_m)) * t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.5d-23) then
        tmp = ((l * l) / ((k ** 4.0d0) * t_m)) * 2.0d0
    else
        tmp = (l * l) / (((k * t_m) * (k * t_m)) * t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.5e-23) {
		tmp = ((l * l) / (Math.pow(k, 4.0) * t_m)) * 2.0;
	} else {
		tmp = (l * l) / (((k * t_m) * (k * t_m)) * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.5e-23:
		tmp = ((l * l) / (math.pow(k, 4.0) * t_m)) * 2.0
	else:
		tmp = (l * l) / (((k * t_m) * (k * t_m)) * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.5e-23)
		tmp = Float64(Float64(Float64(l * l) / Float64((k ^ 4.0) * t_m)) * 2.0);
	else
		tmp = Float64(Float64(l * l) / Float64(Float64(Float64(k * t_m) * Float64(k * t_m)) * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.5e-23)
		tmp = ((l * l) / ((k ^ 4.0) * t_m)) * 2.0;
	else
		tmp = (l * l) / (((k * t_m) * (k * t_m)) * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.5e-23], N[(N[(N[(l * l), $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.49999999999999975e-23

   1. Initial program 51.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. 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. lift-pow.f64 50.6

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   7. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      4. +-commutative N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      6. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      7. associate-*r* N/A

         \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

   8. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot 2 \]

       2. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

       5. lower-pow.f64 54.3

          \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

   11. Applied rewrites 54.3%

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2 \]

   if 4.49999999999999975e-23 < t

   1. Initial program 69.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. 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. lift-pow.f64 47.9

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   5. Applied rewrites 47.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. pow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      4. lift-*.f64 47.9

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   7. Applied rewrites 47.9%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \]

      10. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

      12. pow2 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

      13. lift-*.f64 51.2

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

   9. Applied rewrites 51.2%

      \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

       4. unswap-sqr N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \]

       7. lower-*.f64 65.1

          \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \]

   11. Applied rewrites 65.1%

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{k}^{4} \cdot t} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 61.5% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\ell \cdot \ell}{\left(\left(k \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)\right) \cdot t\_m} \end{array} \]

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 t_m) (* k t_m)) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) / (((k * t_m) * (k * t_m)) * t_m));
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * l) / (((k * t_m) * (k * t_m)) * t_m))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) / (((k * t_m) * (k * t_m)) * t_m));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * l) / (((k * t_m) * (k * t_m)) * t_m))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) / Float64(Float64(Float64(k * t_m) * Float64(k * t_m)) * t_m)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * l) / (((k * t_m) * (k * t_m)) * t_m));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] / N[(N[(N[(k * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. 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. lift-pow.f64 50.0

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

5. Applied rewrites 50.0%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   2. pow3 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   4. lift-*.f64 50.0

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

7. Applied rewrites 50.0%

   \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   5. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

   6. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

   7. associate-*r* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \]

   10. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

   12. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

   13. lift-*.f64 52.0

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

9. Applied rewrites 52.0%

   \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

    4. unswap-sqr N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \]

    5. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \]

    6. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \]

    7. lower-*.f64 60.6

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \]

11. Applied rewrites 60.6%

    \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right) \cdot t} \]
12. Add Preprocessing

Alternative 19: 58.2% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\ell \cdot \ell}{\left(k \cdot \left(k \cdot \left(t\_m \cdot t\_m\right)\right)\right) \cdot t\_m} \end{array} \]

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 t_m))) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) / ((k * (k * (t_m * t_m))) * t_m));
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * l) / ((k * (k * (t_m * t_m))) * t_m))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) / ((k * (k * (t_m * t_m))) * t_m));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * l) / ((k * (k * (t_m * t_m))) * t_m))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) / Float64(Float64(k * Float64(k * Float64(t_m * t_m))) * t_m)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * l) / ((k * (k * (t_m * t_m))) * t_m));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] / N[(N[(k * N[(k * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. 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. lift-pow.f64 50.0

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

5. Applied rewrites 50.0%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   2. pow3 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   4. lift-*.f64 50.0

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

7. Applied rewrites 50.0%

   \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   5. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

   6. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

   7. associate-*r* N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot t} \]

   10. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

   12. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

   13. lift-*.f64 52.0

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

9. Applied rewrites 52.0%

   \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{t}} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t} \]

    4. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(\left(k \cdot k\right) \cdot {t}^{2}\right) \cdot t} \]

    5. associate-*l* N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot \left(k \cdot {t}^{2}\right)\right) \cdot t} \]

    6. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot \left(k \cdot {t}^{2}\right)\right) \cdot t} \]

    7. lower-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot \left(k \cdot {t}^{2}\right)\right) \cdot t} \]

    8. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot \left(k \cdot \left(t \cdot t\right)\right)\right) \cdot t} \]

    9. lift-*.f64 57.2

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot \left(k \cdot \left(t \cdot t\right)\right)\right) \cdot t} \]

11. Applied rewrites 57.2%

    \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot \left(k \cdot \left(t \cdot t\right)\right)\right) \cdot t} \]
12. Add Preprocessing

Alternative 20: 52.0% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (* l l) (* (* k k) (* (* t_m t_m) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) / ((k * k) * ((t_m * t_m) * t_m)));
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * l) / ((k * k) * ((t_m * t_m) * t_m)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) / ((k * k) * ((t_m * t_m) * t_m)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * l) / ((k * k) * ((t_m * t_m) * t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) / Float64(Float64(k * k) * Float64(Float64(t_m * t_m) * t_m))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * l) / ((k * k) * ((t_m * t_m) * t_m)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. 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. lift-pow.f64 50.0

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

5. Applied rewrites 50.0%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   2. pow3 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   4. lift-*.f64 50.0

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

7. Applied rewrites 50.0%

   \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025051 
(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))))