Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.1% → 92.8%

Time: 12.6s

Alternatives: 21

Speedup: 10.7×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.02 \cdot 10^{-82}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left({\sin k}^{2} \cdot t\_m\right) \cdot k\right) \cdot k}{\cos k \cdot \ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \sin k\right) \cdot t\_m\right) \cdot \left(\frac{t\_m}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.02e-82)
    (/ 2.0 (/ (/ (* (* (* (pow (sin k) 2.0) t_m) k) k) (* (cos k) l)) l))
    (/
     2.0
     (*
      (* (* (/ t_m l) (sin k)) t_m)
      (* (/ t_m l) (* (tan k) (+ (pow (/ k t_m) 2.0) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.02e-82) {
		tmp = 2.0 / (((((pow(sin(k), 2.0) * t_m) * k) * k) / (cos(k) * l)) / l);
	} else {
		tmp = 2.0 / ((((t_m / l) * sin(k)) * t_m) * ((t_m / l) * (tan(k) * (pow((k / t_m), 2.0) + 2.0))));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.02d-82) then
        tmp = 2.0d0 / ((((((sin(k) ** 2.0d0) * t_m) * k) * k) / (cos(k) * l)) / l)
    else
        tmp = 2.0d0 / ((((t_m / l) * sin(k)) * t_m) * ((t_m / l) * (tan(k) * (((k / t_m) ** 2.0d0) + 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.02e-82) {
		tmp = 2.0 / (((((Math.pow(Math.sin(k), 2.0) * t_m) * k) * k) / (Math.cos(k) * l)) / l);
	} else {
		tmp = 2.0 / ((((t_m / l) * Math.sin(k)) * t_m) * ((t_m / l) * (Math.tan(k) * (Math.pow((k / t_m), 2.0) + 2.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.02e-82:
		tmp = 2.0 / (((((math.pow(math.sin(k), 2.0) * t_m) * k) * k) / (math.cos(k) * l)) / l)
	else:
		tmp = 2.0 / ((((t_m / l) * math.sin(k)) * t_m) * ((t_m / l) * (math.tan(k) * (math.pow((k / t_m), 2.0) + 2.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.02e-82)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64((sin(k) ^ 2.0) * t_m) * k) * k) / Float64(cos(k) * l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * sin(k)) * t_m) * Float64(Float64(t_m / l) * Float64(tan(k) * Float64((Float64(k / t_m) ^ 2.0) + 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.02e-82)
		tmp = 2.0 / ((((((sin(k) ^ 2.0) * t_m) * k) * k) / (cos(k) * l)) / l);
	else
		tmp = 2.0 / ((((t_m / l) * sin(k)) * t_m) * ((t_m / l) * (tan(k) * (((k / t_m) ^ 2.0) + 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.02e-82], N[(2.0 / N[(N[(N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.02000000000000007e-82

   1. Initial program 54.0%

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 77.8

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

   7. Applied rewrites 77.8%

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

   if 1.02000000000000007e-82 < t

   1. Initial program 71.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 87.2

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

   4. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 94.2

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

   8. Applied rewrites 94.2%

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

Alternative 2: 58.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(k \cdot k\right) \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 10^{+188}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(t\_m \cdot t\_m\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 2}{t\_2 \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (* k k) t_m)))
   (*
    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)))
         1e+188)
      (/ (* l l) (* (* t_m t_m) t_2))
      (/ (* (* l l) 2.0) (* t_2 (* k k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (k * k) * t_m;
	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))) <= 1e+188) {
		tmp = (l * l) / ((t_m * t_m) * t_2);
	} else {
		tmp = ((l * l) * 2.0) / (t_2 * (k * k));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k * k) * t_m
    if ((2.0d0 / (((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 1d+188) then
        tmp = (l * l) / ((t_m * t_m) * t_2)
    else
        tmp = ((l * l) * 2.0d0) / (t_2 * (k * k))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (k * k) * t_m;
	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))) <= 1e+188) {
		tmp = (l * l) / ((t_m * t_m) * t_2);
	} else {
		tmp = ((l * l) * 2.0) / (t_2 * (k * k));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (k * k) * t_m
	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))) <= 1e+188:
		tmp = (l * l) / ((t_m * t_m) * t_2)
	else:
		tmp = ((l * l) * 2.0) / (t_2 * (k * k))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(k * k) * t_m)
	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))) <= 1e+188)
		tmp = Float64(Float64(l * l) / Float64(Float64(t_m * t_m) * t_2));
	else
		tmp = Float64(Float64(Float64(l * l) * 2.0) / Float64(t_2 * Float64(k * k)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (k * k) * t_m;
	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))) <= 1e+188)
		tmp = (l * l) / ((t_m * t_m) * t_2);
	else
		tmp = ((l * l) * 2.0) / (t_2 * (k * k));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]}, 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], 1e+188], N[(N[(l * l), $MachinePrecision] / N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t$95$2 * N[(k * k), $MachinePrecision]), $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)))) < 1e188

   1. Initial program 80.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 70.4

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

   5. Applied rewrites 70.4%

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

   7. Applied rewrites 70.4%

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

   9. Applied rewrites 70.4%

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

   11. Applied rewrites 70.3%

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

   if 1e188 < (/.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 27.6%

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

   4. Applied rewrites 34.1%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{t}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{4}}} \cdot \frac{{\ell}^{2}}{t} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{4}}} \cdot \frac{{\ell}^{2}}{t} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

      8. lower-*.f64 50.2

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

   7. Applied rewrites 50.2%

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

   9. Applied rewrites 49.2%

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

   11. Applied rewrites 50.2%

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

Alternative 3: 90.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sin k \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.9 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left({\sin k}^{2} \cdot t\_m\right) \cdot k\right) \cdot k}{\cos k \cdot \ell}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 3.4 \cdot 10^{+158}:\\ \;\;\;\;\frac{\frac{2}{t\_2 \cdot t\_m} \cdot \ell}{\left(t\_m \cdot \left({\left(\frac{k}{t\_m}\right)}^{2} + 2\right)\right) \cdot \tan k} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t\_2 \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (sin k) t_m)))
   (*
    t_s
    (if (<= t_m 4.9e-60)
      (/ 2.0 (/ (/ (* (* (* (pow (sin k) 2.0) t_m) k) k) (* (cos k) l)) l))
      (if (<= t_m 3.4e+158)
        (*
         (/
          (* (/ 2.0 (* t_2 t_m)) l)
          (* (* t_m (+ (pow (/ k t_m) 2.0) 2.0)) (tan k)))
         l)
        (/
         2.0
         (*
          (* (* t_2 (* (/ t_m l) (/ t_m l))) (tan k))
          (fma (/ k t_m) (/ k t_m) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sin(k) * t_m;
	double tmp;
	if (t_m <= 4.9e-60) {
		tmp = 2.0 / (((((pow(sin(k), 2.0) * t_m) * k) * k) / (cos(k) * l)) / l);
	} else if (t_m <= 3.4e+158) {
		tmp = (((2.0 / (t_2 * t_m)) * l) / ((t_m * (pow((k / t_m), 2.0) + 2.0)) * tan(k))) * l;
	} else {
		tmp = 2.0 / (((t_2 * ((t_m / l) * (t_m / l))) * tan(k)) * fma((k / t_m), (k / t_m), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(sin(k) * t_m)
	tmp = 0.0
	if (t_m <= 4.9e-60)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64((sin(k) ^ 2.0) * t_m) * k) * k) / Float64(cos(k) * l)) / l));
	elseif (t_m <= 3.4e+158)
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(t_2 * t_m)) * l) / Float64(Float64(t_m * Float64((Float64(k / t_m) ^ 2.0) + 2.0)) * tan(k))) * l);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * Float64(Float64(t_m / l) * Float64(t_m / l))) * tan(k)) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.9e-60], N[(2.0 / N[(N[(N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.4e+158], N[(N[(N[(N[(2.0 / N[(t$95$2 * t$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(N[(t$95$m * N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$2 * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $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 3 regimes

2. if t < 4.89999999999999988e-60

   1. Initial program 54.2%

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

   4. Applied rewrites 48.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 77.6

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

   7. Applied rewrites 77.6%

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

   if 4.89999999999999988e-60 < t < 3.3999999999999999e158

   1. Initial program 71.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 84.9

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

   4. Applied rewrites 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 91.5%

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

   8. Applied rewrites 95.3%

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

   if 3.3999999999999999e158 < t

   1. Initial program 74.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 96.8

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

   4. Applied rewrites 96.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 96.8

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

   6. Applied rewrites 96.8%

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

Alternative 4: 87.6% accurate, 1.3× speedup

Math

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

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-58)
    (/ 2.0 (/ (/ (* (* (* (pow (sin k) 2.0) t_m) k) k) (* (cos k) l)) l))
    (/
     2.0
     (*
      (* (* (* (sin k) t_m) (* (/ t_m l) (/ t_m l))) (tan k))
      (fma (/ k t_m) (/ k t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1e-58) {
		tmp = 2.0 / (((((pow(sin(k), 2.0) * t_m) * k) * k) / (cos(k) * l)) / l);
	} else {
		tmp = 2.0 / ((((sin(k) * t_m) * ((t_m / l) * (t_m / l))) * tan(k)) * fma((k / t_m), (k / t_m), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1e-58)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64((sin(k) ^ 2.0) * t_m) * k) * k) / Float64(cos(k) * l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) * t_m) * Float64(Float64(t_m / l) * Float64(t_m / l))) * tan(k)) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1e-58], N[(2.0 / N[(N[(N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $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 < 1e-58

   1. Initial program 54.2%

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

   4. Applied rewrites 48.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 77.6

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

   7. Applied rewrites 77.6%

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

   if 1e-58 < t

   1. Initial program 72.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 89.9

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

   4. Applied rewrites 89.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 89.9

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

   6. Applied rewrites 89.9%

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

Alternative 5: 84.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\sin k \cdot t\_m\right) \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 3.8e-45)
    (/ 2.0 (/ (* (* (/ (sin k) l) (tan k)) (* (* k k) t_m)) l))
    (/
     2.0
     (*
      (* (* (* (sin k) t_m) (* (/ t_m l) (/ t_m l))) (tan k))
      (fma (/ k t_m) (/ k t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.8e-45) {
		tmp = 2.0 / ((((sin(k) / l) * tan(k)) * ((k * k) * t_m)) / l);
	} else {
		tmp = 2.0 / ((((sin(k) * t_m) * ((t_m / l) * (t_m / l))) * tan(k)) * fma((k / t_m), (k / t_m), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.8e-45)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) / l) * tan(k)) * Float64(Float64(k * k) * t_m)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) * t_m) * Float64(Float64(t_m / l) * Float64(t_m / l))) * tan(k)) * fma(Float64(k / t_m), Float64(k / t_m), 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.8e-45], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $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 < 3.79999999999999997e-45

   1. Initial program 54.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 69.6

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

   5. Applied rewrites 69.6%

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

   7. Applied rewrites 74.6%

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

   if 3.79999999999999997e-45 < t

   1. Initial program 72.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 90.3

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

   4. Applied rewrites 90.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 90.3

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

   6. Applied rewrites 90.3%

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

Alternative 6: 84.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\sin k \cdot t\_m\right) \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right)\right) \cdot \tan k\right) \cdot \mathsf{fma}\left(k, \frac{k}{t\_m \cdot t\_m}, 2\right)}\\ \end{array} \end{array} \]

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 3.8e-45)
    (/ 2.0 (/ (* (* (/ (sin k) l) (tan k)) (* (* k k) t_m)) l))
    (/
     2.0
     (*
      (* (* (* (sin k) t_m) (* (/ t_m l) (/ t_m l))) (tan k))
      (fma k (/ k (* t_m 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 <= 3.8e-45) {
		tmp = 2.0 / ((((sin(k) / l) * tan(k)) * ((k * k) * t_m)) / l);
	} else {
		tmp = 2.0 / ((((sin(k) * t_m) * ((t_m / l) * (t_m / l))) * tan(k)) * fma(k, (k / (t_m * 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 <= 3.8e-45)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) / l) * tan(k)) * Float64(Float64(k * k) * t_m)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) * t_m) * Float64(Float64(t_m / l) * Float64(t_m / l))) * tan(k)) * fma(k, Float64(k / Float64(t_m * 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, 3.8e-45], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(k * N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.79999999999999997e-45

   1. Initial program 54.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 69.6

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

   5. Applied rewrites 69.6%

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

   7. Applied rewrites 74.6%

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

   if 3.79999999999999997e-45 < t

   1. Initial program 72.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 90.3

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 46.0

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

   7. Applied rewrites 46.0%

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

   9. Applied rewrites 87.6%

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

Alternative 7: 77.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.7 \cdot 10^{-86}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(t\_m \cdot t\_m\right) \cdot 2\right)}{\ell} \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \left(t\_m \cdot \sin k\right)\right) \cdot \left(\frac{t\_m}{\ell} \cdot \left(\mathsf{fma}\left(0.6666666666666666 + {\left(t\_m \cdot t\_m\right)}^{-1}, k \cdot k, 2\right) \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.7e-86)
    (/
     2.0
     (*
      (/ t_m l)
      (*
       (/
        (fma
         (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
         k
         (* (* t_m t_m) 2.0))
        l)
       (* k k))))
    (/
     2.0
     (*
      (* (/ t_m l) (* t_m (sin k)))
      (*
       (/ t_m l)
       (*
        (fma (+ 0.6666666666666666 (pow (* t_m t_m) -1.0)) (* k k) 2.0)
        k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.7e-86) {
		tmp = 2.0 / ((t_m / l) * ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((t_m * t_m) * 2.0)) / l) * (k * k)));
	} else {
		tmp = 2.0 / (((t_m / l) * (t_m * sin(k))) * ((t_m / l) * (fma((0.6666666666666666 + pow((t_m * t_m), -1.0)), (k * k), 2.0) * k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.7e-86)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(t_m * t_m) * 2.0)) / l) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(t_m * sin(k))) * Float64(Float64(t_m / l) * Float64(fma(Float64(0.6666666666666666 + (Float64(t_m * t_m) ^ -1.0)), Float64(k * k), 2.0) * k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.7e-86], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(0.6666666666666666 + N[Power[N[(t$95$m * t$95$m), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision] + 2.0), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.7000000000000004e-86

   1. Initial program 54.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 69.6%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(t \cdot t\right) \cdot 2\right)}{\ell} \cdot \left(k \cdot k\right)\right)}} \]

   if 5.7000000000000004e-86 < t

   1. Initial program 71.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 87.2

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

   4. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 94.2%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 86.2

          \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\mathsf{fma}\left(0.6666666666666666 + \frac{1}{t \cdot t}, \color{blue}{k \cdot k}, 2\right) \cdot k\right)\right)} \]

   9. Applied rewrites 86.2%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\mathsf{fma}\left(0.6666666666666666 + \frac{1}{t \cdot t}, k \cdot k, 2\right) \cdot k\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.7 \cdot 10^{-86}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(t \cdot t\right) \cdot 2\right)}{\ell} \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \left(\mathsf{fma}\left(0.6666666666666666 + {\left(t \cdot t\right)}^{-1}, k \cdot k, 2\right) \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \sin k\right) \cdot t\_m\right) \cdot \left(\frac{k \cdot t\_m}{\ell} \cdot 2\right)}\\ \mathbf{elif}\;k \leq 0.48:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(t\_m \cdot t\_m\right) \cdot 2\right)}{\ell} \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;k \leq 2.05 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{\sin k}{\ell} \cdot \tan k\right) \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(t\_m \cdot k\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell \cdot \ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6.5e-161)
    (/ 2.0 (* (* (* (/ t_m l) (sin k)) t_m) (* (/ (* k t_m) l) 2.0)))
    (if (<= k 0.48)
      (/
       2.0
       (*
        (/ t_m l)
        (*
         (/
          (fma
           (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
           k
           (* (* t_m t_m) 2.0))
          l)
         (* k k))))
      (if (<= k 2.05e+148)
        (/ 2.0 (/ (* (* (/ (sin k) l) (tan k)) (* (* k k) t_m)) l))
        (/ 2.0 (* k (* (* t_m k) (* (tan k) (/ (sin k) (* l l)))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.5e-161) {
		tmp = 2.0 / ((((t_m / l) * sin(k)) * t_m) * (((k * t_m) / l) * 2.0));
	} else if (k <= 0.48) {
		tmp = 2.0 / ((t_m / l) * ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((t_m * t_m) * 2.0)) / l) * (k * k)));
	} else if (k <= 2.05e+148) {
		tmp = 2.0 / ((((sin(k) / l) * tan(k)) * ((k * k) * t_m)) / l);
	} else {
		tmp = 2.0 / (k * ((t_m * k) * (tan(k) * (sin(k) / (l * l)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6.5e-161)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * sin(k)) * t_m) * Float64(Float64(Float64(k * t_m) / l) * 2.0)));
	elseif (k <= 0.48)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(t_m * t_m) * 2.0)) / l) * Float64(k * k))));
	elseif (k <= 2.05e+148)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) / l) * tan(k)) * Float64(Float64(k * k) * t_m)) / l));
	else
		tmp = Float64(2.0 / Float64(k * Float64(Float64(t_m * k) * Float64(tan(k) * Float64(sin(k) / Float64(l * l))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6.5e-161], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.48], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.05e+148], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(N[(t$95$m * k), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if k < 6.50000000000000008e-161

   1. Initial program 59.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 75.1

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

   4. Applied rewrites 75.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 83.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 84.5

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

   8. Applied rewrites 84.5%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 72.4

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

   11. Applied rewrites 72.4%

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

   if 6.50000000000000008e-161 < k < 0.47999999999999998

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 85.2%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 98.0%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(t \cdot t\right) \cdot 2\right)}{\ell} \cdot \left(k \cdot k\right)\right)}} \]

   if 0.47999999999999998 < k < 2.0499999999999999e148

   1. Initial program 50.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 75.7

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

   5. Applied rewrites 75.7%

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

   7. Applied rewrites 84.0%

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

   if 2.0499999999999999e148 < k

   1. Initial program 61.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 71.6

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

   5. Applied rewrites 71.6%

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

   7. Applied rewrites 84.3%

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

Alternative 9: 76.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \sin k\right) \cdot t\_m\right) \cdot \left(\frac{k \cdot t\_m}{\ell} \cdot 2\right)}\\ \mathbf{elif}\;k \leq 900000:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(t\_m \cdot t\_m\right) \cdot 2\right)}{\ell} \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(t\_m \cdot k\right) \cdot \left(\tan k \cdot \frac{\sin k}{\ell \cdot \ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6.5e-161)
    (/ 2.0 (* (* (* (/ t_m l) (sin k)) t_m) (* (/ (* k t_m) l) 2.0)))
    (if (<= k 900000.0)
      (/
       2.0
       (*
        (/ t_m l)
        (*
         (/
          (fma
           (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
           k
           (* (* t_m t_m) 2.0))
          l)
         (* k k))))
      (/ 2.0 (* k (* (* t_m k) (* (tan k) (/ (sin k) (* l l))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.5e-161) {
		tmp = 2.0 / ((((t_m / l) * sin(k)) * t_m) * (((k * t_m) / l) * 2.0));
	} else if (k <= 900000.0) {
		tmp = 2.0 / ((t_m / l) * ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((t_m * t_m) * 2.0)) / l) * (k * k)));
	} else {
		tmp = 2.0 / (k * ((t_m * k) * (tan(k) * (sin(k) / (l * l)))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6.5e-161)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * sin(k)) * t_m) * Float64(Float64(Float64(k * t_m) / l) * 2.0)));
	elseif (k <= 900000.0)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(t_m * t_m) * 2.0)) / l) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(k * Float64(Float64(t_m * k) * Float64(tan(k) * Float64(sin(k) / Float64(l * l))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6.5e-161], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 900000.0], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(N[(t$95$m * k), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 6.50000000000000008e-161

   1. Initial program 59.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 75.1

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

   4. Applied rewrites 75.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 83.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 84.5

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

   8. Applied rewrites 84.5%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 72.4

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

   11. Applied rewrites 72.4%

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

   if 6.50000000000000008e-161 < k < 9e5

   1. Initial program 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 85.7%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 95.1%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(t \cdot t\right) \cdot 2\right)}{\ell} \cdot \left(k \cdot k\right)\right)}} \]

   if 9e5 < k

   1. Initial program 56.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 74.9

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

   5. Applied rewrites 74.9%

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

   7. Applied rewrites 80.9%

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

Alternative 10: 75.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \sin k\right) \cdot t\_m\right) \cdot \left(\frac{k \cdot t\_m}{\ell} \cdot 2\right)}\\ \mathbf{elif}\;k \leq 900000:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(t\_m \cdot t\_m\right) \cdot 2\right)}{\ell} \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\left(\tan k \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot t\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6.5e-161)
    (/ 2.0 (* (* (* (/ t_m l) (sin k)) t_m) (* (/ (* k t_m) l) 2.0)))
    (if (<= k 900000.0)
      (/
       2.0
       (*
        (/ t_m l)
        (*
         (/
          (fma
           (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
           k
           (* (* t_m t_m) 2.0))
          l)
         (* k k))))
      (/ 2.0 (* k (* k (* (* (tan k) (/ (sin k) (* l 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 (k <= 6.5e-161) {
		tmp = 2.0 / ((((t_m / l) * sin(k)) * t_m) * (((k * t_m) / l) * 2.0));
	} else if (k <= 900000.0) {
		tmp = 2.0 / ((t_m / l) * ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((t_m * t_m) * 2.0)) / l) * (k * k)));
	} else {
		tmp = 2.0 / (k * (k * ((tan(k) * (sin(k) / (l * 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 (k <= 6.5e-161)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * sin(k)) * t_m) * Float64(Float64(Float64(k * t_m) / l) * 2.0)));
	elseif (k <= 900000.0)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(t_m * t_m) * 2.0)) / l) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(tan(k) * Float64(sin(k) / Float64(l * l))) * t_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6.5e-161], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 900000.0], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(k * N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 6.50000000000000008e-161

   1. Initial program 59.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 75.1

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

   4. Applied rewrites 75.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 83.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 84.5

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

   8. Applied rewrites 84.5%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 72.4

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

   11. Applied rewrites 72.4%

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

   if 6.50000000000000008e-161 < k < 9e5

   1. Initial program 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 85.7%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 95.1%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(t \cdot t\right) \cdot 2\right)}{\ell} \cdot \left(k \cdot k\right)\right)}} \]

   if 9e5 < k

   1. Initial program 56.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 74.9

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

   5. Applied rewrites 74.9%

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

   7. Applied rewrites 79.6%

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

Alternative 11: 76.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(t\_m \cdot t\_m\right) \cdot 2\right)}{\ell} \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t\_m}{\ell} \cdot \sin k\right) \cdot t\_m\right) \cdot \left(\frac{k \cdot t\_m}{\ell} \cdot 2\right)}\\ \end{array} \end{array} \]

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 3.6e-13)
    (/
     2.0
     (*
      (/ t_m l)
      (*
       (/
        (fma
         (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
         k
         (* (* t_m t_m) 2.0))
        l)
       (* k k))))
    (/ 2.0 (* (* (* (/ t_m l) (sin k)) t_m) (* (/ (* k t_m) l) 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 <= 3.6e-13) {
		tmp = 2.0 / ((t_m / l) * ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((t_m * t_m) * 2.0)) / l) * (k * k)));
	} else {
		tmp = 2.0 / ((((t_m / l) * sin(k)) * t_m) * (((k * t_m) / l) * 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 <= 3.6e-13)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(t_m * t_m) * 2.0)) / l) * Float64(k * k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * sin(k)) * t_m) * Float64(Float64(Float64(k * t_m) / l) * 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, 3.6e-13], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.5999999999999998e-13

   1. Initial program 56.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 68.7%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 70.8%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(t \cdot t\right) \cdot 2\right)}{\ell} \cdot \left(k \cdot k\right)\right)}} \]

   if 3.5999999999999998e-13 < t

   1. Initial program 68.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. cube-mult N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 89.2

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

   4. Applied rewrites 89.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 95.1

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

   8. Applied rewrites 95.1%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 84.9

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

   11. Applied rewrites 84.9%

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

Alternative 12: 73.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2200000000000:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(t\_m \cdot t\_m\right) \cdot 2\right)}{\ell} \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\ell}{t\_m}\right)}^{2}}{k \cdot t\_m}}{k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2200000000000.0)
    (/
     2.0
     (*
      (/ t_m l)
      (*
       (/
        (fma
         (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
         k
         (* (* t_m t_m) 2.0))
        l)
       (* k k))))
    (/ (/ (pow (/ l t_m) 2.0) (* k t_m)) k))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2200000000000.0) {
		tmp = 2.0 / ((t_m / l) * ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((t_m * t_m) * 2.0)) / l) * (k * k)));
	} else {
		tmp = (pow((l / t_m), 2.0) / (k * t_m)) / k;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2200000000000.0)
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(t_m * t_m) * 2.0)) / l) * Float64(k * k))));
	else
		tmp = Float64(Float64((Float64(l / t_m) ^ 2.0) / Float64(k * t_m)) / k);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2200000000000.0], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(l / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.2e12

   1. Initial program 56.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 68.4%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 70.4%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(t \cdot t\right) \cdot 2\right)}{\ell} \cdot \left(k \cdot k\right)\right)}} \]

   if 2.2e12 < t

   1. Initial program 70.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 54.5

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

   5. Applied rewrites 54.5%

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

   7. Applied rewrites 54.5%

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

   9. Applied rewrites 61.9%

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

   11. Applied rewrites 79.6%

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

Alternative 13: 68.1% accurate, 5.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{k \cdot t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t\_m \cdot t\_m, 1\right) \cdot k, k, \left(t\_m \cdot t\_m\right) \cdot 2\right)}{\ell} \cdot \left(k \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6.5e-161)
    (* (/ (/ l (* t_m t_m)) k) (/ l (* k t_m)))
    (/
     2.0
     (*
      (/ t_m l)
      (*
       (/
        (fma
         (* (fma 0.3333333333333333 (* t_m t_m) 1.0) k)
         k
         (* (* t_m t_m) 2.0))
        l)
       (* k k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.5e-161) {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	} else {
		tmp = 2.0 / ((t_m / l) * ((fma((fma(0.3333333333333333, (t_m * t_m), 1.0) * k), k, ((t_m * t_m) * 2.0)) / l) * (k * k)));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6.5e-161)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(k * t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(fma(Float64(fma(0.3333333333333333, Float64(t_m * t_m), 1.0) * k), k, Float64(Float64(t_m * t_m) * 2.0)) / l) * Float64(k * k))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6.5e-161], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(N[(N[(N[(0.3333333333333333 * N[(t$95$m * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * k), $MachinePrecision] * k + N[(N[(t$95$m * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.50000000000000008e-161

   1. Initial program 59.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 56.7

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

   5. Applied rewrites 56.7%

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

   7. Applied rewrites 56.7%

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

   9. Applied rewrites 58.3%

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

   11. Applied rewrites 68.3%

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

   if 6.50000000000000008e-161 < k

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. cube-mult N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 72.8%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 74.2%

      \[\leadsto \frac{2}{\frac{t}{\ell} \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, t \cdot t, 1\right) \cdot k, k, \left(t \cdot t\right) \cdot 2\right)}{\ell} \cdot \left(k \cdot k\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 65.5% accurate, 7.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{t\_m}\\ \mathbf{elif}\;t\_m \leq 5.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot t\_m} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m} \cdot \ell}{\left(k \cdot t\_m\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9.6e-99)
    (* (/ 2.0 (* (* k k) (* k k))) (/ (* l l) t_m))
    (if (<= t_m 5.5e+76)
      (* (/ l (* (* t_m t_m) t_m)) (/ (/ l k) k))
      (/ (* (/ l (* t_m t_m)) l) (* (* k t_m) k))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9.6e-99) {
		tmp = (2.0 / ((k * k) * (k * k))) * ((l * l) / t_m);
	} else if (t_m <= 5.5e+76) {
		tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k);
	} else {
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k);
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 9.6d-99) then
        tmp = (2.0d0 / ((k * k) * (k * k))) * ((l * l) / t_m)
    else if (t_m <= 5.5d+76) then
        tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k)
    else
        tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 9.6e-99) {
		tmp = (2.0 / ((k * k) * (k * k))) * ((l * l) / t_m);
	} else if (t_m <= 5.5e+76) {
		tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k);
	} else {
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 9.6e-99:
		tmp = (2.0 / ((k * k) * (k * k))) * ((l * l) / t_m)
	elif t_m <= 5.5e+76:
		tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k)
	else:
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 9.6e-99)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * Float64(k * k))) * Float64(Float64(l * l) / t_m));
	elseif (t_m <= 5.5e+76)
		tmp = Float64(Float64(l / Float64(Float64(t_m * t_m) * t_m)) * Float64(Float64(l / k) / k));
	else
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) * l) / Float64(Float64(k * t_m) * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 9.6e-99)
		tmp = (2.0 / ((k * k) * (k * k))) * ((l * l) / t_m);
	elseif (t_m <= 5.5e+76)
		tmp = (l / ((t_m * t_m) * t_m)) * ((l / k) / k);
	else
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 9.6e-99], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.5e+76], N[(N[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 9.6000000000000002e-99

   1. Initial program 54.6%

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{t}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{4}}} \cdot \frac{{\ell}^{2}}{t} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{4}}} \cdot \frac{{\ell}^{2}}{t} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

      8. lower-*.f64 57.3

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

   7. Applied rewrites 57.3%

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

   9. Applied rewrites 57.3%

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

   if 9.6000000000000002e-99 < t < 5.5000000000000001e76

   1. Initial program 70.9%

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

   3. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 63.7

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

   5. Applied rewrites 63.7%

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

   7. Applied rewrites 63.7%

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

   9. Applied rewrites 70.5%

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

   if 5.5000000000000001e76 < 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. unpow2 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 57.2

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

   5. Applied rewrites 57.2%

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

   7. Applied rewrites 57.2%

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

   9. Applied rewrites 66.3%

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

   11. Applied rewrites 79.6%

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

Alternative 15: 69.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{k \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.3e-80)
    (/ 2.0 (* (* (* k k) t_m) (* (/ k l) (/ k l))))
    (* (/ (/ l (* t_m t_m)) k) (/ l (* k t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e-80) {
		tmp = 2.0 / (((k * k) * t_m) * ((k / l) * (k / l)));
	} else {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	}
	return t_s * tmp;
}

Fortran

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

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.3d-80) then
        tmp = 2.0d0 / (((k * k) * t_m) * ((k / l) * (k / l)))
    else
        tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e-80) {
		tmp = 2.0 / (((k * k) * t_m) * ((k / l) * (k / l)));
	} else {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.3e-80:
		tmp = 2.0 / (((k * k) * t_m) * ((k / l) * (k / l)))
	else:
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.3e-80)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) * t_m) * Float64(Float64(k / l) * Float64(k / l))));
	else
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(k * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.3e-80)
		tmp = 2.0 / (((k * k) * t_m) * ((k / l) * (k / l)));
	else
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.3e-80], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.2999999999999998e-80

   1. Initial program 54.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot \frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{\color{blue}{{\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\color{blue}{\sin k}}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\cos k \cdot {\ell}^{2}}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\color{blue}{\left(\cos k \cdot \ell\right)} \cdot \ell}} \]

      15. lower-cos.f64 69.9

          \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}} \]

   5. Applied rewrites 69.9%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{\sin k}^{2}}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{{k}^{2}}{\color{blue}{{\ell}^{2}}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.4%

      \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot \color{blue}{\frac{k}{\ell}}\right)} \]

   if 2.2999999999999998e-80 < t

   1. Initial program 71.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 60.0

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 60.0%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 64.7%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 79.7%

       \[\leadsto \frac{\frac{\ell}{t \cdot t}}{k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 67.6% accurate, 7.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{t\_m} \cdot 2}{k \cdot k}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{k \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.2e-80)
    (/ (/ (* (/ (* l l) t_m) 2.0) (* k k)) (* k k))
    (* (/ (/ l (* t_m t_m)) k) (/ l (* k t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.2e-80) {
		tmp = ((((l * l) / t_m) * 2.0) / (k * k)) / (k * k);
	} else {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.2d-80) then
        tmp = ((((l * l) / t_m) * 2.0d0) / (k * k)) / (k * k)
    else
        tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.2e-80) {
		tmp = ((((l * l) / t_m) * 2.0) / (k * k)) / (k * k);
	} else {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.2e-80:
		tmp = ((((l * l) / t_m) * 2.0) / (k * k)) / (k * k)
	else:
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.2e-80)
		tmp = Float64(Float64(Float64(Float64(Float64(l * l) / t_m) * 2.0) / Float64(k * k)) / Float64(k * k));
	else
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(k * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.2e-80)
		tmp = ((((l * l) / t_m) * 2.0) / (k * k)) / (k * k);
	else
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.2e-80], N[(N[(N[(N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.2000000000000001e-80

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 48.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{{t}^{3}}{\ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot {\left(\frac{k}{t}\right)}^{2}}{\ell}}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{t}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{4}}} \cdot \frac{{\ell}^{2}}{t} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{4}}} \cdot \frac{{\ell}^{2}}{t} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

      8. lower-*.f64 56.9

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

   7. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t}} \]
   8. Step-by-step derivation

   9. Applied rewrites 61.0%

      \[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{t} \cdot 2}{k \cdot k}}{\color{blue}{k \cdot k}} \]

   if 2.2000000000000001e-80 < t

   1. Initial program 71.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 60.0

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 60.0%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 64.7%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 79.7%

       \[\leadsto \frac{\frac{\ell}{t \cdot t}}{k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 67.1% accurate, 8.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m}}{k} \cdot \frac{\ell}{k \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.2e-88)
    (* (/ 2.0 (* (* k k) (* k k))) (/ (* l l) t_m))
    (* (/ (/ l (* t_m t_m)) k) (/ l (* k t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.2e-88) {
		tmp = (2.0 / ((k * k) * (k * k))) * ((l * l) / t_m);
	} else {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 5.2d-88) then
        tmp = (2.0d0 / ((k * k) * (k * k))) * ((l * l) / t_m)
    else
        tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.2e-88) {
		tmp = (2.0 / ((k * k) * (k * k))) * ((l * l) / t_m);
	} else {
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 5.2e-88:
		tmp = (2.0 / ((k * k) * (k * k))) * ((l * l) / t_m)
	else:
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.2e-88)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * Float64(k * k))) * Float64(Float64(l * l) / t_m));
	else
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) / k) * Float64(l / Float64(k * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 5.2e-88)
		tmp = (2.0 / ((k * k) * (k * k))) * ((l * l) / t_m);
	else
		tmp = ((l / (t_m * t_m)) / k) * (l / (k * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.2e-88], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.20000000000000027e-88

   1. Initial program 54.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 48.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{{t}^{3}}{\ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot {\left(\frac{k}{t}\right)}^{2}}{\ell}}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{t}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{4}}} \cdot \frac{{\ell}^{2}}{t} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{4}}} \cdot \frac{{\ell}^{2}}{t} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

      8. lower-*.f64 57.0

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

   7. Applied rewrites 57.0%

      \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t}} \]
   8. Step-by-step derivation

   9. Applied rewrites 57.0%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \color{blue}{\ell}}{t} \]

   if 5.20000000000000027e-88 < t

   1. Initial program 70.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 61.0

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 61.0%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 61.0%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 65.6%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 80.2%

       \[\leadsto \frac{\frac{\ell}{t \cdot t}}{k} \cdot \color{blue}{\frac{\ell}{k \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 62.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{-137}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m \cdot t\_m} \cdot \ell}{\left(k \cdot t\_m\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \ell}{t\_m \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1e-137)
    (/ (* (/ l (* t_m t_m)) l) (* (* k t_m) k))
    (/ (* (/ l t_m) l) (* t_m (* (* k k) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1e-137) {
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k);
	} else {
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m));
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1d-137) then
        tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k)
    else
        tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1e-137) {
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k);
	} else {
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1e-137:
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k)
	else:
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1e-137)
		tmp = Float64(Float64(Float64(l / Float64(t_m * t_m)) * l) / Float64(Float64(k * t_m) * k));
	else
		tmp = Float64(Float64(Float64(l / t_m) * l) / Float64(t_m * Float64(Float64(k * k) * t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1e-137)
		tmp = ((l / (t_m * t_m)) * l) / ((k * t_m) * k);
	else
		tmp = ((l / t_m) * l) / (t_m * ((k * k) * t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1e-137], N[(N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k * t$95$m), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(t$95$m * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.99999999999999978e-138

   1. Initial program 60.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 57.5

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 57.5%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 59.1%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 64.3%

       \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\left(k \cdot t\right) \cdot \color{blue}{k}} \]

   if 9.99999999999999978e-138 < k

   1. Initial program 57.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 59.1

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 59.1%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 63.1%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 66.3%

       \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{t \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 61.0% accurate, 9.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(k \cdot k\right) \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-46}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 2}{t\_2 \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{t\_m \cdot t\_m}}{t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (* k k) t_m)))
   (*
    t_s
    (if (<= t_m 4e-46)
      (/ (* (* l l) 2.0) (* t_2 (* k k)))
      (* l (/ (/ l (* t_m t_m)) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = (k * k) * t_m;
	double tmp;
	if (t_m <= 4e-46) {
		tmp = ((l * l) * 2.0) / (t_2 * (k * k));
	} else {
		tmp = l * ((l / (t_m * t_m)) / t_2);
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k * k) * t_m
    if (t_m <= 4d-46) then
        tmp = ((l * l) * 2.0d0) / (t_2 * (k * k))
    else
        tmp = l * ((l / (t_m * t_m)) / t_2)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = (k * k) * t_m;
	double tmp;
	if (t_m <= 4e-46) {
		tmp = ((l * l) * 2.0) / (t_2 * (k * k));
	} else {
		tmp = l * ((l / (t_m * t_m)) / t_2);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = (k * k) * t_m
	tmp = 0
	if t_m <= 4e-46:
		tmp = ((l * l) * 2.0) / (t_2 * (k * k))
	else:
		tmp = l * ((l / (t_m * t_m)) / t_2)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(k * k) * t_m)
	tmp = 0.0
	if (t_m <= 4e-46)
		tmp = Float64(Float64(Float64(l * l) * 2.0) / Float64(t_2 * Float64(k * k)));
	else
		tmp = Float64(l * Float64(Float64(l / Float64(t_m * t_m)) / t_2));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (k * k) * t_m;
	tmp = 0.0;
	if (t_m <= 4e-46)
		tmp = ((l * l) * 2.0) / (t_2 * (k * k));
	else
		tmp = l * ((l / (t_m * t_m)) / t_2);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4e-46], N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t$95$2 * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(l / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.00000000000000009e-46

   1. Initial program 54.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   4. Applied rewrites 49.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{{t}^{3}}{\ell} \cdot \left(\tan k \cdot \sin k\right)\right) \cdot {\left(\frac{k}{t}\right)}^{2}}{\ell}}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{t}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{k}^{4}}} \cdot \frac{{\ell}^{2}}{t} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{{k}^{4}}} \cdot \frac{{\ell}^{2}}{t} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \color{blue}{\frac{{\ell}^{2}}{t}} \]

      7. unpow2 N/A

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

      8. lower-*.f64 57.3

         \[\leadsto \frac{2}{{k}^{4}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

   7. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\frac{2}{{k}^{4}} \cdot \frac{\ell \cdot \ell}{t}} \]
   8. Step-by-step derivation

   9. Applied rewrites 57.6%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\color{blue}{{k}^{4} \cdot t}} \]
   10. Step-by-step derivation

   11. Applied rewrites 59.3%

       \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

   if 4.00000000000000009e-46 < t

   1. Initial program 72.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

      9. lower-*.f64 59.4

         \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
   6. Step-by-step derivation

   7. Applied rewrites 59.4%

      \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
   8. Step-by-step derivation

   9. Applied rewrites 65.3%

      \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 65.4%

       \[\leadsto \ell \cdot \color{blue}{\frac{\frac{\ell}{t \cdot t}}{\left(k \cdot k\right) \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 62.0% accurate, 10.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\frac{\ell}{t\_m} \cdot \ell}{t\_m \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (* (/ l t_m) l) (* t_m (* (* k k) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (((l / t_m) * l) / (t_m * ((k * k) * t_m)));
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (((l / t_m) * l) / (t_m * ((k * k) * t_m)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (((l / t_m) * l) / (t_m * ((k * k) * t_m)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (((l / t_m) * l) / (t_m * ((k * k) * t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(Float64(l / t_m) * l) / Float64(t_m * Float64(Float64(k * k) * t_m))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (((l / t_m) * l) / (t_m * ((k * k) * t_m)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(N[(l / t$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(t$95$m * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   9. lower-*.f64 58.1

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

5. Applied rewrites 58.1%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
6. Step-by-step derivation

7. Applied rewrites 58.1%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
8. Step-by-step derivation

9. Applied rewrites 60.6%

   \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
10. Step-by-step derivation

11. Applied rewrites 64.1%

    \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{t \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
12. Add Preprocessing

Alternative 21: 54.1% accurate, 12.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\ell \cdot \ell}{\left(t\_m \cdot t\_m\right) \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (* l l) (* (* t_m t_m) (* (* k k) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) / ((t_m * t_m) * ((k * k) * t_m)));
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l * l) / ((t_m * t_m) * ((k * k) * t_m)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((l * l) / ((t_m * t_m) * ((k * k) * t_m)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((l * l) / ((t_m * t_m) * ((k * k) * t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(Float64(l * l) / Float64(Float64(t_m * t_m) * Float64(Float64(k * k) * t_m))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l * l) / ((t_m * t_m) * ((k * k) * t_m)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]

   3. times-frac N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\ell}{\color{blue}{{t}^{3}}} \cdot \frac{\ell}{{k}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \color{blue}{\frac{\ell}{{k}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   9. lower-*.f64 58.1

      \[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

5. Applied rewrites 58.1%

   \[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}} \]
6. Step-by-step derivation

7. Applied rewrites 58.1%

   \[\leadsto \frac{\ell}{\left(t \cdot t\right) \cdot t} \cdot \frac{\ell}{k \cdot k} \]
8. Step-by-step derivation

9. Applied rewrites 60.6%

   \[\leadsto \frac{\frac{\ell}{t \cdot t} \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]
10. Step-by-step derivation

11. Applied rewrites 57.3%

    \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot t\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024347 
(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))))