Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.2% → 90.4%

Time: 8.0s

Alternatives: 20

Speedup: 7.8×

• 27.0% 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: 55.2% 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: 90.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.95 \cdot 10^{+70}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\left(\sin k\_m \cdot t\right)}^{2}, 2, {\left(\sin k\_m \cdot k\_m\right)}^{2}\right)}{\cos k\_m \cdot \ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\cos k\_m}{{\sin k\_m}^{2} \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.95e+70)
   (/
    2.0
    (*
     (/
      (fma (pow (* (sin k_m) t) 2.0) 2.0 (pow (* (sin k_m) k_m) 2.0))
      (* (cos k_m) l))
     (/ t l)))
   (*
    (* (* (/ l k_m) (/ l k_m)) (/ (cos k_m) (* (pow (sin k_m) 2.0) t)))
    2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.95e+70) {
		tmp = 2.0 / ((fma(pow((sin(k_m) * t), 2.0), 2.0, pow((sin(k_m) * k_m), 2.0)) / (cos(k_m) * l)) * (t / l));
	} else {
		tmp = (((l / k_m) * (l / k_m)) * (cos(k_m) / (pow(sin(k_m), 2.0) * t))) * 2.0;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.95e+70)
		tmp = Float64(2.0 / Float64(Float64(fma((Float64(sin(k_m) * t) ^ 2.0), 2.0, (Float64(sin(k_m) * k_m) ^ 2.0)) / Float64(cos(k_m) * l)) * Float64(t / l)));
	else
		tmp = Float64(Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * Float64(cos(k_m) / Float64((sin(k_m) ^ 2.0) * t))) * 2.0);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.95e+70], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.94999999999999987e70

   1. Initial program 60.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.5%

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

   7. Applied rewrites 78.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 86.5

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

   9. Applied rewrites 86.5%

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

   11. Applied rewrites 91.5%

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

   if 1.94999999999999987e70 < k

   1. Initial program 46.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 88.6

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

   7. Applied rewrites 88.6%

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

Alternative 2: 83.6% accurate, 0.5× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k_m)) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))
      1e+105)
   (/
    2.0
    (*
     (fma (pow (* (sin k_m) t) 2.0) 2.0 (pow (* (sin k_m) k_m) 2.0))
     (/ t (* (* (cos k_m) l) l))))
   (*
    (/ (* (* (/ l k_m) (/ l k_m)) (cos k_m)) (* (pow (sin k_m) 2.0) t))
    2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) + 1.0))) <= 1e+105) {
		tmp = 2.0 / (fma(pow((sin(k_m) * t), 2.0), 2.0, pow((sin(k_m) * k_m), 2.0)) * (t / ((cos(k_m) * l) * l)));
	} else {
		tmp = ((((l / k_m) * (l / k_m)) * cos(k_m)) / (pow(sin(k_m), 2.0) * t)) * 2.0;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + 1.0))) <= 1e+105)
		tmp = Float64(2.0 / Float64(fma((Float64(sin(k_m) * t) ^ 2.0), 2.0, (Float64(sin(k_m) * k_m) ^ 2.0)) * Float64(t / Float64(Float64(cos(k_m) * l) * l))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * cos(k_m)) / Float64((sin(k_m) ^ 2.0) * t)) * 2.0);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+105], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t / N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $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)))) < 9.9999999999999994e104

   1. Initial program 81.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.6%

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

   7. Applied rewrites 89.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 90.3

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

   9. Applied rewrites 90.3%

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

   11. Applied rewrites 89.7%

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

   if 9.9999999999999994e104 < (/.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 22.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 75.9%

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

Alternative 3: 64.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, 1 - \left(-t\right) \cdot t\right), k\_m \cdot k\_m, \left(t \cdot t\right) \cdot 2\right)\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)} \leq 10^{+105}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\left(t\_1 \cdot k\_m\right) \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left(t\_1 \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1
         (fma
          (fma (* -0.6666666666666666 t) t (- 1.0 (* (- t) t)))
          (* k_m k_m)
          (* (* t t) 2.0))))
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k_m)) (tan k_m))
          (+ (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))
        1e+105)
     (/ 2.0 (* (/ t (* l l)) (* (* t_1 k_m) k_m)))
     (/ 2.0 (* (/ (/ t l) l) (* t_1 (* k_m k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = fma(fma((-0.6666666666666666 * t), t, (1.0 - (-t * t))), (k_m * k_m), ((t * t) * 2.0));
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) + 1.0))) <= 1e+105) {
		tmp = 2.0 / ((t / (l * l)) * ((t_1 * k_m) * k_m));
	} else {
		tmp = 2.0 / (((t / l) / l) * (t_1 * (k_m * k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = fma(fma(Float64(-0.6666666666666666 * t), t, Float64(1.0 - Float64(Float64(-t) * t))), Float64(k_m * k_m), Float64(Float64(t * t) * 2.0))
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + 1.0))) <= 1e+105)
		tmp = Float64(2.0 / Float64(Float64(t / Float64(l * l)) * Float64(Float64(t_1 * k_m) * k_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64(t_1 * Float64(k_m * k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(N[(-0.6666666666666666 * t), $MachinePrecision] * t + N[(1.0 - N[((-t) * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+105], N[(2.0 / N[(N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$1 * N[(k$95$m * k$95$m), $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)))) < 9.9999999999999994e104

   1. Initial program 81.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.6%

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

   7. Applied rewrites 89.7%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 47.8%

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

   12. Applied rewrites 75.3%

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

   if 9.9999999999999994e104 < (/.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 22.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 55.8%

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

   7. Applied rewrites 55.7%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 43.4%

       \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right) - \left(-t \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 49.1

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

   12. Applied rewrites 52.0%

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

Alternative 4: 59.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{t}{\ell \cdot \ell}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)} \leq 4 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left(\left(\mathsf{fma}\left(k\_m \cdot k\_m, 0.3333333333333333, 2\right) \cdot \left(t \cdot t\right)\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left(\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ t (* l l))))
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k_m)) (tan k_m))
          (+ (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))
        4e+89)
     (/
      2.0
      (*
       t_1
       (* (* (fma (* k_m k_m) 0.3333333333333333 2.0) (* t t)) (* k_m k_m))))
     (/ 2.0 (* t_1 (* (* k_m k_m) (* k_m k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = t / (l * l);
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) + 1.0))) <= 4e+89) {
		tmp = 2.0 / (t_1 * ((fma((k_m * k_m), 0.3333333333333333, 2.0) * (t * t)) * (k_m * k_m)));
	} else {
		tmp = 2.0 / (t_1 * ((k_m * k_m) * (k_m * k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(t / Float64(l * l))
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + 1.0))) <= 4e+89)
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(fma(Float64(k_m * k_m), 0.3333333333333333, 2.0) * Float64(t * t)) * Float64(k_m * k_m))));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(k_m * k_m) * Float64(k_m * k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+89], N[(2.0 / N[(t$95$1 * N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $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)))) < 3.99999999999999998e89

   1. Initial program 81.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.6%

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

   7. Applied rewrites 89.7%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 47.7%

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

   11. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 71.5

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

   13. Applied rewrites 71.5%

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

   if 3.99999999999999998e89 < (/.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 22.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 55.8%

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

   7. Applied rewrites 55.8%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 43.4%

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

   11. Taylor expanded in t around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 43.9

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

   13. Applied rewrites 43.9%

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

Alternative 5: 58.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)} \leq 4 \cdot 10^{+89}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(k\_m \cdot k\_m\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k_m)) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))
      4e+89)
   (/ (* l l) (* (* k_m k_m) (* (* t t) t)))
   (/ 2.0 (* (/ t (* l l)) (* (* k_m k_m) (* k_m k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) + 1.0))) <= 4e+89) {
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	} else {
		tmp = 2.0 / ((t / (l * l)) * ((k_m * k_m) * (k_m * k_m)));
	}
	return tmp;
}

Fortran

k_m =     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, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0d0 + ((k_m / t) ** 2.0d0)) + 1.0d0))) <= 4d+89) then
        tmp = (l * l) / ((k_m * k_m) * ((t * t) * t))
    else
        tmp = 2.0d0 / ((t / (l * l)) * ((k_m * k_m) * (k_m * k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k_m)) * Math.tan(k_m)) * ((1.0 + Math.pow((k_m / t), 2.0)) + 1.0))) <= 4e+89) {
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	} else {
		tmp = 2.0 / ((t / (l * l)) * ((k_m * k_m) * (k_m * k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k_m)) * math.tan(k_m)) * ((1.0 + math.pow((k_m / t), 2.0)) + 1.0))) <= 4e+89:
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t))
	else:
		tmp = 2.0 / ((t / (l * l)) * ((k_m * k_m) * (k_m * k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + 1.0))) <= 4e+89)
		tmp = Float64(Float64(l * l) / Float64(Float64(k_m * k_m) * Float64(Float64(t * t) * t)));
	else
		tmp = Float64(2.0 / Float64(Float64(t / Float64(l * l)) * Float64(Float64(k_m * k_m) * Float64(k_m * k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + ((k_m / t) ^ 2.0)) + 1.0))) <= 4e+89)
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	else
		tmp = 2.0 / ((t / (l * l)) * ((k_m * k_m) * (k_m * k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+89], N[(N[(l * l), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $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)))) < 3.99999999999999998e89

   1. Initial program 81.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 70.1

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

   5. Applied rewrites 70.1%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 70.1

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

   7. Applied rewrites 70.1%

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

   if 3.99999999999999998e89 < (/.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 22.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 55.8%

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

   7. Applied rewrites 55.8%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 43.4%

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

   11. Taylor expanded in t around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 43.9

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

   13. Applied rewrites 43.9%

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

Alternative 6: 76.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.02 \cdot 10^{+21}:\\ \;\;\;\;\left(\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\cos k\_m}{{\sin k\_m}^{2} \cdot t}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left({\left(k\_m \cdot t\right)}^{2} \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.02e+21)
   (* (* (* (/ l k_m) (/ l k_m)) (/ (cos k_m) (* (pow (sin k_m) 2.0) t))) 2.0)
   (/ 2.0 (* (/ (/ t l) l) (* (pow (* k_m t) 2.0) 2.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.02e+21) {
		tmp = (((l / k_m) * (l / k_m)) * (cos(k_m) / (pow(sin(k_m), 2.0) * t))) * 2.0;
	} else {
		tmp = 2.0 / (((t / l) / l) * (pow((k_m * t), 2.0) * 2.0));
	}
	return tmp;
}

Fortran

k_m =     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, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 1.02d+21) then
        tmp = (((l / k_m) * (l / k_m)) * (cos(k_m) / ((sin(k_m) ** 2.0d0) * t))) * 2.0d0
    else
        tmp = 2.0d0 / (((t / l) / l) * (((k_m * t) ** 2.0d0) * 2.0d0))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.02e+21) {
		tmp = (((l / k_m) * (l / k_m)) * (Math.cos(k_m) / (Math.pow(Math.sin(k_m), 2.0) * t))) * 2.0;
	} else {
		tmp = 2.0 / (((t / l) / l) * (Math.pow((k_m * t), 2.0) * 2.0));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 1.02e+21:
		tmp = (((l / k_m) * (l / k_m)) * (math.cos(k_m) / (math.pow(math.sin(k_m), 2.0) * t))) * 2.0
	else:
		tmp = 2.0 / (((t / l) / l) * (math.pow((k_m * t), 2.0) * 2.0))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.02e+21)
		tmp = Float64(Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * Float64(cos(k_m) / Float64((sin(k_m) ^ 2.0) * t))) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64((Float64(k_m * t) ^ 2.0) * 2.0)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 1.02e+21)
		tmp = (((l / k_m) * (l / k_m)) * (cos(k_m) / ((sin(k_m) ^ 2.0) * t))) * 2.0;
	else
		tmp = 2.0 / (((t / l) / l) * (((k_m * t) ^ 2.0) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.02e+21], N[(N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.02e21

   1. Initial program 52.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 64.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 75.2

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

   7. Applied rewrites 75.2%

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

   if 1.02e21 < t

   1. Initial program 64.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 76.1%

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

   7. Applied rewrites 75.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 83.8

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

   9. Applied rewrites 83.8%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow-prod-down N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-*.f64 79.4

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

   12. Applied rewrites 79.4%

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

Alternative 7: 70.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 60000000:\\ \;\;\;\;\left(\left(\ell \cdot \frac{\ell}{k\_m \cdot k\_m}\right) \cdot \frac{\cos k\_m}{{\sin k\_m}^{2} \cdot t}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left({\left(k\_m \cdot t\right)}^{2} \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 60000000.0)
   (* (* (* l (/ l (* k_m k_m))) (/ (cos k_m) (* (pow (sin k_m) 2.0) t))) 2.0)
   (/ 2.0 (* (/ (/ t l) l) (* (pow (* k_m t) 2.0) 2.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 60000000.0) {
		tmp = ((l * (l / (k_m * k_m))) * (cos(k_m) / (pow(sin(k_m), 2.0) * t))) * 2.0;
	} else {
		tmp = 2.0 / (((t / l) / l) * (pow((k_m * t), 2.0) * 2.0));
	}
	return tmp;
}

Fortran

k_m =     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, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 60000000.0d0) then
        tmp = ((l * (l / (k_m * k_m))) * (cos(k_m) / ((sin(k_m) ** 2.0d0) * t))) * 2.0d0
    else
        tmp = 2.0d0 / (((t / l) / l) * (((k_m * t) ** 2.0d0) * 2.0d0))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 60000000.0) {
		tmp = ((l * (l / (k_m * k_m))) * (Math.cos(k_m) / (Math.pow(Math.sin(k_m), 2.0) * t))) * 2.0;
	} else {
		tmp = 2.0 / (((t / l) / l) * (Math.pow((k_m * t), 2.0) * 2.0));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 60000000.0:
		tmp = ((l * (l / (k_m * k_m))) * (math.cos(k_m) / (math.pow(math.sin(k_m), 2.0) * t))) * 2.0
	else:
		tmp = 2.0 / (((t / l) / l) * (math.pow((k_m * t), 2.0) * 2.0))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 60000000.0)
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(k_m * k_m))) * Float64(cos(k_m) / Float64((sin(k_m) ^ 2.0) * t))) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64((Float64(k_m * t) ^ 2.0) * 2.0)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 60000000.0)
		tmp = ((l * (l / (k_m * k_m))) * (cos(k_m) / ((sin(k_m) ^ 2.0) * t))) * 2.0;
	else
		tmp = 2.0 / (((t / l) / l) * (((k_m * t) ^ 2.0) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 60000000.0], N[(N[(N[(l * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6e7

   1. Initial program 51.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 64.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 68.0

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

   7. Applied rewrites 68.0%

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

   if 6e7 < t

   1. Initial program 65.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 76.2%

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

   7. Applied rewrites 75.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 83.5

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

   9. Applied rewrites 83.5%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow-prod-down N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-*.f64 78.8

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

   12. Applied rewrites 78.8%

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

Alternative 8: 78.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.45 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left({\left(k\_m \cdot t\right)}^{2} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\cos k\_m \cdot \ell\right) \cdot \ell\right)}{\left(k\_m \cdot \left(k\_m \cdot t\right)\right) \cdot {\sin k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.45e-13)
   (/ 2.0 (* (/ (/ t l) l) (* (pow (* k_m t) 2.0) 2.0)))
   (/
    (* 2.0 (* (* (cos k_m) l) l))
    (* (* k_m (* k_m t)) (pow (sin k_m) 2.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.45e-13) {
		tmp = 2.0 / (((t / l) / l) * (pow((k_m * t), 2.0) * 2.0));
	} else {
		tmp = (2.0 * ((cos(k_m) * l) * l)) / ((k_m * (k_m * t)) * pow(sin(k_m), 2.0));
	}
	return tmp;
}

Fortran

k_m =     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, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.45d-13) then
        tmp = 2.0d0 / (((t / l) / l) * (((k_m * t) ** 2.0d0) * 2.0d0))
    else
        tmp = (2.0d0 * ((cos(k_m) * l) * l)) / ((k_m * (k_m * t)) * (sin(k_m) ** 2.0d0))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.45e-13) {
		tmp = 2.0 / (((t / l) / l) * (Math.pow((k_m * t), 2.0) * 2.0));
	} else {
		tmp = (2.0 * ((Math.cos(k_m) * l) * l)) / ((k_m * (k_m * t)) * Math.pow(Math.sin(k_m), 2.0));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.45e-13:
		tmp = 2.0 / (((t / l) / l) * (math.pow((k_m * t), 2.0) * 2.0))
	else:
		tmp = (2.0 * ((math.cos(k_m) * l) * l)) / ((k_m * (k_m * t)) * math.pow(math.sin(k_m), 2.0))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.45e-13)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64((Float64(k_m * t) ^ 2.0) * 2.0)));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k_m) * l) * l)) / Float64(Float64(k_m * Float64(k_m * t)) * (sin(k_m) ^ 2.0)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.45e-13)
		tmp = 2.0 / (((t / l) / l) * (((k_m * t) ^ 2.0) * 2.0));
	else
		tmp = (2.0 * ((cos(k_m) * l) * l)) / ((k_m * (k_m * t)) * (sin(k_m) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.45e-13], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.4499999999999999e-13

   1. Initial program 62.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

   7. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 86.1

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

   9. Applied rewrites 86.1%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow-prod-down N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-*.f64 82.8

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

   12. Applied rewrites 82.8%

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

   if 1.4499999999999999e-13 < k

   1. Initial program 48.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. frac-times N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

   7. Applied rewrites 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 74.1

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

   9. Applied rewrites 74.1%

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

Alternative 9: 76.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.45 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left({\left(k\_m \cdot t\right)}^{2} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\ell \cdot \ell\right) \cdot \frac{\cos k\_m}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot {\sin k\_m}^{2}}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.45e-13)
   (/ 2.0 (* (/ (/ t l) l) (* (pow (* k_m t) 2.0) 2.0)))
   (*
    (* (* l l) (/ (cos k_m) (* (* (* k_m k_m) t) (pow (sin k_m) 2.0))))
    2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.45e-13) {
		tmp = 2.0 / (((t / l) / l) * (pow((k_m * t), 2.0) * 2.0));
	} else {
		tmp = ((l * l) * (cos(k_m) / (((k_m * k_m) * t) * pow(sin(k_m), 2.0)))) * 2.0;
	}
	return tmp;
}

Fortran

k_m =     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, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.45d-13) then
        tmp = 2.0d0 / (((t / l) / l) * (((k_m * t) ** 2.0d0) * 2.0d0))
    else
        tmp = ((l * l) * (cos(k_m) / (((k_m * k_m) * t) * (sin(k_m) ** 2.0d0)))) * 2.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.45e-13) {
		tmp = 2.0 / (((t / l) / l) * (Math.pow((k_m * t), 2.0) * 2.0));
	} else {
		tmp = ((l * l) * (Math.cos(k_m) / (((k_m * k_m) * t) * Math.pow(Math.sin(k_m), 2.0)))) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.45e-13:
		tmp = 2.0 / (((t / l) / l) * (math.pow((k_m * t), 2.0) * 2.0))
	else:
		tmp = ((l * l) * (math.cos(k_m) / (((k_m * k_m) * t) * math.pow(math.sin(k_m), 2.0)))) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.45e-13)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64((Float64(k_m * t) ^ 2.0) * 2.0)));
	else
		tmp = Float64(Float64(Float64(l * l) * Float64(cos(k_m) / Float64(Float64(Float64(k_m * k_m) * t) * (sin(k_m) ^ 2.0)))) * 2.0);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.45e-13)
		tmp = 2.0 / (((t / l) / l) * (((k_m * t) ^ 2.0) * 2.0));
	else
		tmp = ((l * l) * (cos(k_m) / (((k_m * k_m) * t) * (sin(k_m) ^ 2.0)))) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.45e-13], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.4499999999999999e-13

   1. Initial program 62.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.5%

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

   7. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 86.1

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

   9. Applied rewrites 86.1%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow-prod-down N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-*.f64 82.8

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

   12. Applied rewrites 82.8%

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

   if 1.4499999999999999e-13 < k

   1. Initial program 48.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 N/A

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

   7. Applied rewrites 70.0%

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

Alternative 10: 76.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left({\left(k\_m \cdot t\right)}^{2} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\cos k\_m \cdot \ell\right) \cdot \ell\right)}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.75e-6)
   (/ 2.0 (* (/ (/ t l) l) (* (pow (* k_m t) 2.0) 2.0)))
   (/
    (* 2.0 (* (* (cos k_m) l) l))
    (* (* (* k_m k_m) t) (- 0.5 (* 0.5 (cos (* 2.0 k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.75e-6) {
		tmp = 2.0 / (((t / l) / l) * (pow((k_m * t), 2.0) * 2.0));
	} else {
		tmp = (2.0 * ((cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (0.5 - (0.5 * cos((2.0 * k_m)))));
	}
	return tmp;
}

Fortran

k_m =     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, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3.75d-6) then
        tmp = 2.0d0 / (((t / l) / l) * (((k_m * t) ** 2.0d0) * 2.0d0))
    else
        tmp = (2.0d0 * ((cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (0.5d0 - (0.5d0 * cos((2.0d0 * k_m)))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.75e-6) {
		tmp = 2.0 / (((t / l) / l) * (Math.pow((k_m * t), 2.0) * 2.0));
	} else {
		tmp = (2.0 * ((Math.cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (0.5 - (0.5 * Math.cos((2.0 * k_m)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3.75e-6:
		tmp = 2.0 / (((t / l) / l) * (math.pow((k_m * t), 2.0) * 2.0))
	else:
		tmp = (2.0 * ((math.cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (0.5 - (0.5 * math.cos((2.0 * k_m)))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.75e-6)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64((Float64(k_m * t) ^ 2.0) * 2.0)));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k_m) * l) * l)) / Float64(Float64(Float64(k_m * k_m) * t) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 3.75e-6)
		tmp = 2.0 / (((t / l) / l) * (((k_m * t) ^ 2.0) * 2.0));
	else
		tmp = (2.0 * ((cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (0.5 - (0.5 * cos((2.0 * k_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3.75e-6], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.7500000000000001e-6

   1. Initial program 62.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.6%

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

   7. Applied rewrites 77.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 86.2

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

   9. Applied rewrites 86.2%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow-prod-down N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-*.f64 82.4

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

   12. Applied rewrites 82.4%

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

   if 3.7500000000000001e-6 < k

   1. Initial program 48.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. frac-times N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

   7. Applied rewrites 70.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 69.9

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

   9. Applied rewrites 69.9%

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

Alternative 11: 62.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.006349206349206349, 0.044444444444444446\right), k\_m \cdot k\_m, \left(t \cdot t\right) \cdot 0.08888888888888889 - 0.3333333333333333\right), k\_m \cdot k\_m, \mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right)\right), k\_m \cdot k\_m, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)}{\cos k\_m \cdot \ell} \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left({\left(k\_m \cdot t\right)}^{2} \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.4e-62)
   (/
    2.0
    (*
     (/
      (*
       (fma
        (fma
         (fma
          (fma (* t t) -0.006349206349206349 0.044444444444444446)
          (* k_m k_m)
          (- (* (* t t) 0.08888888888888889) 0.3333333333333333))
         (* k_m k_m)
         (fma (* t t) -0.6666666666666666 1.0))
        (* k_m k_m)
        (* (* t t) 2.0))
       (* k_m k_m))
      (* (cos k_m) l))
     (/ t l)))
   (if (<= t 3.6e+87)
     (/ (* (/ l k_m) (/ l k_m)) (pow t 3.0))
     (/ 2.0 (* (/ (/ t l) l) (* (pow (* k_m t) 2.0) 2.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.4e-62) {
		tmp = 2.0 / (((fma(fma(fma(fma((t * t), -0.006349206349206349, 0.044444444444444446), (k_m * k_m), (((t * t) * 0.08888888888888889) - 0.3333333333333333)), (k_m * k_m), fma((t * t), -0.6666666666666666, 1.0)), (k_m * k_m), ((t * t) * 2.0)) * (k_m * k_m)) / (cos(k_m) * l)) * (t / l));
	} else if (t <= 3.6e+87) {
		tmp = ((l / k_m) * (l / k_m)) / pow(t, 3.0);
	} else {
		tmp = 2.0 / (((t / l) / l) * (pow((k_m * t), 2.0) * 2.0));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.4e-62)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(fma(fma(Float64(t * t), -0.006349206349206349, 0.044444444444444446), Float64(k_m * k_m), Float64(Float64(Float64(t * t) * 0.08888888888888889) - 0.3333333333333333)), Float64(k_m * k_m), fma(Float64(t * t), -0.6666666666666666, 1.0)), Float64(k_m * k_m), Float64(Float64(t * t) * 2.0)) * Float64(k_m * k_m)) / Float64(cos(k_m) * l)) * Float64(t / l)));
	elseif (t <= 3.6e+87)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64((Float64(k_m * t) ^ 2.0) * 2.0)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.4e-62], N[(2.0 / N[(N[(N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] * -0.006349206349206349 + 0.044444444444444446), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(N[(t * t), $MachinePrecision] * 0.08888888888888889), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * -0.6666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.6e+87], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.39999999999999984e-62

   1. Initial program 50.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\left(\frac{4}{45} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{2}{45} + \frac{-2}{315} \cdot {t}^{2}\right)\right) - \frac{1}{3}\right)\right)\right)\right) \cdot {k}^{2}}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   8. Applied rewrites 49.8%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k \cdot k, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k \cdot k, -0.6666666666666666 \cdot \left(t \cdot t\right)\right) + 1, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t} \]

   10. Applied rewrites 49.1%

       \[\leadsto \frac{2}{\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.006349206349206349, 0.044444444444444446\right) \cdot k, k, \left(t \cdot t\right) \cdot 0.08888888888888889 - 0.3333333333333333\right), k \cdot k, \mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right)\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

   12. Applied rewrites 55.2%

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.006349206349206349, 0.044444444444444446\right), k \cdot k, \left(t \cdot t\right) \cdot 0.08888888888888889 - 0.3333333333333333\right), k \cdot k, \mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right)\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)}{\cos k \cdot \ell} \cdot \color{blue}{\frac{t}{\ell}}} \]

   if 2.39999999999999984e-62 < t < 3.59999999999999994e87

   1. Initial program 75.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 62.8

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

   5. Applied rewrites 62.8%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 62.8

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

   7. Applied rewrites 62.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow3 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-pow.f64 72.6

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

   9. Applied rewrites 72.6%

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

   if 3.59999999999999994e87 < t

   1. Initial program 61.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 75.7%

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

   7. Applied rewrites 75.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 85.4

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

   9. Applied rewrites 85.4%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow-prod-down N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-*.f64 83.0

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

   12. Applied rewrites 83.0%

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

Alternative 12: 68.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\frac{t}{\ell}}{\ell}\\ \mathbf{if}\;t \leq 2.4 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, 1 - \left(-t\right) \cdot t\right), k\_m \cdot k\_m, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left({\left(k\_m \cdot t\right)}^{2} \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (/ t l) l)))
   (if (<= t 2.4e-62)
     (/
      2.0
      (*
       t_1
       (*
        (fma
         (fma (* -0.6666666666666666 t) t (- 1.0 (* (- t) t)))
         (* k_m k_m)
         (* (* t t) 2.0))
        (* k_m k_m))))
     (if (<= t 3.6e+87)
       (/ (* (/ l k_m) (/ l k_m)) (pow t 3.0))
       (/ 2.0 (* t_1 (* (pow (* k_m t) 2.0) 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (t / l) / l;
	double tmp;
	if (t <= 2.4e-62) {
		tmp = 2.0 / (t_1 * (fma(fma((-0.6666666666666666 * t), t, (1.0 - (-t * t))), (k_m * k_m), ((t * t) * 2.0)) * (k_m * k_m)));
	} else if (t <= 3.6e+87) {
		tmp = ((l / k_m) * (l / k_m)) / pow(t, 3.0);
	} else {
		tmp = 2.0 / (t_1 * (pow((k_m * t), 2.0) * 2.0));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(t / l) / l)
	tmp = 0.0
	if (t <= 2.4e-62)
		tmp = Float64(2.0 / Float64(t_1 * Float64(fma(fma(Float64(-0.6666666666666666 * t), t, Float64(1.0 - Float64(Float64(-t) * t))), Float64(k_m * k_m), Float64(Float64(t * t) * 2.0)) * Float64(k_m * k_m))));
	elseif (t <= 3.6e+87)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64((Float64(k_m * t) ^ 2.0) * 2.0)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[t, 2.4e-62], N[(2.0 / N[(t$95$1 * N[(N[(N[(N[(-0.6666666666666666 * t), $MachinePrecision] * t + N[(1.0 - N[((-t) * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.6e+87], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.39999999999999984e-62

   1. Initial program 50.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.8%

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

   7. Applied rewrites 74.1%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 50.0%

       \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right) - \left(-t \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 53.6

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

   12. Applied rewrites 64.1%

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

   if 2.39999999999999984e-62 < t < 3.59999999999999994e87

   1. Initial program 75.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 62.8

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

   5. Applied rewrites 62.8%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 62.8

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

   7. Applied rewrites 62.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow3 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-pow.f64 72.6

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

   9. Applied rewrites 72.6%

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

   if 3.59999999999999994e87 < t

   1. Initial program 61.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 75.7%

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

   7. Applied rewrites 75.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 85.4

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

   9. Applied rewrites 85.4%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow-prod-down N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-*.f64 83.0

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

   12. Applied rewrites 83.0%

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

Alternative 13: 67.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, 1 - \left(-t\right) \cdot t\right), k\_m \cdot k\_m, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left({\left(k\_m \cdot t\right)}^{2} \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.4e-62)
   (/
    2.0
    (*
     (/ (/ t l) l)
     (*
      (fma
       (fma (* -0.6666666666666666 t) t (- 1.0 (* (- t) t)))
       (* k_m k_m)
       (* (* t t) 2.0))
      (* k_m k_m))))
   (if (<= t 5.5e+87)
     (/ (* (/ l k_m) (/ l k_m)) (pow t 3.0))
     (/ 2.0 (* (/ t (* l l)) (* (pow (* k_m t) 2.0) 2.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.4e-62) {
		tmp = 2.0 / (((t / l) / l) * (fma(fma((-0.6666666666666666 * t), t, (1.0 - (-t * t))), (k_m * k_m), ((t * t) * 2.0)) * (k_m * k_m)));
	} else if (t <= 5.5e+87) {
		tmp = ((l / k_m) * (l / k_m)) / pow(t, 3.0);
	} else {
		tmp = 2.0 / ((t / (l * l)) * (pow((k_m * t), 2.0) * 2.0));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.4e-62)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64(fma(fma(Float64(-0.6666666666666666 * t), t, Float64(1.0 - Float64(Float64(-t) * t))), Float64(k_m * k_m), Float64(Float64(t * t) * 2.0)) * Float64(k_m * k_m))));
	elseif (t <= 5.5e+87)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(t / Float64(l * l)) * Float64((Float64(k_m * t) ^ 2.0) * 2.0)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.4e-62], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(-0.6666666666666666 * t), $MachinePrecision] * t + N[(1.0 - N[((-t) * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e+87], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.39999999999999984e-62

   1. Initial program 50.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.8%

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

   7. Applied rewrites 74.1%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 50.0%

       \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right) - \left(-t \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 53.6

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

   12. Applied rewrites 64.1%

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

   if 2.39999999999999984e-62 < t < 5.50000000000000022e87

   1. Initial program 75.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 62.9

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

   5. Applied rewrites 62.9%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 62.9

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

   7. Applied rewrites 62.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow3 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-pow.f64 72.6

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

   9. Applied rewrites 72.6%

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

   if 5.50000000000000022e87 < t

   1. Initial program 61.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 75.7%

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

   7. Applied rewrites 75.7%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 74.6

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

   10. Applied rewrites 74.6%

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

Alternative 14: 67.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, 1 - \left(-t\right) \cdot t\right), k\_m \cdot k\_m, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.4e-62)
   (/
    2.0
    (*
     (/ (/ t l) l)
     (*
      (fma
       (fma (* -0.6666666666666666 t) t (- 1.0 (* (- t) t)))
       (* k_m k_m)
       (* (* t t) 2.0))
      (* k_m k_m))))
   (if (<= t 6.4e+87)
     (/ (* (/ l k_m) (/ l k_m)) (pow t 3.0))
     (/ 2.0 (* (* (/ (pow (* k_m t) 2.0) (* l l)) 2.0) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.4e-62) {
		tmp = 2.0 / (((t / l) / l) * (fma(fma((-0.6666666666666666 * t), t, (1.0 - (-t * t))), (k_m * k_m), ((t * t) * 2.0)) * (k_m * k_m)));
	} else if (t <= 6.4e+87) {
		tmp = ((l / k_m) * (l / k_m)) / pow(t, 3.0);
	} else {
		tmp = 2.0 / (((pow((k_m * t), 2.0) / (l * l)) * 2.0) * t);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.4e-62)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64(fma(fma(Float64(-0.6666666666666666 * t), t, Float64(1.0 - Float64(Float64(-t) * t))), Float64(k_m * k_m), Float64(Float64(t * t) * 2.0)) * Float64(k_m * k_m))));
	elseif (t <= 6.4e+87)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / Float64(l * l)) * 2.0) * t));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.4e-62], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(-0.6666666666666666 * t), $MachinePrecision] * t + N[(1.0 - N[((-t) * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.4e+87], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.39999999999999984e-62

   1. Initial program 50.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.8%

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

   7. Applied rewrites 74.1%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 50.0%

       \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right) - \left(-t \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 53.6

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

   12. Applied rewrites 64.1%

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

   if 2.39999999999999984e-62 < t < 6.4000000000000001e87

   1. Initial program 75.5%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 63.0

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

   5. Applied rewrites 63.0%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 62.9

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

   7. Applied rewrites 62.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow3 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-pow.f64 72.6

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

   9. Applied rewrites 72.6%

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

   if 6.4000000000000001e87 < t

   1. Initial program 61.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 74.5

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

   8. Applied rewrites 74.5%

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

Alternative 15: 66.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, 1 - \left(-t\right) \cdot t\right), k\_m \cdot k\_m, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(k\_m \cdot t\right)}^{2} \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.4e-62)
   (/
    2.0
    (*
     (/ (/ t l) l)
     (*
      (fma
       (fma (* -0.6666666666666666 t) t (- 1.0 (* (- t) t)))
       (* k_m k_m)
       (* (* t t) 2.0))
      (* k_m k_m))))
   (if (<= t 5.5e+87)
     (/ (* (/ l k_m) (/ l k_m)) (pow t 3.0))
     (/ (* l l) (* (pow (* k_m t) 2.0) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.4e-62) {
		tmp = 2.0 / (((t / l) / l) * (fma(fma((-0.6666666666666666 * t), t, (1.0 - (-t * t))), (k_m * k_m), ((t * t) * 2.0)) * (k_m * k_m)));
	} else if (t <= 5.5e+87) {
		tmp = ((l / k_m) * (l / k_m)) / pow(t, 3.0);
	} else {
		tmp = (l * l) / (pow((k_m * t), 2.0) * t);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.4e-62)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64(fma(fma(Float64(-0.6666666666666666 * t), t, Float64(1.0 - Float64(Float64(-t) * t))), Float64(k_m * k_m), Float64(Float64(t * t) * 2.0)) * Float64(k_m * k_m))));
	elseif (t <= 5.5e+87)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t ^ 3.0));
	else
		tmp = Float64(Float64(l * l) / Float64((Float64(k_m * t) ^ 2.0) * t));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.4e-62], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(-0.6666666666666666 * t), $MachinePrecision] * t + N[(1.0 - N[((-t) * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e+87], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.39999999999999984e-62

   1. Initial program 50.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.8%

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

   7. Applied rewrites 74.1%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 50.0%

       \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right) - \left(-t \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 53.6

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

   12. Applied rewrites 64.1%

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

   if 2.39999999999999984e-62 < t < 5.50000000000000022e87

   1. Initial program 75.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 62.9

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

   5. Applied rewrites 62.9%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 62.9

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

   7. Applied rewrites 62.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow3 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-pow.f64 72.6

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

   9. Applied rewrites 72.6%

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

   if 5.50000000000000022e87 < t

   1. Initial program 61.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 52.7

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

   5. Applied rewrites 52.7%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 52.7

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

   7. Applied rewrites 52.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. pow-prod-down N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 73.7

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

   9. Applied rewrites 73.7%

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

Alternative 16: 64.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, 1 - \left(-t\right) \cdot t\right), k\_m \cdot k\_m, \left(t \cdot t\right) \cdot 2\right)\\ \mathbf{if}\;t \leq 5 \cdot 10^{-130}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left(t\_1 \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\left(t\_1 \cdot k\_m\right) \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k\_m \cdot \left(k\_m \cdot {t}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1
         (fma
          (fma (* -0.6666666666666666 t) t (- 1.0 (* (- t) t)))
          (* k_m k_m)
          (* (* t t) 2.0))))
   (if (<= t 5e-130)
     (/ 2.0 (* (/ (/ t l) l) (* t_1 (* k_m k_m))))
     (if (<= t 1.9e+152)
       (/ 2.0 (* (/ t (* l l)) (* (* t_1 k_m) k_m)))
       (/ (* l l) (* k_m (* k_m (pow t 3.0))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = fma(fma((-0.6666666666666666 * t), t, (1.0 - (-t * t))), (k_m * k_m), ((t * t) * 2.0));
	double tmp;
	if (t <= 5e-130) {
		tmp = 2.0 / (((t / l) / l) * (t_1 * (k_m * k_m)));
	} else if (t <= 1.9e+152) {
		tmp = 2.0 / ((t / (l * l)) * ((t_1 * k_m) * k_m));
	} else {
		tmp = (l * l) / (k_m * (k_m * pow(t, 3.0)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = fma(fma(Float64(-0.6666666666666666 * t), t, Float64(1.0 - Float64(Float64(-t) * t))), Float64(k_m * k_m), Float64(Float64(t * t) * 2.0))
	tmp = 0.0
	if (t <= 5e-130)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64(t_1 * Float64(k_m * k_m))));
	elseif (t <= 1.9e+152)
		tmp = Float64(2.0 / Float64(Float64(t / Float64(l * l)) * Float64(Float64(t_1 * k_m) * k_m)));
	else
		tmp = Float64(Float64(l * l) / Float64(k_m * Float64(k_m * (t ^ 3.0))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(N[(-0.6666666666666666 * t), $MachinePrecision] * t + N[(1.0 - N[((-t) * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 5e-130], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$1 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.9e+152], N[(2.0 / N[(N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(k$95$m * N[(k$95$m * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.9999999999999996e-130

   1. Initial program 49.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.2%

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

   7. Applied rewrites 74.3%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 49.3%

       \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right) - \left(-t \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 53.0

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

   12. Applied rewrites 64.2%

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

   if 4.9999999999999996e-130 < t < 1.9e152

   1. Initial program 65.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.4%

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

   7. Applied rewrites 74.5%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 61.6%

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

   12. Applied rewrites 65.7%

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

   if 1.9e152 < t

   1. Initial program 63.9%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 54.5

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

   5. Applied rewrites 54.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 63.9

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

   7. Applied rewrites 63.9%

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

Alternative 17: 65.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.45 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, 1 - \left(-t\right) \cdot t\right), k\_m \cdot k\_m, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(k\_m \cdot t\right)}^{2} \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.45e-62)
   (/
    2.0
    (*
     (/ (/ t l) l)
     (*
      (fma
       (fma (* -0.6666666666666666 t) t (- 1.0 (* (- t) t)))
       (* k_m k_m)
       (* (* t t) 2.0))
      (* k_m k_m))))
   (/ (* l l) (* (pow (* k_m t) 2.0) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.45e-62) {
		tmp = 2.0 / (((t / l) / l) * (fma(fma((-0.6666666666666666 * t), t, (1.0 - (-t * t))), (k_m * k_m), ((t * t) * 2.0)) * (k_m * k_m)));
	} else {
		tmp = (l * l) / (pow((k_m * t), 2.0) * t);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.45e-62)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64(fma(fma(Float64(-0.6666666666666666 * t), t, Float64(1.0 - Float64(Float64(-t) * t))), Float64(k_m * k_m), Float64(Float64(t * t) * 2.0)) * Float64(k_m * k_m))));
	else
		tmp = Float64(Float64(l * l) / Float64((Float64(k_m * t) ^ 2.0) * t));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.45e-62], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(-0.6666666666666666 * t), $MachinePrecision] * t + N[(1.0 - N[((-t) * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.4500000000000002e-62

   1. Initial program 50.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.8%

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

   7. Applied rewrites 74.1%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 50.0%

       \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right) - \left(-t \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 53.6

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

   12. Applied rewrites 64.1%

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

   if 2.4500000000000002e-62 < t

   1. Initial program 67.0%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 56.6

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

   5. Applied rewrites 56.6%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 56.6

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

   7. Applied rewrites 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. pow-prod-down N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 70.2

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

   9. Applied rewrites 70.2%

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

Alternative 18: 61.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, 1 - \left(-t\right) \cdot t\right), k\_m \cdot k\_m, \left(t \cdot t\right) \cdot 2\right) \cdot k\_m\right) \cdot k\_m\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/
  2.0
  (*
   (/ t (* l l))
   (*
    (*
     (fma
      (fma (* -0.6666666666666666 t) t (- 1.0 (* (- t) t)))
      (* k_m k_m)
      (* (* t t) 2.0))
     k_m)
    k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / ((t / (l * l)) * ((fma(fma((-0.6666666666666666 * t), t, (1.0 - (-t * t))), (k_m * k_m), ((t * t) * 2.0)) * k_m) * k_m));
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64(Float64(t / Float64(l * l)) * Float64(Float64(fma(fma(Float64(-0.6666666666666666 * t), t, Float64(1.0 - Float64(Float64(-t) * t))), Float64(k_m * k_m), Float64(Float64(t * t) * 2.0)) * k_m) * k_m)))
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-0.6666666666666666 * t), $MachinePrecision] * t + N[(1.0 - N[((-t) * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 74.5%

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

7. Applied rewrites 74.6%

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

8. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 45.8%

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

12. Applied rewrites 61.2%

    \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, 1 - \left(-t\right) \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot k\right) \cdot k\right)} \]
13. Add Preprocessing

Alternative 19: 56.1% accurate, 7.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{t}{\ell \cdot \ell}\\ \mathbf{if}\;t \leq 4.7 \cdot 10^{-60}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left(\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left(\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ t (* l l))))
   (if (<= t 4.7e-60)
     (/ 2.0 (* t_1 (* (* k_m k_m) (* k_m k_m))))
     (/ 2.0 (* t_1 (* (* (* t t) 2.0) (* k_m k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = t / (l * l);
	double tmp;
	if (t <= 4.7e-60) {
		tmp = 2.0 / (t_1 * ((k_m * k_m) * (k_m * k_m)));
	} else {
		tmp = 2.0 / (t_1 * (((t * t) * 2.0) * (k_m * k_m)));
	}
	return tmp;
}

Fortran

k_m =     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, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / (l * l)
    if (t <= 4.7d-60) then
        tmp = 2.0d0 / (t_1 * ((k_m * k_m) * (k_m * k_m)))
    else
        tmp = 2.0d0 / (t_1 * (((t * t) * 2.0d0) * (k_m * k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = t / (l * l);
	double tmp;
	if (t <= 4.7e-60) {
		tmp = 2.0 / (t_1 * ((k_m * k_m) * (k_m * k_m)));
	} else {
		tmp = 2.0 / (t_1 * (((t * t) * 2.0) * (k_m * k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = t / (l * l)
	tmp = 0
	if t <= 4.7e-60:
		tmp = 2.0 / (t_1 * ((k_m * k_m) * (k_m * k_m)))
	else:
		tmp = 2.0 / (t_1 * (((t * t) * 2.0) * (k_m * k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(t / Float64(l * l))
	tmp = 0.0
	if (t <= 4.7e-60)
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(k_m * k_m) * Float64(k_m * k_m))));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(Float64(t * t) * 2.0) * Float64(k_m * k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = t / (l * l);
	tmp = 0.0;
	if (t <= 4.7e-60)
		tmp = 2.0 / (t_1 * ((k_m * k_m) * (k_m * k_m)));
	else
		tmp = 2.0 / (t_1 * (((t * t) * 2.0) * (k_m * k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 4.7e-60], N[(2.0 / N[(t$95$1 * N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 4.7e-60

   1. Initial program 50.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.8%

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

   7. Applied rewrites 74.0%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 50.0%

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

   11. Taylor expanded in t around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 55.1

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

   13. Applied rewrites 55.1%

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

   if 4.7e-60 < t

   1. Initial program 67.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 76.2%

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left({k}^{2} \cdot \color{blue}{\left(2 \cdot {t}^{2} + {k}^{2} \cdot \left(\left(1 + \left(\frac{-2}{3} \cdot {t}^{2} + {k}^{2} \cdot \left(\frac{4}{45} \cdot {t}^{2} - \left(\frac{1}{3} + \left(\frac{-1}{2} \cdot \left(\left(1 + \frac{-2}{3} \cdot {t}^{2}\right) - -1 \cdot {t}^{2}\right) + \frac{1}{12} \cdot {t}^{2}\right)\right)\right)\right)\right) - -1 \cdot {t}^{2}\right)\right)}\right)} \]

   10. Applied rewrites 35.9%

       \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.6666666666666666 \cdot t, t, \left(\left(\left(t \cdot t\right) \cdot 0.08888888888888889 - 0.3333333333333333\right) - \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, -0.6666666666666666, 1\right) - \left(-t \cdot t\right), -0.5, 0.08333333333333333 \cdot \left(t \cdot t\right)\right)\right) \cdot \left(k \cdot k\right)\right) + 1\right) - \left(-t \cdot t\right), k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   11. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 58.6

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

   13. Applied rewrites 58.6%

       \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \left(\left(\left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 51.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell \cdot \ell}{\left(k\_m \cdot k\_m\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ (* l l) (* (* k_m k_m) (* (* t t) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l * l) / ((k_m * k_m) * ((t * t) * t));
}

Fortran

k_m =     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, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l * l) / ((k_m * k_m) * ((t * t) * t))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return (l * l) / ((k_m * k_m) * ((t * t) * t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l * l) / ((k_m * k_m) * ((t * t) * t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l * l) / Float64(Float64(k_m * k_m) * Float64(Float64(t * t) * t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(l * l), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   7. lift-pow.f64 51.8

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

5. Applied rewrites 51.8%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   2. pow3 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   4. lift-*.f64 51.8

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

7. Applied rewrites 51.8%

   \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025089 
(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))))