Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.4% → 88.9%

Time: 10.2s

Alternatives: 21

Speedup: 12.5×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot \ell\\ t_2 := {\left(\sin k\_m \cdot t\right)}^{2}\\ \mathbf{if}\;k\_m \leq 1.6 \cdot 10^{+207}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\sin k\_m \cdot k\_m, \frac{\tan k\_m \cdot k\_m}{\ell}, \frac{t\_2}{\ell} \cdot \frac{2}{\cos k\_m}\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{{\sin k\_m}^{2} \cdot k\_m}{\ell}, \frac{k\_m}{t\_1}, t\_2 \cdot \frac{2}{t\_1 \cdot \ell}\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l)) (t_2 (pow (* (sin k_m) t) 2.0)))
   (if (<= k_m 1.6e+207)
     (/
      2.0
      (*
       (fma
        (* (sin k_m) k_m)
        (/ (* (tan k_m) k_m) l)
        (* (/ t_2 l) (/ 2.0 (cos k_m))))
       (/ t l)))
     (/
      2.0
      (*
       (fma
        (/ (* (pow (sin k_m) 2.0) k_m) l)
        (/ k_m t_1)
        (* t_2 (/ 2.0 (* t_1 l))))
       t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * l;
	double t_2 = pow((sin(k_m) * t), 2.0);
	double tmp;
	if (k_m <= 1.6e+207) {
		tmp = 2.0 / (fma((sin(k_m) * k_m), ((tan(k_m) * k_m) / l), ((t_2 / l) * (2.0 / cos(k_m)))) * (t / l));
	} else {
		tmp = 2.0 / (fma(((pow(sin(k_m), 2.0) * k_m) / l), (k_m / t_1), (t_2 * (2.0 / (t_1 * l)))) * t);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * l)
	t_2 = Float64(sin(k_m) * t) ^ 2.0
	tmp = 0.0
	if (k_m <= 1.6e+207)
		tmp = Float64(2.0 / Float64(fma(Float64(sin(k_m) * k_m), Float64(Float64(tan(k_m) * k_m) / l), Float64(Float64(t_2 / l) * Float64(2.0 / cos(k_m)))) * Float64(t / l)));
	else
		tmp = Float64(2.0 / Float64(fma(Float64(Float64((sin(k_m) ^ 2.0) * k_m) / l), Float64(k_m / t_1), Float64(t_2 * Float64(2.0 / Float64(t_1 * l)))) * t));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k$95$m, 1.6e+207], N[(2.0 / N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision] * N[(N[(N[Tan[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] + N[(N[(t$95$2 / l), $MachinePrecision] * N[(2.0 / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * k$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m / t$95$1), $MachinePrecision] + N[(t$95$2 * N[(2.0 / N[(t$95$1 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.6000000000000001e207

   1. Initial program 54.0%

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

   3. 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

   5. Applied rewrites 67.7%

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

   7. Applied rewrites 83.7%

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

   9. Applied rewrites 87.4%

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

   11. Applied rewrites 87.4%

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

   if 1.6000000000000001e207 < k

   1. Initial program 42.9%

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

   5. Applied rewrites 62.3%

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

   7. Applied rewrites 81.2%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{+207}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\sin k \cdot k, \frac{\tan k \cdot k}{\ell}, \frac{{\left(\sin k \cdot t\right)}^{2}}{\ell} \cdot \frac{2}{\cos k}\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\frac{{\sin k}^{2} \cdot k}{\ell}, \frac{k}{\cos k \cdot \ell}, {\left(\sin k \cdot t\right)}^{2} \cdot \frac{2}{\left(\cos k \cdot \ell\right) \cdot \ell}\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{\mathsf{fma}\left(\sin k\_m \cdot k\_m, \frac{\tan k\_m \cdot k\_m}{\ell}, \frac{{\left(\sin k\_m \cdot t\right)}^{2}}{\ell} \cdot \frac{2}{\cos k\_m}\right) \cdot \frac{t}{\ell}} \end{array} \]

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / (fma((sin(k_m) * k_m), ((tan(k_m) * k_m) / l), ((pow((sin(k_m) * t), 2.0) / l) * (2.0 / cos(k_m)))) * (t / l));
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64(fma(Float64(sin(k_m) * k_m), Float64(Float64(tan(k_m) * k_m) / l), Float64(Float64((Float64(sin(k_m) * t) ^ 2.0) / l) * Float64(2.0 / cos(k_m)))) * Float64(t / l)))
end

Wolfram

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

Derivation

1. Initial program 53.1%

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

5. Applied rewrites 67.2%

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

7. Applied rewrites 82.4%

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

9. Applied rewrites 86.5%

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

11. Applied rewrites 86.5%

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

12. Final simplification 86.5%

    \[\leadsto \frac{2}{\mathsf{fma}\left(\sin k \cdot k, \frac{\tan k \cdot k}{\ell}, \frac{{\left(\sin k \cdot t\right)}^{2}}{\ell} \cdot \frac{2}{\cos k}\right) \cdot \frac{t}{\ell}} \]
13. Add Preprocessing

Alternative 3: 87.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{\mathsf{fma}\left({\left(t \cdot \sin k\_m\right)}^{2}, \frac{2}{\ell \cdot \cos k\_m}, \frac{\sin k\_m \cdot k\_m}{\ell} \cdot \left(k\_m \cdot \tan k\_m\right)\right) \cdot \frac{t}{\ell}} \end{array} \]

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / (fma(pow((t * sin(k_m)), 2.0), (2.0 / (l * cos(k_m))), (((sin(k_m) * k_m) / l) * (k_m * tan(k_m)))) * (t / l));
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64(fma((Float64(t * sin(k_m)) ^ 2.0), Float64(2.0 / Float64(l * cos(k_m))), Float64(Float64(Float64(sin(k_m) * k_m) / l) * Float64(k_m * tan(k_m)))) * Float64(t / l)))
end

Wolfram

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

Derivation

1. Initial program 53.1%

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

5. Applied rewrites 67.2%

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

7. Applied rewrites 82.4%

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

9. Applied rewrites 86.5%

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

11. Applied rewrites 86.5%

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

12. Final simplification 86.5%

    \[\leadsto \frac{2}{\mathsf{fma}\left({\left(t \cdot \sin k\right)}^{2}, \frac{2}{\ell \cdot \cos k}, \frac{\sin k \cdot k}{\ell} \cdot \left(k \cdot \tan k\right)\right) \cdot \frac{t}{\ell}} \]
13. Add Preprocessing

Alternative 4: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot t\\ \mathbf{if}\;t \leq 6.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \mathsf{fma}\left({t\_1}^{2}, 2, {\left(\sin k\_m \cdot k\_m\right)}^{2}\right)}{\ell \cdot \cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \left(\frac{t}{\ell} \cdot \frac{t\_1}{\ell}\right)\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) t)))
   (if (<= t 6.4e-22)
     (/
      2.0
      (/
       (* (/ t l) (fma (pow t_1 2.0) 2.0 (pow (* (sin k_m) k_m) 2.0)))
       (* l (cos k_m))))
     (/
      2.0
      (*
       (* (* t (* (/ t l) (/ t_1 l))) (tan k_m))
       (+ (+ 1.0 (pow (/ k_m t) 2.0)) 1.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * t;
	double tmp;
	if (t <= 6.4e-22) {
		tmp = 2.0 / (((t / l) * fma(pow(t_1, 2.0), 2.0, pow((sin(k_m) * k_m), 2.0))) / (l * cos(k_m)));
	} else {
		tmp = 2.0 / (((t * ((t / l) * (t_1 / l))) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) + 1.0));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * t)
	tmp = 0.0
	if (t <= 6.4e-22)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * fma((t_1 ^ 2.0), 2.0, (Float64(sin(k_m) * k_m) ^ 2.0))) / Float64(l * cos(k_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(Float64(t / l) * Float64(t_1 / l))) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + 1.0)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, 6.4e-22], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[(N[Power[t$95$1, 2.0], $MachinePrecision] * 2.0 + N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t * N[(N[(t / l), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $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]]]

Derivation

1. Split input into 2 regimes

2. if t < 6.39999999999999975e-22

   1. Initial program 48.1%

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

   5. Applied rewrites 66.6%

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

   7. Applied rewrites 81.2%

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

   9. Applied rewrites 82.6%

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

   if 6.39999999999999975e-22 < t

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow3 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 78.6

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

   4. Applied rewrites 78.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 88.2

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

   6. Applied rewrites 88.2%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \mathsf{fma}\left({\left(\sin k \cdot t\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \left(\frac{t}{\ell} \cdot \frac{\sin k \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \sin k\_m \cdot t\\ t_2 := {\left(\frac{k\_m}{t}\right)}^{2}\\ \mathbf{if}\;t \leq 9 \cdot 10^{-103}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left({t\_1}^{2}, 2, {\left(\sin k\_m \cdot k\_m\right)}^{2}\right) \cdot \frac{\frac{t}{\ell}}{\cos k\_m \cdot \ell}}\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{+94}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_2 + 2\right) \cdot \left(\left({t}^{3} \cdot \sin k\_m\right) \cdot \frac{\tan k\_m}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \left(\frac{t}{\ell} \cdot \frac{t\_1}{\ell}\right)\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + t\_2\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (sin k_m) t)) (t_2 (pow (/ k_m t) 2.0)))
   (if (<= t 9e-103)
     (/
      2.0
      (*
       (fma (pow t_1 2.0) 2.0 (pow (* (sin k_m) k_m) 2.0))
       (/ (/ t l) (* (cos k_m) l))))
     (if (<= t 1.26e+94)
       (/
        2.0
        (/ (* (+ t_2 2.0) (* (* (pow t 3.0) (sin k_m)) (/ (tan k_m) l))) l))
       (/
        2.0
        (* (* (* t (* (/ t l) (/ t_1 l))) (tan k_m)) (+ (+ 1.0 t_2) 1.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = sin(k_m) * t;
	double t_2 = pow((k_m / t), 2.0);
	double tmp;
	if (t <= 9e-103) {
		tmp = 2.0 / (fma(pow(t_1, 2.0), 2.0, pow((sin(k_m) * k_m), 2.0)) * ((t / l) / (cos(k_m) * l)));
	} else if (t <= 1.26e+94) {
		tmp = 2.0 / (((t_2 + 2.0) * ((pow(t, 3.0) * sin(k_m)) * (tan(k_m) / l))) / l);
	} else {
		tmp = 2.0 / (((t * ((t / l) * (t_1 / l))) * tan(k_m)) * ((1.0 + t_2) + 1.0));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(sin(k_m) * t)
	t_2 = Float64(k_m / t) ^ 2.0
	tmp = 0.0
	if (t <= 9e-103)
		tmp = Float64(2.0 / Float64(fma((t_1 ^ 2.0), 2.0, (Float64(sin(k_m) * k_m) ^ 2.0)) * Float64(Float64(t / l) / Float64(cos(k_m) * l))));
	elseif (t <= 1.26e+94)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 + 2.0) * Float64(Float64((t ^ 3.0) * sin(k_m)) * Float64(tan(k_m) / l))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(Float64(t / l) * Float64(t_1 / l))) * tan(k_m)) * Float64(Float64(1.0 + t_2) + 1.0)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, 9e-103], N[(2.0 / N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] * 2.0 + N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.26e+94], N[(2.0 / N[(N[(N[(t$95$2 + 2.0), $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t * N[(N[(t / l), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + t$95$2), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 9e-103

   1. Initial program 48.2%

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

   5. Applied rewrites 68.2%

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

   7. Applied rewrites 83.8%

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

   9. Applied rewrites 82.7%

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

   if 9e-103 < t < 1.25999999999999997e94

   1. Initial program 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 74.8%

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

   if 1.25999999999999997e94 < t

   1. Initial program 65.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow3 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 77.2

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

   4. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 90.5

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

   6. Applied rewrites 90.5%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{-103}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left({\left(\sin k \cdot t\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right) \cdot \frac{\frac{t}{\ell}}{\cos k \cdot \ell}}\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{+94}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \frac{\tan k}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \left(\frac{t}{\ell} \cdot \frac{\sin k \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2100000000000:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\left(t \cdot k\_m\right) \cdot \left(t \cdot k\_m\right), 2, {\left(\sin k\_m \cdot k\_m\right)}^{2}\right)}{\cos k\_m \cdot \ell} \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k\_m \leq 1.9 \cdot 10^{+145}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\mathsf{fma}\left(t \cdot t, 2, k\_m \cdot k\_m\right) \cdot \left(\tan k\_m \cdot \sin k\_m\right)\right) \cdot t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\frac{\sin k\_m \cdot t}{\ell} \cdot \left(\tan k\_m \cdot \left({\left(\frac{k\_m}{t}\right)}^{2} + 2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2100000000000.0)
   (/
    2.0
    (*
     (/
      (fma (* (* t k_m) (* t k_m)) 2.0 (pow (* (sin k_m) k_m) 2.0))
      (* (cos k_m) l))
     (/ t l)))
   (if (<= k_m 1.9e+145)
     (/
      2.0
      (/
       (/ (* (* (fma (* t t) 2.0 (* k_m k_m)) (* (tan k_m) (sin k_m))) t) l)
       l))
     (/
      2.0
      (*
       (* (/ t l) t)
       (* (/ (* (sin k_m) t) l) (* (tan k_m) (+ (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 (k_m <= 2100000000000.0) {
		tmp = 2.0 / ((fma(((t * k_m) * (t * k_m)), 2.0, pow((sin(k_m) * k_m), 2.0)) / (cos(k_m) * l)) * (t / l));
	} else if (k_m <= 1.9e+145) {
		tmp = 2.0 / ((((fma((t * t), 2.0, (k_m * k_m)) * (tan(k_m) * sin(k_m))) * t) / l) / l);
	} else {
		tmp = 2.0 / (((t / l) * t) * (((sin(k_m) * t) / l) * (tan(k_m) * (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 (k_m <= 2100000000000.0)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(Float64(t * k_m) * Float64(t * k_m)), 2.0, (Float64(sin(k_m) * k_m) ^ 2.0)) / Float64(cos(k_m) * l)) * Float64(t / l)));
	elseif (k_m <= 1.9e+145)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(t * t), 2.0, Float64(k_m * k_m)) * Float64(tan(k_m) * sin(k_m))) * t) / l) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * t) * Float64(Float64(Float64(sin(k_m) * t) / l) * Float64(tan(k_m) * 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[k$95$m, 2100000000000.0], N[(2.0 / N[(N[(N[(N[(N[(t * k$95$m), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $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], If[LessEqual[k$95$m, 1.9e+145], N[(2.0 / N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * t), $MachinePrecision] * N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.1e12

   1. Initial program 57.9%

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

   3. Taylor expanded in 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

   5. Applied rewrites 68.1%

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

   7. Applied rewrites 85.7%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 82.3%

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

   if 2.1e12 < k < 1.90000000000000006e145

   1. Initial program 26.9%

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 81.2%

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

   10. Applied rewrites 88.7%

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

   if 1.90000000000000006e145 < k

   1. Initial program 44.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow3 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 54.2

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

   4. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lift-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 66.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2100000000000:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\left(t \cdot k\right) \cdot \left(t \cdot k\right), 2, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \ell} \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{+145}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right) \cdot \left(\tan k \cdot \sin k\right)\right) \cdot t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\frac{\sin k \cdot t}{\ell} \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 2\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 6.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{\mathsf{fma}\left(t \cdot t, 2, k\_m \cdot k\_m\right) \cdot \left(\tan k\_m \cdot \sin k\_m\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t \cdot \left(\frac{t}{\ell} \cdot \frac{\sin k\_m \cdot t}{\ell}\right)\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 6.4e-22)
   (/
    2.0
    (*
     (/ t l)
     (/ (* (fma (* t t) 2.0 (* k_m k_m)) (* (tan k_m) (sin k_m))) l)))
   (/
    2.0
    (*
     (* (* t (* (/ t l) (/ (* (sin k_m) t) l))) (tan k_m))
     (+ (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 6.4e-22) {
		tmp = 2.0 / ((t / l) * ((fma((t * t), 2.0, (k_m * k_m)) * (tan(k_m) * sin(k_m))) / l));
	} else {
		tmp = 2.0 / (((t * ((t / l) * ((sin(k_m) * t) / l))) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) + 1.0));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 6.4e-22)
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(Float64(fma(Float64(t * t), 2.0, Float64(k_m * k_m)) * Float64(tan(k_m) * sin(k_m))) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(Float64(t / l) * Float64(Float64(sin(k_m) * t) / l))) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + 1.0)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 6.4e-22], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[(N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t * N[(N[(t / l), $MachinePrecision] * N[(N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision]), $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]]

Derivation

1. Split input into 2 regimes

2. if t < 6.39999999999999975e-22

   1. Initial program 48.1%

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 67.9%

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

   10. Applied rewrites 77.2%

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

   if 6.39999999999999975e-22 < t

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow3 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 78.6

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

   4. Applied rewrites 78.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 88.2

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

   6. Applied rewrites 88.2%

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

Alternative 8: 83.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2100000000000:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\left(t \cdot k\_m\right) \cdot \left(t \cdot k\_m\right), 2, {\left(\sin k\_m \cdot k\_m\right)}^{2}\right)}{\cos k\_m \cdot \ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\mathsf{fma}\left(t \cdot t, 2, k\_m \cdot k\_m\right) \cdot \left(\tan k\_m \cdot \sin k\_m\right)\right) \cdot t}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2100000000000.0)
   (/
    2.0
    (*
     (/
      (fma (* (* t k_m) (* t k_m)) 2.0 (pow (* (sin k_m) k_m) 2.0))
      (* (cos k_m) l))
     (/ t l)))
   (/
    2.0
    (/
     (/ (* (* (fma (* t t) 2.0 (* k_m k_m)) (* (tan k_m) (sin k_m))) t) l)
     l))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2100000000000.0) {
		tmp = 2.0 / ((fma(((t * k_m) * (t * k_m)), 2.0, pow((sin(k_m) * k_m), 2.0)) / (cos(k_m) * l)) * (t / l));
	} else {
		tmp = 2.0 / ((((fma((t * t), 2.0, (k_m * k_m)) * (tan(k_m) * sin(k_m))) * t) / l) / l);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2100000000000.0)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(Float64(t * k_m) * Float64(t * k_m)), 2.0, (Float64(sin(k_m) * k_m) ^ 2.0)) / Float64(cos(k_m) * l)) * Float64(t / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(t * t), 2.0, Float64(k_m * k_m)) * Float64(tan(k_m) * sin(k_m))) * t) / l) / l));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2100000000000.0], N[(2.0 / N[(N[(N[(N[(N[(t * k$95$m), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $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[(2.0 / N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.1e12

   1. Initial program 57.9%

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

   3. Taylor expanded in 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

   5. Applied rewrites 68.1%

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

   7. Applied rewrites 85.7%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 82.3%

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

   if 2.1e12 < k

   1. Initial program 36.9%

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 67.5%

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

   10. Applied rewrites 74.8%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2100000000000:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\left(t \cdot k\right) \cdot \left(t \cdot k\right), 2, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right) \cdot \left(\tan k \cdot \sin k\right)\right) \cdot t}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{t \cdot t}{\ell}\\ \mathbf{if}\;k\_m \leq 1.85 \cdot 10^{-144}:\\ \;\;\;\;\frac{2}{\left(\frac{2}{\cos k\_m} \cdot \frac{{\left(\sin k\_m \cdot t\right)}^{2}}{\ell}\right) \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k\_m \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_1, 0.3333333333333333, \frac{1}{\ell}\right), k\_m \cdot k\_m, t\_1 \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\mathsf{fma}\left(t \cdot t, 2, k\_m \cdot k\_m\right) \cdot \left(\tan k\_m \cdot \sin k\_m\right)\right) \cdot t}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* t t) l)))
   (if (<= k_m 1.85e-144)
     (/ 2.0 (* (* (/ 2.0 (cos k_m)) (/ (pow (* (sin k_m) t) 2.0) l)) (/ t l)))
     (if (<= k_m 4e-16)
       (/
        2.0
        (*
         (*
          (fma (fma t_1 0.3333333333333333 (/ 1.0 l)) (* k_m k_m) (* t_1 2.0))
          (* k_m k_m))
         (/ t l)))
       (/
        2.0
        (/
         (/ (* (* (fma (* t t) 2.0 (* k_m k_m)) (* (tan k_m) (sin k_m))) t) l)
         l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (t * t) / l;
	double tmp;
	if (k_m <= 1.85e-144) {
		tmp = 2.0 / (((2.0 / cos(k_m)) * (pow((sin(k_m) * t), 2.0) / l)) * (t / l));
	} else if (k_m <= 4e-16) {
		tmp = 2.0 / ((fma(fma(t_1, 0.3333333333333333, (1.0 / l)), (k_m * k_m), (t_1 * 2.0)) * (k_m * k_m)) * (t / l));
	} else {
		tmp = 2.0 / ((((fma((t * t), 2.0, (k_m * k_m)) * (tan(k_m) * sin(k_m))) * t) / l) / l);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(t * t) / l)
	tmp = 0.0
	if (k_m <= 1.85e-144)
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 / cos(k_m)) * Float64((Float64(sin(k_m) * t) ^ 2.0) / l)) * Float64(t / l)));
	elseif (k_m <= 4e-16)
		tmp = Float64(2.0 / Float64(Float64(fma(fma(t_1, 0.3333333333333333, Float64(1.0 / l)), Float64(k_m * k_m), Float64(t_1 * 2.0)) * Float64(k_m * k_m)) * Float64(t / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(t * t), 2.0, Float64(k_m * k_m)) * Float64(tan(k_m) * sin(k_m))) * t) / l) / l));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[k$95$m, 1.85e-144], N[(2.0 / N[(N[(N[(2.0 / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 4e-16], N[(2.0 / N[(N[(N[(N[(t$95$1 * 0.3333333333333333 + N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(t$95$1 * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.8500000000000001e-144

   1. Initial program 57.6%

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

   5. Applied rewrites 68.1%

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

   7. Applied rewrites 85.1%

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

   9. Applied rewrites 87.5%

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

   10. Taylor expanded in t around inf

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

   12. Applied rewrites 73.3%

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

   if 1.8500000000000001e-144 < k < 3.9999999999999999e-16

   1. Initial program 58.7%

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

   3. Taylor expanded in 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

   5. Applied rewrites 67.9%

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

   7. Applied rewrites 87.2%

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

   9. Applied rewrites 93.4%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 93.3%

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

   if 3.9999999999999999e-16 < k

   1. Initial program 38.3%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 64.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 67.5%

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

   10. Applied rewrites 76.0%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{-144}:\\ \;\;\;\;\frac{2}{\left(\frac{2}{\cos k} \cdot \frac{{\left(\sin k \cdot t\right)}^{2}}{\ell}\right) \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.3333333333333333, \frac{1}{\ell}\right), k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right) \cdot \left(\tan k \cdot \sin k\right)\right) \cdot t}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{t \cdot t}{\ell}\\ \mathbf{if}\;k\_m \leq 1.8 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k\_m} \cdot \frac{\frac{\ell}{k\_m}}{t}}{t}\\ \mathbf{elif}\;k\_m \leq 0.0032 \lor \neg \left(k\_m \leq 1.9 \cdot 10^{+145}\right):\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_1, 0.3333333333333333, \frac{1}{\ell}\right), k\_m \cdot k\_m, t\_1 \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(t \cdot t, 2, k\_m \cdot k\_m\right) \cdot \left(\tan k\_m \cdot \sin k\_m\right)\right) \cdot t}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* t t) l)))
   (if (<= k_m 1.8e-102)
     (/ (* (/ l (* t k_m)) (/ (/ l k_m) t)) t)
     (if (or (<= k_m 0.0032) (not (<= k_m 1.9e+145)))
       (/
        2.0
        (*
         (*
          (fma (fma t_1 0.3333333333333333 (/ 1.0 l)) (* k_m k_m) (* t_1 2.0))
          (* k_m k_m))
         (/ t l)))
       (/
        2.0
        (/
         (* (* (fma (* t t) 2.0 (* k_m k_m)) (* (tan k_m) (sin k_m))) t)
         (* l l)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (t * t) / l;
	double tmp;
	if (k_m <= 1.8e-102) {
		tmp = ((l / (t * k_m)) * ((l / k_m) / t)) / t;
	} else if ((k_m <= 0.0032) || !(k_m <= 1.9e+145)) {
		tmp = 2.0 / ((fma(fma(t_1, 0.3333333333333333, (1.0 / l)), (k_m * k_m), (t_1 * 2.0)) * (k_m * k_m)) * (t / l));
	} else {
		tmp = 2.0 / (((fma((t * t), 2.0, (k_m * k_m)) * (tan(k_m) * sin(k_m))) * t) / (l * l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(t * t) / l)
	tmp = 0.0
	if (k_m <= 1.8e-102)
		tmp = Float64(Float64(Float64(l / Float64(t * k_m)) * Float64(Float64(l / k_m) / t)) / t);
	elseif ((k_m <= 0.0032) || !(k_m <= 1.9e+145))
		tmp = Float64(2.0 / Float64(Float64(fma(fma(t_1, 0.3333333333333333, Float64(1.0 / l)), Float64(k_m * k_m), Float64(t_1 * 2.0)) * Float64(k_m * k_m)) * Float64(t / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(Float64(t * t), 2.0, Float64(k_m * k_m)) * Float64(tan(k_m) * sin(k_m))) * t) / Float64(l * l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[k$95$m, 1.8e-102], N[(N[(N[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[Or[LessEqual[k$95$m, 0.0032], N[Not[LessEqual[k$95$m, 1.9e+145]], $MachinePrecision]], N[(2.0 / N[(N[(N[(N[(t$95$1 * 0.3333333333333333 + N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(t$95$1 * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.8e-102

   1. Initial program 58.1%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 59.6%

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

   7. Applied rewrites 63.5%

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

   9. Applied rewrites 67.1%

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

   11. Applied rewrites 74.9%

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

   if 1.8e-102 < k < 0.00320000000000000015 or 1.90000000000000006e145 < k

   1. Initial program 49.8%

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

   5. Applied rewrites 61.0%

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

   7. Applied rewrites 75.4%

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

   9. Applied rewrites 84.9%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 76.9%

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

   if 0.00320000000000000015 < k < 1.90000000000000006e145

   1. Initial program 29.6%

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 81.9%

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

   10. Applied rewrites 85.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k} \cdot \frac{\frac{\ell}{k}}{t}}{t}\\ \mathbf{elif}\;k \leq 0.0032 \lor \neg \left(k \leq 1.9 \cdot 10^{+145}\right):\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.3333333333333333, \frac{1}{\ell}\right), k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right) \cdot \left(\tan k \cdot \sin k\right)\right) \cdot t}{\ell \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{t \cdot t}{\ell}\\ \mathbf{if}\;k\_m \leq 1.8 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k\_m} \cdot \frac{\frac{\ell}{k\_m}}{t}}{t}\\ \mathbf{elif}\;k\_m \leq 0.0005 \lor \neg \left(k\_m \leq 8.6 \cdot 10^{+148}\right):\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_1, 0.3333333333333333, \frac{1}{\ell}\right), k\_m \cdot k\_m, t\_1 \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \frac{\mathsf{fma}\left(t \cdot t, 2, k\_m \cdot k\_m\right) \cdot \left(\tan k\_m \cdot \sin k\_m\right)}{\ell \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* t t) l)))
   (if (<= k_m 1.8e-102)
     (/ (* (/ l (* t k_m)) (/ (/ l k_m) t)) t)
     (if (or (<= k_m 0.0005) (not (<= k_m 8.6e+148)))
       (/
        2.0
        (*
         (*
          (fma (fma t_1 0.3333333333333333 (/ 1.0 l)) (* k_m k_m) (* t_1 2.0))
          (* k_m k_m))
         (/ t l)))
       (/
        2.0
        (*
         t
         (/
          (* (fma (* t t) 2.0 (* k_m k_m)) (* (tan k_m) (sin k_m)))
          (* l l))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (t * t) / l;
	double tmp;
	if (k_m <= 1.8e-102) {
		tmp = ((l / (t * k_m)) * ((l / k_m) / t)) / t;
	} else if ((k_m <= 0.0005) || !(k_m <= 8.6e+148)) {
		tmp = 2.0 / ((fma(fma(t_1, 0.3333333333333333, (1.0 / l)), (k_m * k_m), (t_1 * 2.0)) * (k_m * k_m)) * (t / l));
	} else {
		tmp = 2.0 / (t * ((fma((t * t), 2.0, (k_m * k_m)) * (tan(k_m) * sin(k_m))) / (l * l)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(t * t) / l)
	tmp = 0.0
	if (k_m <= 1.8e-102)
		tmp = Float64(Float64(Float64(l / Float64(t * k_m)) * Float64(Float64(l / k_m) / t)) / t);
	elseif ((k_m <= 0.0005) || !(k_m <= 8.6e+148))
		tmp = Float64(2.0 / Float64(Float64(fma(fma(t_1, 0.3333333333333333, Float64(1.0 / l)), Float64(k_m * k_m), Float64(t_1 * 2.0)) * Float64(k_m * k_m)) * Float64(t / l)));
	else
		tmp = Float64(2.0 / Float64(t * Float64(Float64(fma(Float64(t * t), 2.0, Float64(k_m * k_m)) * Float64(tan(k_m) * sin(k_m))) / Float64(l * l))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[k$95$m, 1.8e-102], N[(N[(N[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[Or[LessEqual[k$95$m, 0.0005], N[Not[LessEqual[k$95$m, 8.6e+148]], $MachinePrecision]], N[(2.0 / N[(N[(N[(N[(t$95$1 * 0.3333333333333333 + N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(t$95$1 * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t * N[(N[(N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.8e-102

   1. Initial program 58.1%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 59.6%

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

   7. Applied rewrites 63.5%

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

   9. Applied rewrites 67.1%

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

   11. Applied rewrites 74.9%

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

   if 1.8e-102 < k < 5.0000000000000001e-4 or 8.6000000000000003e148 < k

   1. Initial program 49.8%

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

   5. Applied rewrites 61.0%

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

   7. Applied rewrites 75.4%

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

   9. Applied rewrites 84.9%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 76.9%

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

   if 5.0000000000000001e-4 < k < 8.6000000000000003e148

   1. Initial program 29.6%

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 81.9%

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

   10. Applied rewrites 75.0%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k} \cdot \frac{\frac{\ell}{k}}{t}}{t}\\ \mathbf{elif}\;k \leq 0.0005 \lor \neg \left(k \leq 8.6 \cdot 10^{+148}\right):\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.3333333333333333, \frac{1}{\ell}\right), k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \frac{\mathsf{fma}\left(t \cdot t, 2, k \cdot k\right) \cdot \left(\tan k \cdot \sin k\right)}{\ell \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 6.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k\_m} \cdot \frac{\frac{\ell}{k\_m}}{t}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{\mathsf{fma}\left(t \cdot t, 2, k\_m \cdot k\_m\right) \cdot \left(\tan k\_m \cdot \sin k\_m\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 6.5e-101)
   (/ (* (/ l (* t k_m)) (/ (/ l k_m) t)) t)
   (/
    2.0
    (*
     (/ t l)
     (/ (* (fma (* t t) 2.0 (* k_m k_m)) (* (tan k_m) (sin k_m))) l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 6.5e-101) {
		tmp = ((l / (t * k_m)) * ((l / k_m) / t)) / t;
	} else {
		tmp = 2.0 / ((t / l) * ((fma((t * t), 2.0, (k_m * k_m)) * (tan(k_m) * sin(k_m))) / l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 6.5e-101)
		tmp = Float64(Float64(Float64(l / Float64(t * k_m)) * Float64(Float64(l / k_m) / t)) / t);
	else
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(Float64(fma(Float64(t * t), 2.0, Float64(k_m * k_m)) * Float64(tan(k_m) * sin(k_m))) / l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 6.5e-101], N[(N[(N[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[(N[(N[(t * t), $MachinePrecision] * 2.0 + N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.4999999999999996e-101

   1. Initial program 58.1%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 59.6%

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

   7. Applied rewrites 63.5%

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

   9. Applied rewrites 67.1%

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

   11. Applied rewrites 74.9%

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

   if 6.4999999999999996e-101 < k

   1. Initial program 43.6%

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

   5. Applied rewrites 65.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 67.3%

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

   10. Applied rewrites 77.6%

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

Alternative 13: 70.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{t \cdot t}{\ell}\\ \mathbf{if}\;t \leq 1.6 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_1, 0.3333333333333333, \frac{1}{\ell}\right), k\_m \cdot k\_m, t\_1 \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k\_m} \cdot \frac{\frac{\ell}{k\_m}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* t t) l)))
   (if (<= t 1.6e-28)
     (/
      2.0
      (*
       (*
        (fma (fma t_1 0.3333333333333333 (/ 1.0 l)) (* k_m k_m) (* t_1 2.0))
        (* k_m k_m))
       (/ t l)))
     (/ (* (/ l (* t k_m)) (/ (/ l k_m) t)) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (t * t) / l;
	double tmp;
	if (t <= 1.6e-28) {
		tmp = 2.0 / ((fma(fma(t_1, 0.3333333333333333, (1.0 / l)), (k_m * k_m), (t_1 * 2.0)) * (k_m * k_m)) * (t / l));
	} else {
		tmp = ((l / (t * k_m)) * ((l / k_m) / t)) / t;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(t * t) / l)
	tmp = 0.0
	if (t <= 1.6e-28)
		tmp = Float64(2.0 / Float64(Float64(fma(fma(t_1, 0.3333333333333333, Float64(1.0 / l)), Float64(k_m * k_m), Float64(t_1 * 2.0)) * Float64(k_m * k_m)) * Float64(t / l)));
	else
		tmp = Float64(Float64(Float64(l / Float64(t * k_m)) * Float64(Float64(l / k_m) / t)) / t);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[t, 1.6e-28], N[(2.0 / N[(N[(N[(N[(t$95$1 * 0.3333333333333333 + N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(t$95$1 * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.59999999999999991e-28

   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

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

   5. Applied rewrites 66.9%

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

   7. Applied rewrites 81.5%

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

   9. Applied rewrites 86.0%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 70.0%

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

   if 1.59999999999999991e-28 < t

   1. Initial program 68.5%

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

   5. Applied rewrites 60.4%

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

   7. Applied rewrites 67.3%

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

   9. Applied rewrites 67.4%

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

   11. Applied rewrites 79.1%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell}, 0.3333333333333333, \frac{1}{\ell}\right), k \cdot k, \frac{t \cdot t}{\ell} \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k} \cdot \frac{\frac{\ell}{k}}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.9% accurate, 7.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k\_m} \cdot \frac{\frac{\ell}{k\_m}}{t}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.05e-28)
   (/ 2.0 (* (* (* k_m k_m) (/ t (* l l))) (* k_m k_m)))
   (/ (* (/ l (* t k_m)) (/ (/ l k_m) t)) t)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.05e-28) {
		tmp = 2.0 / (((k_m * k_m) * (t / (l * l))) * (k_m * k_m));
	} else {
		tmp = ((l / (t * k_m)) * ((l / k_m) / t)) / t;
	}
	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 <= 1.05d-28) then
        tmp = 2.0d0 / (((k_m * k_m) * (t / (l * l))) * (k_m * k_m))
    else
        tmp = ((l / (t * k_m)) * ((l / k_m) / t)) / t
    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 <= 1.05e-28) {
		tmp = 2.0 / (((k_m * k_m) * (t / (l * l))) * (k_m * k_m));
	} else {
		tmp = ((l / (t * k_m)) * ((l / k_m) / t)) / t;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 1.05e-28:
		tmp = 2.0 / (((k_m * k_m) * (t / (l * l))) * (k_m * k_m))
	else:
		tmp = ((l / (t * k_m)) * ((l / k_m) / t)) / t
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.05e-28)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * Float64(t / Float64(l * l))) * Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(l / Float64(t * k_m)) * Float64(Float64(l / k_m) / t)) / t);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 1.05e-28)
		tmp = 2.0 / (((k_m * k_m) * (t / (l * l))) * (k_m * k_m));
	else
		tmp = ((l / (t * k_m)) * ((l / k_m) / t)) / t;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.05e-28], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(t * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.05000000000000003e-28

   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 k around 0

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

   5. Applied rewrites 37.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\left(\frac{1}{t \cdot t} - -0.3333333333333333\right) \cdot {t}^{3}, k \cdot k, {t}^{3} \cdot 2\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 57.9%

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

   if 1.05000000000000003e-28 < t

   1. Initial program 68.5%

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

   5. Applied rewrites 60.4%

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

   7. Applied rewrites 67.3%

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

   9. Applied rewrites 67.4%

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

   11. Applied rewrites 79.1%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k} \cdot \frac{\frac{\ell}{k}}{t}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.4% accurate, 8.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m}}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.05e-28)
   (/ 2.0 (* (* (* k_m k_m) (/ t (* l l))) (* k_m k_m)))
   (* (/ (/ l k_m) t) (/ l (* (* t t) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.05e-28) {
		tmp = 2.0 / (((k_m * k_m) * (t / (l * l))) * (k_m * k_m));
	} else {
		tmp = ((l / k_m) / t) * (l / ((t * t) * 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 (t <= 1.05d-28) then
        tmp = 2.0d0 / (((k_m * k_m) * (t / (l * l))) * (k_m * k_m))
    else
        tmp = ((l / k_m) / t) * (l / ((t * t) * 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 (t <= 1.05e-28) {
		tmp = 2.0 / (((k_m * k_m) * (t / (l * l))) * (k_m * k_m));
	} else {
		tmp = ((l / k_m) / t) * (l / ((t * t) * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 1.05e-28:
		tmp = 2.0 / (((k_m * k_m) * (t / (l * l))) * (k_m * k_m))
	else:
		tmp = ((l / k_m) / t) * (l / ((t * t) * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.05e-28)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * Float64(t / Float64(l * l))) * Float64(k_m * k_m)));
	else
		tmp = Float64(Float64(Float64(l / k_m) / t) * Float64(l / Float64(Float64(t * t) * k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 1.05e-28)
		tmp = 2.0 / (((k_m * k_m) * (t / (l * l))) * (k_m * k_m));
	else
		tmp = ((l / k_m) / t) * (l / ((t * t) * k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.05e-28], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k$95$m), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[(N[(t * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.05000000000000003e-28

   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 k around 0

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

   5. Applied rewrites 37.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\left(\frac{1}{t \cdot t} - -0.3333333333333333\right) \cdot {t}^{3}, k \cdot k, {t}^{3} \cdot 2\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 57.9%

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

   if 1.05000000000000003e-28 < t

   1. Initial program 68.5%

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

   5. Applied rewrites 60.4%

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

   7. Applied rewrites 67.3%

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

   9. Applied rewrites 67.4%

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

   11. Applied rewrites 72.3%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t} \cdot \frac{\ell}{\left(t \cdot t\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\_m\right) \cdot \left(t \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.05e-28)
   (/ 2.0 (* (* (* k_m k_m) (/ t (* l l))) (* k_m k_m)))
   (* l (/ l (* (* (* t t) k_m) (* t k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.05e-28) {
		tmp = 2.0 / (((k_m * k_m) * (t / (l * l))) * (k_m * k_m));
	} else {
		tmp = l * (l / (((t * t) * k_m) * (t * 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 (t <= 1.05d-28) then
        tmp = 2.0d0 / (((k_m * k_m) * (t / (l * l))) * (k_m * k_m))
    else
        tmp = l * (l / (((t * t) * k_m) * (t * 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 (t <= 1.05e-28) {
		tmp = 2.0 / (((k_m * k_m) * (t / (l * l))) * (k_m * k_m));
	} else {
		tmp = l * (l / (((t * t) * k_m) * (t * k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 1.05e-28:
		tmp = 2.0 / (((k_m * k_m) * (t / (l * l))) * (k_m * k_m))
	else:
		tmp = l * (l / (((t * t) * k_m) * (t * k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.05e-28)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * Float64(t / Float64(l * l))) * Float64(k_m * k_m)));
	else
		tmp = Float64(l * Float64(l / Float64(Float64(Float64(t * t) * k_m) * Float64(t * k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 1.05e-28)
		tmp = 2.0 / (((k_m * k_m) * (t / (l * l))) * (k_m * k_m));
	else
		tmp = l * (l / (((t * t) * k_m) * (t * k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.05e-28], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(N[(N[(t * t), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(t * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.05000000000000003e-28

   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 k around 0

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

   5. Applied rewrites 37.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(\left(\frac{1}{t \cdot t} - -0.3333333333333333\right) \cdot {t}^{3}, k \cdot k, {t}^{3} \cdot 2\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 57.9%

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

   if 1.05000000000000003e-28 < t

   1. Initial program 68.5%

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

   5. Applied rewrites 60.4%

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

   7. Applied rewrites 67.5%

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

   9. Applied rewrites 69.2%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(\left(t \cdot t\right) \cdot k\right) \cdot \left(t \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 63.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \ell}{\left(\left(t \cdot k\_m\right) \cdot t\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot t} \cdot \ell}{t \cdot \left(k\_m \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= l 4e-50)
   (/ (* (/ l k_m) l) (* (* (* t k_m) t) t))
   (/ (* (/ l (* t t)) l) (* t (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 4e-50) {
		tmp = ((l / k_m) * l) / (((t * k_m) * t) * t);
	} else {
		tmp = ((l / (t * t)) * l) / (t * (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 (l <= 4d-50) then
        tmp = ((l / k_m) * l) / (((t * k_m) * t) * t)
    else
        tmp = ((l / (t * t)) * l) / (t * (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 (l <= 4e-50) {
		tmp = ((l / k_m) * l) / (((t * k_m) * t) * t);
	} else {
		tmp = ((l / (t * t)) * l) / (t * (k_m * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if l <= 4e-50:
		tmp = ((l / k_m) * l) / (((t * k_m) * t) * t)
	else:
		tmp = ((l / (t * t)) * l) / (t * (k_m * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (l <= 4e-50)
		tmp = Float64(Float64(Float64(l / k_m) * l) / Float64(Float64(Float64(t * k_m) * t) * t));
	else
		tmp = Float64(Float64(Float64(l / Float64(t * t)) * l) / Float64(t * 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 (l <= 4e-50)
		tmp = ((l / k_m) * l) / (((t * k_m) * t) * t);
	else
		tmp = ((l / (t * t)) * l) / (t * (k_m * k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[l, 4e-50], N[(N[(N[(l / k$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(t * k$95$m), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(t * t), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(t * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.00000000000000003e-50

   1. Initial program 55.1%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 57.6%

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

   7. Applied rewrites 61.8%

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

   9. Applied rewrites 65.2%

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

   11. Applied rewrites 66.3%

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

   if 4.00000000000000003e-50 < l

   1. Initial program 48.6%

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

   5. Applied rewrites 53.0%

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

   7. Applied rewrites 53.0%

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

   9. Applied rewrites 62.9%

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

Alternative 18: 64.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \ell}{\left(\left(t \cdot k\_m\right) \cdot t\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(k\_m \cdot \left(\left(t \cdot t\right) \cdot k\_m\right)\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= l 7.5e-41)
   (/ (* (/ l k_m) l) (* (* (* t k_m) t) t))
   (* l (/ l (* (* k_m (* (* t t) k_m)) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 7.5e-41) {
		tmp = ((l / k_m) * l) / (((t * k_m) * t) * t);
	} else {
		tmp = l * (l / ((k_m * ((t * t) * k_m)) * t));
	}
	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 (l <= 7.5d-41) then
        tmp = ((l / k_m) * l) / (((t * k_m) * t) * t)
    else
        tmp = l * (l / ((k_m * ((t * t) * k_m)) * t))
    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 (l <= 7.5e-41) {
		tmp = ((l / k_m) * l) / (((t * k_m) * t) * t);
	} else {
		tmp = l * (l / ((k_m * ((t * t) * k_m)) * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if l <= 7.5e-41:
		tmp = ((l / k_m) * l) / (((t * k_m) * t) * t)
	else:
		tmp = l * (l / ((k_m * ((t * t) * k_m)) * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (l <= 7.5e-41)
		tmp = Float64(Float64(Float64(l / k_m) * l) / Float64(Float64(Float64(t * k_m) * t) * t));
	else
		tmp = Float64(l * Float64(l / Float64(Float64(k_m * Float64(Float64(t * t) * k_m)) * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (l <= 7.5e-41)
		tmp = ((l / k_m) * l) / (((t * k_m) * t) * t);
	else
		tmp = l * (l / ((k_m * ((t * t) * k_m)) * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[l, 7.5e-41], N[(N[(N[(l / k$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(t * k$95$m), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(N[(k$95$m * N[(N[(t * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.50000000000000049e-41

   1. Initial program 55.4%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 57.9%

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

   7. Applied rewrites 62.0%

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

   9. Applied rewrites 65.4%

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

   11. Applied rewrites 66.5%

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

   if 7.50000000000000049e-41 < l

   1. Initial program 47.9%

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

   5. Applied rewrites 52.4%

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

   7. Applied rewrites 52.8%

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

   9. Applied rewrites 59.1%

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

Alternative 19: 63.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-43}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \ell}{\left(k\_m \cdot \left(t \cdot t\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(k\_m \cdot \left(\left(t \cdot t\right) \cdot k\_m\right)\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= l 1e-43)
   (/ (* (/ l k_m) l) (* (* k_m (* t t)) t))
   (* l (/ l (* (* k_m (* (* t t) k_m)) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 1e-43) {
		tmp = ((l / k_m) * l) / ((k_m * (t * t)) * t);
	} else {
		tmp = l * (l / ((k_m * ((t * t) * k_m)) * t));
	}
	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 (l <= 1d-43) then
        tmp = ((l / k_m) * l) / ((k_m * (t * t)) * t)
    else
        tmp = l * (l / ((k_m * ((t * t) * k_m)) * t))
    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 (l <= 1e-43) {
		tmp = ((l / k_m) * l) / ((k_m * (t * t)) * t);
	} else {
		tmp = l * (l / ((k_m * ((t * t) * k_m)) * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if l <= 1e-43:
		tmp = ((l / k_m) * l) / ((k_m * (t * t)) * t)
	else:
		tmp = l * (l / ((k_m * ((t * t) * k_m)) * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (l <= 1e-43)
		tmp = Float64(Float64(Float64(l / k_m) * l) / Float64(Float64(k_m * Float64(t * t)) * t));
	else
		tmp = Float64(l * Float64(l / Float64(Float64(k_m * Float64(Float64(t * t) * k_m)) * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (l <= 1e-43)
		tmp = ((l / k_m) * l) / ((k_m * (t * t)) * t);
	else
		tmp = l * (l / ((k_m * ((t * t) * k_m)) * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[l, 1e-43], N[(N[(N[(l / k$95$m), $MachinePrecision] * l), $MachinePrecision] / N[(N[(k$95$m * N[(t * t), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(N[(k$95$m * N[(N[(t * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.00000000000000008e-43

   1. Initial program 55.4%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 57.9%

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

   7. Applied rewrites 62.0%

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

   9. Applied rewrites 65.4%

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

   if 1.00000000000000008e-43 < l

   1. Initial program 47.9%

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

   5. Applied rewrites 52.4%

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

   7. Applied rewrites 52.8%

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

   9. Applied rewrites 59.1%

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

Alternative 20: 62.6% accurate, 12.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \frac{\ell}{\left(k\_m \cdot \left(\left(t \cdot t\right) \cdot k\_m\right)\right) \cdot t} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* l (/ l (* (* k_m (* (* t t) k_m)) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (l / ((k_m * ((t * t) * k_m)) * 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 * ((t * t) * k_m)) * 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 * ((t * t) * k_m)) * t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (l / ((k_m * ((t * t) * k_m)) * t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(l / Float64(Float64(k_m * Float64(Float64(t * t) * k_m)) * t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (l / ((k_m * ((t * t) * k_m)) * t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(l / N[(N[(k$95$m * N[(N[(t * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.1%

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

3. Taylor expanded in k around 0

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

5. Applied rewrites 56.2%

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

7. Applied rewrites 58.6%

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

9. Applied rewrites 61.7%

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

Alternative 21: 58.2% accurate, 12.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \ell \cdot \frac{\ell}{t \cdot \left(\left(t \cdot t\right) \cdot \left(k\_m \cdot k\_m\right)\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* l (/ l (* t (* (* t t) (* k_m k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return l * (l / (t * ((t * t) * (k_m * k_m))));
}

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 / (t * ((t * t) * (k_m * k_m))))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return l * (l / (t * ((t * t) * (k_m * k_m))));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return l * (l / (t * ((t * t) * (k_m * k_m))))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(l * Float64(l / Float64(t * Float64(Float64(t * t) * Float64(k_m * k_m)))))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = l * (l / (t * ((t * t) * (k_m * k_m))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(l * N[(l / N[(t * N[(N[(t * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.1%

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

3. Taylor expanded in k around 0

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

5. Applied rewrites 56.2%

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

7. Applied rewrites 58.6%

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

9. Applied rewrites 60.4%

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

Reproduce

herbie shell --seed 2025024 
(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))))