Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.8% → 91.7%

Time: 7.8s

Alternatives: 19

Speedup: 2.9×

• 27.4% 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: 54.8% 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: 91.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2.2 \cdot 10^{+84}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \mathsf{fma}\left({\left(\sin k\_m \cdot t\right)}^{2}, 2, {\left(\sin k\_m \cdot k\_m\right)}^{2}\right)}{\ell \cdot \cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\cos k\_m}{t}}{{\sin k\_m}^{2}} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\ell}{k\_m}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2.2e+84)
   (/
    2.0
    (/
     (*
      (/ t l)
      (fma (pow (* (sin k_m) t) 2.0) 2.0 (pow (* (sin k_m) k_m) 2.0)))
     (* l (cos k_m))))
   (*
    (* (* (/ (/ (cos k_m) t) (pow (sin k_m) 2.0)) (/ l k_m)) (/ l k_m))
    2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 2.2e+84) {
		tmp = 2.0 / (((t / l) * fma(pow((sin(k_m) * t), 2.0), 2.0, pow((sin(k_m) * k_m), 2.0))) / (l * cos(k_m)));
	} else {
		tmp = ((((cos(k_m) / t) / pow(sin(k_m), 2.0)) * (l / k_m)) * (l / k_m)) * 2.0;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 2.2e+84)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * fma((Float64(sin(k_m) * t) ^ 2.0), 2.0, (Float64(sin(k_m) * k_m) ^ 2.0))) / Float64(l * cos(k_m))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(cos(k_m) / t) / (sin(k_m) ^ 2.0)) * Float64(l / k_m)) * Float64(l / k_m)) * 2.0);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2.2e+84], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.1999999999999998e84

   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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.1%

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

   7. Applied rewrites 76.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 81.7

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

   9. Applied rewrites 81.7%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-fma.f64 N/A

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

       6. lift-pow.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-sin.f64 N/A

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

       9. lift-pow.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-sin.f64 N/A

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

       12. lift-cos.f64 N/A

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

       13. frac-times N/A

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

   11. Applied rewrites 84.6%

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

   if 2.1999999999999998e84 < k

   1. Initial program 52.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 67.4%

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

   7. Applied rewrites 86.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-/.f64 96.0

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

   9. Applied rewrites 96.0%

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

Alternative 2: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)} \leq 10^{+239}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(2, {\left(\sin k\_m \cdot t\right)}^{2}, {\left(\sin k\_m \cdot k\_m\right)}^{2}\right)}{\cos k\_m \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\cos k\_m}{t}}{{\sin k\_m}^{2}} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\ell}{k\_m}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k_m)) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))
      1e+239)
   (/
    2.0
    (*
     (/
      (fma 2.0 (pow (* (sin k_m) t) 2.0) (pow (* (sin k_m) k_m) 2.0))
      (* (cos k_m) (* l l)))
     t))
   (*
    (* (* (/ (/ (cos k_m) t) (pow (sin k_m) 2.0)) (/ l k_m)) (/ l k_m))
    2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) + 1.0))) <= 1e+239) {
		tmp = 2.0 / ((fma(2.0, pow((sin(k_m) * t), 2.0), pow((sin(k_m) * k_m), 2.0)) / (cos(k_m) * (l * l))) * t);
	} else {
		tmp = ((((cos(k_m) / t) / pow(sin(k_m), 2.0)) * (l / k_m)) * (l / k_m)) * 2.0;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + 1.0))) <= 1e+239)
		tmp = Float64(2.0 / Float64(Float64(fma(2.0, (Float64(sin(k_m) * t) ^ 2.0), (Float64(sin(k_m) * k_m) ^ 2.0)) / Float64(cos(k_m) * Float64(l * l))) * t));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(cos(k_m) / t) / (sin(k_m) ^ 2.0)) * Float64(l / k_m)) * Float64(l / k_m)) * 2.0);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+239], N[(2.0 / N[(N[(N[(2.0 * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 9.99999999999999991e238

   1. Initial program 76.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 88.3%

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

   if 9.99999999999999991e238 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 30.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 57.7%

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

   7. Applied rewrites 75.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-/.f64 82.1

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

   9. Applied rewrites 82.1%

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

Alternative 3: 68.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)} \leq 5 \cdot 10^{+224}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot \frac{\frac{t}{\ell}}{\ell}\right) \cdot \left(k\_m \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k_m)) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))
      5e+224)
   (/ 2.0 (* (* (/ (pow (* k_m t) 2.0) (* l l)) 2.0) t))
   (/ 2.0 (* (* (* k_m k_m) (/ (/ t l) l)) (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) + 1.0))) <= 5e+224) {
		tmp = 2.0 / (((pow((k_m * t), 2.0) / (l * l)) * 2.0) * t);
	} else {
		tmp = 2.0 / (((k_m * k_m) * ((t / l) / l)) * (k_m * k_m));
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0d0 + ((k_m / t) ** 2.0d0)) + 1.0d0))) <= 5d+224) then
        tmp = 2.0d0 / (((((k_m * t) ** 2.0d0) / (l * l)) * 2.0d0) * t)
    else
        tmp = 2.0d0 / (((k_m * k_m) * ((t / l) / l)) * (k_m * k_m))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k_m)) * Math.tan(k_m)) * ((1.0 + Math.pow((k_m / t), 2.0)) + 1.0))) <= 5e+224) {
		tmp = 2.0 / (((Math.pow((k_m * t), 2.0) / (l * l)) * 2.0) * t);
	} else {
		tmp = 2.0 / (((k_m * k_m) * ((t / l) / l)) * (k_m * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k_m)) * math.tan(k_m)) * ((1.0 + math.pow((k_m / t), 2.0)) + 1.0))) <= 5e+224:
		tmp = 2.0 / (((math.pow((k_m * t), 2.0) / (l * l)) * 2.0) * t)
	else:
		tmp = 2.0 / (((k_m * k_m) * ((t / l) / l)) * (k_m * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + 1.0))) <= 5e+224)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / Float64(l * l)) * 2.0) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * Float64(Float64(t / l) / l)) * Float64(k_m * k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + ((k_m / t) ^ 2.0)) + 1.0))) <= 5e+224)
		tmp = 2.0 / (((((k_m * t) ^ 2.0) / (l * l)) * 2.0) * t);
	else
		tmp = 2.0 / (((k_m * k_m) * ((t / l) / l)) * (k_m * k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+224], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 4.99999999999999964e224

   1. Initial program 76.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 80.1

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

   8. Applied rewrites 80.1%

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

   if 4.99999999999999964e224 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 30.3%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + {k}^{2} \cdot \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{{\ell}^{2}} + \frac{{t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{{\ell}^{2}}\right)\right)}} \]

   5. Applied rewrites 18.2%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.6666666666666666 + {t}^{-2}, -0.16666666666666666, {t}^{-2} \cdot 0.3333333333333333\right) + 0.2833333333333333\right) \cdot {t}^{3}, k \cdot k, \mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right)\right)}{\ell \cdot \ell}, k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{6} \cdot {k}^{2} + 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, {k}^{2}, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      12. lift-*.f64 50.3

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 50.3%

      \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(\color{blue}{k} \cdot k\right)} \]

   9. Taylor expanded in k around 0

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

       1. associate-/l* N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. pow2 N/A

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

       6. associate-/l/ N/A

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

       7. lift-/.f64 N/A

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

       8. lift-/.f64 57.4

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

   11. Applied rewrites 57.4%

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

Alternative 4: 68.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)} \leq 5 \cdot 10^{+224}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(k\_m \cdot t\right)}^{2} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot \frac{\frac{t}{\ell}}{\ell}\right) \cdot \left(k\_m \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k_m)) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))
      5e+224)
   (/ (* l l) (* (pow (* k_m t) 2.0) t))
   (/ 2.0 (* (* (* k_m k_m) (/ (/ t l) l)) (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) + 1.0))) <= 5e+224) {
		tmp = (l * l) / (pow((k_m * t), 2.0) * t);
	} else {
		tmp = 2.0 / (((k_m * k_m) * ((t / l) / l)) * (k_m * k_m));
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0d0 + ((k_m / t) ** 2.0d0)) + 1.0d0))) <= 5d+224) then
        tmp = (l * l) / (((k_m * t) ** 2.0d0) * t)
    else
        tmp = 2.0d0 / (((k_m * k_m) * ((t / l) / l)) * (k_m * k_m))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k_m)) * Math.tan(k_m)) * ((1.0 + Math.pow((k_m / t), 2.0)) + 1.0))) <= 5e+224) {
		tmp = (l * l) / (Math.pow((k_m * t), 2.0) * t);
	} else {
		tmp = 2.0 / (((k_m * k_m) * ((t / l) / l)) * (k_m * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k_m)) * math.tan(k_m)) * ((1.0 + math.pow((k_m / t), 2.0)) + 1.0))) <= 5e+224:
		tmp = (l * l) / (math.pow((k_m * t), 2.0) * t)
	else:
		tmp = 2.0 / (((k_m * k_m) * ((t / l) / l)) * (k_m * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + 1.0))) <= 5e+224)
		tmp = Float64(Float64(l * l) / Float64((Float64(k_m * t) ^ 2.0) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * Float64(Float64(t / l) / l)) * Float64(k_m * k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + ((k_m / t) ^ 2.0)) + 1.0))) <= 5e+224)
		tmp = (l * l) / (((k_m * t) ^ 2.0) * t);
	else
		tmp = 2.0 / (((k_m * k_m) * ((t / l) / l)) * (k_m * k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+224], N[(N[(l * l), $MachinePrecision] / N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 4.99999999999999964e224

   1. Initial program 76.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 61.4

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

   5. Applied rewrites 61.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 61.4

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

   7. Applied rewrites 61.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. pow-prod-down N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 78.9

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

   9. Applied rewrites 78.9%

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

   if 4.99999999999999964e224 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 30.3%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + {k}^{2} \cdot \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{{\ell}^{2}} + \frac{{t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{{\ell}^{2}}\right)\right)}} \]

   5. Applied rewrites 18.2%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.6666666666666666 + {t}^{-2}, -0.16666666666666666, {t}^{-2} \cdot 0.3333333333333333\right) + 0.2833333333333333\right) \cdot {t}^{3}, k \cdot k, \mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right)\right)}{\ell \cdot \ell}, k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{6} \cdot {k}^{2} + 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, {k}^{2}, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      12. lift-*.f64 50.3

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 50.3%

      \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(\color{blue}{k} \cdot k\right)} \]

   9. Taylor expanded in k around 0

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

       1. associate-/l* N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. pow2 N/A

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

       6. associate-/l/ N/A

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

       7. lift-/.f64 N/A

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

       8. lift-/.f64 57.4

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

   11. Applied rewrites 57.4%

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

Alternative 5: 65.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)} \leq 5 \cdot 10^{+224}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k\_m \cdot \left(k\_m \cdot {t}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot \frac{\frac{t}{\ell}}{\ell}\right) \cdot \left(k\_m \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k_m)) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))
      5e+224)
   (/ (* l l) (* k_m (* k_m (pow t 3.0))))
   (/ 2.0 (* (* (* k_m k_m) (/ (/ t l) l)) (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) + 1.0))) <= 5e+224) {
		tmp = (l * l) / (k_m * (k_m * pow(t, 3.0)));
	} else {
		tmp = 2.0 / (((k_m * k_m) * ((t / l) / l)) * (k_m * k_m));
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0d0 + ((k_m / t) ** 2.0d0)) + 1.0d0))) <= 5d+224) then
        tmp = (l * l) / (k_m * (k_m * (t ** 3.0d0)))
    else
        tmp = 2.0d0 / (((k_m * k_m) * ((t / l) / l)) * (k_m * k_m))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k_m)) * Math.tan(k_m)) * ((1.0 + Math.pow((k_m / t), 2.0)) + 1.0))) <= 5e+224) {
		tmp = (l * l) / (k_m * (k_m * Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 / (((k_m * k_m) * ((t / l) / l)) * (k_m * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k_m)) * math.tan(k_m)) * ((1.0 + math.pow((k_m / t), 2.0)) + 1.0))) <= 5e+224:
		tmp = (l * l) / (k_m * (k_m * math.pow(t, 3.0)))
	else:
		tmp = 2.0 / (((k_m * k_m) * ((t / l) / l)) * (k_m * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + 1.0))) <= 5e+224)
		tmp = Float64(Float64(l * l) / Float64(k_m * Float64(k_m * (t ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * Float64(Float64(t / l) / l)) * Float64(k_m * k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + ((k_m / t) ^ 2.0)) + 1.0))) <= 5e+224)
		tmp = (l * l) / (k_m * (k_m * (t ^ 3.0)));
	else
		tmp = 2.0 / (((k_m * k_m) * ((t / l) / l)) * (k_m * k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+224], N[(N[(l * l), $MachinePrecision] / N[(k$95$m * N[(k$95$m * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 4.99999999999999964e224

   1. Initial program 76.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 61.4

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

   5. Applied rewrites 61.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 71.0

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

   7. Applied rewrites 71.0%

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

   if 4.99999999999999964e224 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 30.3%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + {k}^{2} \cdot \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{{\ell}^{2}} + \frac{{t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{{\ell}^{2}}\right)\right)}} \]

   5. Applied rewrites 18.2%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.6666666666666666 + {t}^{-2}, -0.16666666666666666, {t}^{-2} \cdot 0.3333333333333333\right) + 0.2833333333333333\right) \cdot {t}^{3}, k \cdot k, \mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right)\right)}{\ell \cdot \ell}, k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{6} \cdot {k}^{2} + 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, {k}^{2}, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      12. lift-*.f64 50.3

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 50.3%

      \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(\color{blue}{k} \cdot k\right)} \]

   9. Taylor expanded in k around 0

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

       1. associate-/l* N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. pow2 N/A

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

       6. associate-/l/ N/A

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

       7. lift-/.f64 N/A

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

       8. lift-/.f64 57.4

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

   11. Applied rewrites 57.4%

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

Alternative 6: 61.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)} \leq 5 \cdot 10^{+224}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(k\_m \cdot k\_m\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot \frac{\frac{t}{\ell}}{\ell}\right) \cdot \left(k\_m \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k_m)) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))
      5e+224)
   (/ (* l l) (* (* k_m k_m) (* (* t t) t)))
   (/ 2.0 (* (* (* k_m k_m) (/ (/ t l) l)) (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) + 1.0))) <= 5e+224) {
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	} else {
		tmp = 2.0 / (((k_m * k_m) * ((t / l) / l)) * (k_m * k_m));
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0d0 + ((k_m / t) ** 2.0d0)) + 1.0d0))) <= 5d+224) then
        tmp = (l * l) / ((k_m * k_m) * ((t * t) * t))
    else
        tmp = 2.0d0 / (((k_m * k_m) * ((t / l) / l)) * (k_m * k_m))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k_m)) * Math.tan(k_m)) * ((1.0 + Math.pow((k_m / t), 2.0)) + 1.0))) <= 5e+224) {
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	} else {
		tmp = 2.0 / (((k_m * k_m) * ((t / l) / l)) * (k_m * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k_m)) * math.tan(k_m)) * ((1.0 + math.pow((k_m / t), 2.0)) + 1.0))) <= 5e+224:
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t))
	else:
		tmp = 2.0 / (((k_m * k_m) * ((t / l) / l)) * (k_m * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + 1.0))) <= 5e+224)
		tmp = Float64(Float64(l * l) / Float64(Float64(k_m * k_m) * Float64(Float64(t * t) * t)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * Float64(Float64(t / l) / l)) * Float64(k_m * k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + ((k_m / t) ^ 2.0)) + 1.0))) <= 5e+224)
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	else
		tmp = 2.0 / (((k_m * k_m) * ((t / l) / l)) * (k_m * k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+224], N[(N[(l * l), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 4.99999999999999964e224

   1. Initial program 76.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 61.4

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

   5. Applied rewrites 61.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 61.4

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

   7. Applied rewrites 61.4%

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

   if 4.99999999999999964e224 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 30.3%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + {k}^{2} \cdot \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{{\ell}^{2}} + \frac{{t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{{\ell}^{2}}\right)\right)}} \]

   5. Applied rewrites 18.2%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.6666666666666666 + {t}^{-2}, -0.16666666666666666, {t}^{-2} \cdot 0.3333333333333333\right) + 0.2833333333333333\right) \cdot {t}^{3}, k \cdot k, \mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right)\right)}{\ell \cdot \ell}, k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{6} \cdot {k}^{2} + 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, {k}^{2}, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      12. lift-*.f64 50.3

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 50.3%

      \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(\color{blue}{k} \cdot k\right)} \]

   9. Taylor expanded in k around 0

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

       1. associate-/l* N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. pow2 N/A

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

       6. associate-/l/ N/A

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

       7. lift-/.f64 N/A

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

       8. lift-/.f64 57.4

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

   11. Applied rewrites 57.4%

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

Alternative 7: 57.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)} \leq 5 \cdot 10^{+224}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(k\_m \cdot k\_m\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k\_m \cdot k\_m} \cdot \frac{1}{\left(k\_m \cdot k\_m\right) \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k_m)) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))
      5e+224)
   (/ (* l l) (* (* k_m k_m) (* (* t t) t)))
   (* (* (/ (* l l) (* k_m k_m)) (/ 1.0 (* (* k_m k_m) t))) 2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) + 1.0))) <= 5e+224) {
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	} else {
		tmp = (((l * l) / (k_m * k_m)) * (1.0 / ((k_m * k_m) * t))) * 2.0;
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0d0 + ((k_m / t) ** 2.0d0)) + 1.0d0))) <= 5d+224) then
        tmp = (l * l) / ((k_m * k_m) * ((t * t) * t))
    else
        tmp = (((l * l) / (k_m * k_m)) * (1.0d0 / ((k_m * k_m) * t))) * 2.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k_m)) * Math.tan(k_m)) * ((1.0 + Math.pow((k_m / t), 2.0)) + 1.0))) <= 5e+224) {
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	} else {
		tmp = (((l * l) / (k_m * k_m)) * (1.0 / ((k_m * k_m) * t))) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k_m)) * math.tan(k_m)) * ((1.0 + math.pow((k_m / t), 2.0)) + 1.0))) <= 5e+224:
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t))
	else:
		tmp = (((l * l) / (k_m * k_m)) * (1.0 / ((k_m * k_m) * t))) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + 1.0))) <= 5e+224)
		tmp = Float64(Float64(l * l) / Float64(Float64(k_m * k_m) * Float64(Float64(t * t) * t)));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k_m * k_m)) * Float64(1.0 / Float64(Float64(k_m * k_m) * t))) * 2.0);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + ((k_m / t) ^ 2.0)) + 1.0))) <= 5e+224)
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	else
		tmp = (((l * l) / (k_m * k_m)) * (1.0 / ((k_m * k_m) * t))) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+224], N[(N[(l * l), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 4.99999999999999964e224

   1. Initial program 76.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 61.4

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

   5. Applied rewrites 61.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 61.4

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

   7. Applied rewrites 61.4%

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

   if 4.99999999999999964e224 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 30.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 57.7%

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

   6. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 53.3

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

   8. Applied rewrites 53.3%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 50.3%

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

Alternative 8: 57.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)} \leq 5 \cdot 10^{+224}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(k\_m \cdot k\_m\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k\_m \cdot k\_m\right) \cdot t}{\ell \cdot \ell} \cdot \left(k\_m \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k_m)) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))
      5e+224)
   (/ (* l l) (* (* k_m k_m) (* (* t t) t)))
   (/ 2.0 (* (/ (* (* k_m k_m) t) (* l l)) (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) + 1.0))) <= 5e+224) {
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	} else {
		tmp = 2.0 / ((((k_m * k_m) * t) / (l * l)) * (k_m * k_m));
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0d0 + ((k_m / t) ** 2.0d0)) + 1.0d0))) <= 5d+224) then
        tmp = (l * l) / ((k_m * k_m) * ((t * t) * t))
    else
        tmp = 2.0d0 / ((((k_m * k_m) * t) / (l * l)) * (k_m * k_m))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k_m)) * Math.tan(k_m)) * ((1.0 + Math.pow((k_m / t), 2.0)) + 1.0))) <= 5e+224) {
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	} else {
		tmp = 2.0 / ((((k_m * k_m) * t) / (l * l)) * (k_m * k_m));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k_m)) * math.tan(k_m)) * ((1.0 + math.pow((k_m / t), 2.0)) + 1.0))) <= 5e+224:
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t))
	else:
		tmp = 2.0 / ((((k_m * k_m) * t) / (l * l)) * (k_m * k_m))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + 1.0))) <= 5e+224)
		tmp = Float64(Float64(l * l) / Float64(Float64(k_m * k_m) * Float64(Float64(t * t) * t)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m * k_m) * t) / Float64(l * l)) * Float64(k_m * k_m)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + ((k_m / t) ^ 2.0)) + 1.0))) <= 5e+224)
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	else
		tmp = 2.0 / ((((k_m * k_m) * t) / (l * l)) * (k_m * k_m));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+224], N[(N[(l * l), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 4.99999999999999964e224

   1. Initial program 76.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 61.4

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

   5. Applied rewrites 61.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 61.4

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

   7. Applied rewrites 61.4%

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

   if 4.99999999999999964e224 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 30.3%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + {k}^{2} \cdot \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{{\ell}^{2}} + \frac{{t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{{\ell}^{2}}\right)\right)}} \]

   5. Applied rewrites 18.2%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.6666666666666666 + {t}^{-2}, -0.16666666666666666, {t}^{-2} \cdot 0.3333333333333333\right) + 0.2833333333333333\right) \cdot {t}^{3}, k \cdot k, \mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right)\right)}{\ell \cdot \ell}, k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{6} \cdot {k}^{2} + 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, {k}^{2}, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      12. lift-*.f64 50.3

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 50.3%

      \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(\color{blue}{k} \cdot k\right)} \]

   9. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 50.3

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

   11. Applied rewrites 50.3%

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

Alternative 9: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 4.3 \cdot 10^{-28}:\\ \;\;\;\;\left(\left(\frac{\frac{\cos k\_m}{t}}{{\sin k\_m}^{2}} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\ell}{k\_m}\right) \cdot 2\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+90}:\\ \;\;\;\;\frac{2}{\left(\left({t}^{3} \cdot \frac{\sin k\_m}{\ell \cdot \ell}\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left({\left(k\_m \cdot t\right)}^{2} \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 4.3e-28)
   (* (* (* (/ (/ (cos k_m) t) (pow (sin k_m) 2.0)) (/ l k_m)) (/ l k_m)) 2.0)
   (if (<= t 2.9e+90)
     (/
      2.0
      (*
       (* (* (pow t 3.0) (/ (sin k_m) (* l l))) (tan k_m))
       (+ (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))
     (/ 2.0 (* (/ (/ t l) l) (* (pow (* k_m t) 2.0) 2.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 4.3e-28) {
		tmp = ((((cos(k_m) / t) / pow(sin(k_m), 2.0)) * (l / k_m)) * (l / k_m)) * 2.0;
	} else if (t <= 2.9e+90) {
		tmp = 2.0 / (((pow(t, 3.0) * (sin(k_m) / (l * l))) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((t / l) / l) * (pow((k_m * t), 2.0) * 2.0));
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 4.3d-28) then
        tmp = ((((cos(k_m) / t) / (sin(k_m) ** 2.0d0)) * (l / k_m)) * (l / k_m)) * 2.0d0
    else if (t <= 2.9d+90) then
        tmp = 2.0d0 / ((((t ** 3.0d0) * (sin(k_m) / (l * l))) * tan(k_m)) * ((1.0d0 + ((k_m / t) ** 2.0d0)) + 1.0d0))
    else
        tmp = 2.0d0 / (((t / l) / l) * (((k_m * t) ** 2.0d0) * 2.0d0))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 4.3e-28) {
		tmp = ((((Math.cos(k_m) / t) / Math.pow(Math.sin(k_m), 2.0)) * (l / k_m)) * (l / k_m)) * 2.0;
	} else if (t <= 2.9e+90) {
		tmp = 2.0 / (((Math.pow(t, 3.0) * (Math.sin(k_m) / (l * l))) * Math.tan(k_m)) * ((1.0 + Math.pow((k_m / t), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / (((t / l) / l) * (Math.pow((k_m * t), 2.0) * 2.0));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 4.3e-28:
		tmp = ((((math.cos(k_m) / t) / math.pow(math.sin(k_m), 2.0)) * (l / k_m)) * (l / k_m)) * 2.0
	elif t <= 2.9e+90:
		tmp = 2.0 / (((math.pow(t, 3.0) * (math.sin(k_m) / (l * l))) * math.tan(k_m)) * ((1.0 + math.pow((k_m / t), 2.0)) + 1.0))
	else:
		tmp = 2.0 / (((t / l) / l) * (math.pow((k_m * t), 2.0) * 2.0))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 4.3e-28)
		tmp = Float64(Float64(Float64(Float64(Float64(cos(k_m) / t) / (sin(k_m) ^ 2.0)) * Float64(l / k_m)) * Float64(l / k_m)) * 2.0);
	elseif (t <= 2.9e+90)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) * Float64(sin(k_m) / Float64(l * l))) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64((Float64(k_m * t) ^ 2.0) * 2.0)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 4.3e-28)
		tmp = ((((cos(k_m) / t) / (sin(k_m) ^ 2.0)) * (l / k_m)) * (l / k_m)) * 2.0;
	elseif (t <= 2.9e+90)
		tmp = 2.0 / ((((t ^ 3.0) * (sin(k_m) / (l * l))) * tan(k_m)) * ((1.0 + ((k_m / t) ^ 2.0)) + 1.0));
	else
		tmp = 2.0 / (((t / l) / l) * (((k_m * t) ^ 2.0) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 4.3e-28], N[(N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t, 2.9e+90], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] / N[(l * 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], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.3e-28

   1. Initial program 55.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 64.2%

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

   7. Applied rewrites 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-/.f64 77.3

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

   9. Applied rewrites 77.3%

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

   if 4.3e-28 < t < 2.9000000000000001e90

   1. Initial program 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 84.0

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

   4. Applied rewrites 84.0%

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

   if 2.9000000000000001e90 < t

   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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 76.5%

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

   7. Applied rewrites 78.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 86.7

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

   9. Applied rewrites 86.7%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 85.1

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

   12. Applied rewrites 85.1%

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

Alternative 10: 85.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.5:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(\sin k\_m \cdot t\right)}^{2} \cdot 2}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\cos k\_m}{t}}{{\sin k\_m}^{2}} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\ell}{k\_m}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.5)
   (/ 2.0 (* (/ (/ t l) l) (/ (* (pow (* (sin k_m) t) 2.0) 2.0) (cos k_m))))
   (*
    (* (* (/ (/ (cos k_m) t) (pow (sin k_m) 2.0)) (/ l k_m)) (/ l k_m))
    2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.5) {
		tmp = 2.0 / (((t / l) / l) * ((pow((sin(k_m) * t), 2.0) * 2.0) / cos(k_m)));
	} else {
		tmp = ((((cos(k_m) / t) / pow(sin(k_m), 2.0)) * (l / k_m)) * (l / k_m)) * 2.0;
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3.5d0) then
        tmp = 2.0d0 / (((t / l) / l) * ((((sin(k_m) * t) ** 2.0d0) * 2.0d0) / cos(k_m)))
    else
        tmp = ((((cos(k_m) / t) / (sin(k_m) ** 2.0d0)) * (l / k_m)) * (l / k_m)) * 2.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.5) {
		tmp = 2.0 / (((t / l) / l) * ((Math.pow((Math.sin(k_m) * t), 2.0) * 2.0) / Math.cos(k_m)));
	} else {
		tmp = ((((Math.cos(k_m) / t) / Math.pow(Math.sin(k_m), 2.0)) * (l / k_m)) * (l / k_m)) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3.5:
		tmp = 2.0 / (((t / l) / l) * ((math.pow((math.sin(k_m) * t), 2.0) * 2.0) / math.cos(k_m)))
	else:
		tmp = ((((math.cos(k_m) / t) / math.pow(math.sin(k_m), 2.0)) * (l / k_m)) * (l / k_m)) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.5)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64(Float64((Float64(sin(k_m) * t) ^ 2.0) * 2.0) / cos(k_m))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(cos(k_m) / t) / (sin(k_m) ^ 2.0)) * Float64(l / k_m)) * Float64(l / k_m)) * 2.0);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 3.5)
		tmp = 2.0 / (((t / l) / l) * ((((sin(k_m) * t) ^ 2.0) * 2.0) / cos(k_m)));
	else
		tmp = ((((cos(k_m) / t) / (sin(k_m) ^ 2.0)) * (l / k_m)) * (l / k_m)) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3.5], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.5

   1. Initial program 59.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.2%

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

   7. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 80.6

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

   9. Applied rewrites 80.6%

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

   10. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. unpow-prod-down N/A

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

       5. lift-sin.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-pow.f64 71.3

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

   12. Applied rewrites 71.3%

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

   if 3.5 < k

   1. Initial program 52.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 64.2%

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

   7. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-/.f64 90.2

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

   9. Applied rewrites 90.2%

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

Alternative 11: 85.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.25:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left({\left(k\_m \cdot t\right)}^{2} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{\cos k\_m}{t}}{{\sin k\_m}^{2}} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\ell}{k\_m}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.25)
   (/ 2.0 (* (/ (/ t l) l) (* (pow (* k_m t) 2.0) 2.0)))
   (*
    (* (* (/ (/ (cos k_m) t) (pow (sin k_m) 2.0)) (/ l k_m)) (/ l k_m))
    2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.25) {
		tmp = 2.0 / (((t / l) / l) * (pow((k_m * t), 2.0) * 2.0));
	} else {
		tmp = ((((cos(k_m) / t) / pow(sin(k_m), 2.0)) * (l / k_m)) * (l / k_m)) * 2.0;
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.25d0) then
        tmp = 2.0d0 / (((t / l) / l) * (((k_m * t) ** 2.0d0) * 2.0d0))
    else
        tmp = ((((cos(k_m) / t) / (sin(k_m) ** 2.0d0)) * (l / k_m)) * (l / k_m)) * 2.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.25) {
		tmp = 2.0 / (((t / l) / l) * (Math.pow((k_m * t), 2.0) * 2.0));
	} else {
		tmp = ((((Math.cos(k_m) / t) / Math.pow(Math.sin(k_m), 2.0)) * (l / k_m)) * (l / k_m)) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.25:
		tmp = 2.0 / (((t / l) / l) * (math.pow((k_m * t), 2.0) * 2.0))
	else:
		tmp = ((((math.cos(k_m) / t) / math.pow(math.sin(k_m), 2.0)) * (l / k_m)) * (l / k_m)) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.25)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64((Float64(k_m * t) ^ 2.0) * 2.0)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(cos(k_m) / t) / (sin(k_m) ^ 2.0)) * Float64(l / k_m)) * Float64(l / k_m)) * 2.0);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.25)
		tmp = 2.0 / (((t / l) / l) * (((k_m * t) ^ 2.0) * 2.0));
	else
		tmp = ((((cos(k_m) / t) / (sin(k_m) ^ 2.0)) * (l / k_m)) * (l / k_m)) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.25], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.25

   1. Initial program 59.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.2%

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

   7. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 80.6

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

   9. Applied rewrites 80.6%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 70.1

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

   12. Applied rewrites 70.1%

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

   if 1.25 < k

   1. Initial program 52.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 64.2%

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

   7. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-/.f64 90.2

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

   9. Applied rewrites 90.2%

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

Alternative 12: 82.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.25:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left({\left(k\_m \cdot t\right)}^{2} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\cos k\_m}{{\sin k\_m}^{2} \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.25)
   (/ 2.0 (* (/ (/ t l) l) (* (pow (* k_m t) 2.0) 2.0)))
   (*
    (* (* (/ l k_m) (/ l k_m)) (/ (cos k_m) (* (pow (sin k_m) 2.0) t)))
    2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.25) {
		tmp = 2.0 / (((t / l) / l) * (pow((k_m * t), 2.0) * 2.0));
	} else {
		tmp = (((l / k_m) * (l / k_m)) * (cos(k_m) / (pow(sin(k_m), 2.0) * t))) * 2.0;
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.25d0) then
        tmp = 2.0d0 / (((t / l) / l) * (((k_m * t) ** 2.0d0) * 2.0d0))
    else
        tmp = (((l / k_m) * (l / k_m)) * (cos(k_m) / ((sin(k_m) ** 2.0d0) * t))) * 2.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.25) {
		tmp = 2.0 / (((t / l) / l) * (Math.pow((k_m * t), 2.0) * 2.0));
	} else {
		tmp = (((l / k_m) * (l / k_m)) * (Math.cos(k_m) / (Math.pow(Math.sin(k_m), 2.0) * t))) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.25:
		tmp = 2.0 / (((t / l) / l) * (math.pow((k_m * t), 2.0) * 2.0))
	else:
		tmp = (((l / k_m) * (l / k_m)) * (math.cos(k_m) / (math.pow(math.sin(k_m), 2.0) * t))) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.25)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64((Float64(k_m * t) ^ 2.0) * 2.0)));
	else
		tmp = Float64(Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * Float64(cos(k_m) / Float64((sin(k_m) ^ 2.0) * t))) * 2.0);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.25)
		tmp = 2.0 / (((t / l) / l) * (((k_m * t) ^ 2.0) * 2.0));
	else
		tmp = (((l / k_m) * (l / k_m)) * (cos(k_m) / ((sin(k_m) ^ 2.0) * t))) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.25], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.25

   1. Initial program 59.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.2%

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

   7. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 80.6

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

   9. Applied rewrites 80.6%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 70.1

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

   12. Applied rewrites 70.1%

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

   if 1.25 < k

   1. Initial program 52.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 64.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 79.7

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

   7. Applied rewrites 79.7%

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

Alternative 13: 82.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.25:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left({\left(k\_m \cdot t\right)}^{2} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\cos k\_m}{t}}{0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.25)
   (/ 2.0 (* (/ (/ t l) l) (* (pow (* k_m t) 2.0) 2.0)))
   (*
    (*
     (/ (/ (cos k_m) t) (- 0.5 (* 0.5 (cos (* 2.0 k_m)))))
     (* (/ l k_m) (/ l k_m)))
    2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.25) {
		tmp = 2.0 / (((t / l) / l) * (pow((k_m * t), 2.0) * 2.0));
	} else {
		tmp = (((cos(k_m) / t) / (0.5 - (0.5 * cos((2.0 * k_m))))) * ((l / k_m) * (l / k_m))) * 2.0;
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.25d0) then
        tmp = 2.0d0 / (((t / l) / l) * (((k_m * t) ** 2.0d0) * 2.0d0))
    else
        tmp = (((cos(k_m) / t) / (0.5d0 - (0.5d0 * cos((2.0d0 * k_m))))) * ((l / k_m) * (l / k_m))) * 2.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.25) {
		tmp = 2.0 / (((t / l) / l) * (Math.pow((k_m * t), 2.0) * 2.0));
	} else {
		tmp = (((Math.cos(k_m) / t) / (0.5 - (0.5 * Math.cos((2.0 * k_m))))) * ((l / k_m) * (l / k_m))) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.25:
		tmp = 2.0 / (((t / l) / l) * (math.pow((k_m * t), 2.0) * 2.0))
	else:
		tmp = (((math.cos(k_m) / t) / (0.5 - (0.5 * math.cos((2.0 * k_m))))) * ((l / k_m) * (l / k_m))) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.25)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64((Float64(k_m * t) ^ 2.0) * 2.0)));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k_m) / t) / Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m))))) * Float64(Float64(l / k_m) * Float64(l / k_m))) * 2.0);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.25)
		tmp = 2.0 / (((t / l) / l) * (((k_m * t) ^ 2.0) * 2.0));
	else
		tmp = (((cos(k_m) / t) / (0.5 - (0.5 * cos((2.0 * k_m))))) * ((l / k_m) * (l / k_m))) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.25], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] / t), $MachinePrecision] / N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.25

   1. Initial program 59.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.2%

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

   7. Applied rewrites 75.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 80.6

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

   9. Applied rewrites 80.6%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 70.1

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

   12. Applied rewrites 70.1%

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

   if 1.25 < k

   1. Initial program 52.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 64.2%

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

   7. Applied rewrites 79.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 79.7

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

   9. Applied rewrites 79.7%

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

Alternative 14: 64.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 260000:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k\_m \cdot k\_m} \cdot \frac{\cos k\_m}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right) \cdot t}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left({\left(k\_m \cdot t\right)}^{2} \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 260000.0)
   (*
    (*
     (/ (* l l) (* k_m k_m))
     (/ (cos k_m) (* (- 0.5 (* 0.5 (cos (* 2.0 k_m)))) t)))
    2.0)
   (/ 2.0 (* (/ (/ t l) l) (* (pow (* k_m t) 2.0) 2.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 260000.0) {
		tmp = (((l * l) / (k_m * k_m)) * (cos(k_m) / ((0.5 - (0.5 * cos((2.0 * k_m)))) * t))) * 2.0;
	} else {
		tmp = 2.0 / (((t / l) / l) * (pow((k_m * t), 2.0) * 2.0));
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 260000.0d0) then
        tmp = (((l * l) / (k_m * k_m)) * (cos(k_m) / ((0.5d0 - (0.5d0 * cos((2.0d0 * k_m)))) * t))) * 2.0d0
    else
        tmp = 2.0d0 / (((t / l) / l) * (((k_m * t) ** 2.0d0) * 2.0d0))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 260000.0) {
		tmp = (((l * l) / (k_m * k_m)) * (Math.cos(k_m) / ((0.5 - (0.5 * Math.cos((2.0 * k_m)))) * t))) * 2.0;
	} else {
		tmp = 2.0 / (((t / l) / l) * (Math.pow((k_m * t), 2.0) * 2.0));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 260000.0:
		tmp = (((l * l) / (k_m * k_m)) * (math.cos(k_m) / ((0.5 - (0.5 * math.cos((2.0 * k_m)))) * t))) * 2.0
	else:
		tmp = 2.0 / (((t / l) / l) * (math.pow((k_m * t), 2.0) * 2.0))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 260000.0)
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k_m * k_m)) * Float64(cos(k_m) / Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m)))) * t))) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64((Float64(k_m * t) ^ 2.0) * 2.0)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 260000.0)
		tmp = (((l * l) / (k_m * k_m)) * (cos(k_m) / ((0.5 - (0.5 * cos((2.0 * k_m)))) * t))) * 2.0;
	else
		tmp = 2.0 / (((t / l) / l) * (((k_m * t) ^ 2.0) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 260000.0], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.6e5

   1. Initial program 56.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 64.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 61.3

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

   7. Applied rewrites 61.3%

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

   if 2.6e5 < t

   1. Initial program 60.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 76.6%

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

   7. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 83.5

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

   9. Applied rewrites 83.5%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 81.0

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

   12. Applied rewrites 81.0%

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

Alternative 15: 60.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\frac{t}{\ell}}{\ell}\\ \mathbf{if}\;t \leq 5.7 \cdot 10^{-111}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k\_m \cdot k\_m, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)}{\cos k\_m}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left({\left(k\_m \cdot t\right)}^{2} \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (/ t l) l)))
   (if (<= t 5.7e-111)
     (/
      2.0
      (*
       t_1
       (/
        (*
         (fma
          (fma -0.6666666666666666 (* t t) 1.0)
          (* k_m k_m)
          (* (* t t) 2.0))
         (* k_m k_m))
        (cos k_m))))
     (if (<= t 2.3e+106)
       (/ (* (/ l k_m) (/ l k_m)) (pow t 3.0))
       (/ 2.0 (* t_1 (* (pow (* k_m t) 2.0) 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (t / l) / l;
	double tmp;
	if (t <= 5.7e-111) {
		tmp = 2.0 / (t_1 * ((fma(fma(-0.6666666666666666, (t * t), 1.0), (k_m * k_m), ((t * t) * 2.0)) * (k_m * k_m)) / cos(k_m)));
	} else if (t <= 2.3e+106) {
		tmp = ((l / k_m) * (l / k_m)) / pow(t, 3.0);
	} else {
		tmp = 2.0 / (t_1 * (pow((k_m * t), 2.0) * 2.0));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(t / l) / l)
	tmp = 0.0
	if (t <= 5.7e-111)
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(fma(fma(-0.6666666666666666, Float64(t * t), 1.0), Float64(k_m * k_m), Float64(Float64(t * t) * 2.0)) * Float64(k_m * k_m)) / cos(k_m))));
	elseif (t <= 2.3e+106)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64((Float64(k_m * t) ^ 2.0) * 2.0)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[t, 5.7e-111], N[(2.0 / N[(t$95$1 * N[(N[(N[(N[(-0.6666666666666666 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.3e+106], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 5.7e-111

   1. Initial program 51.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.5%

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

   7. Applied rewrites 73.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 77.9

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

   9. Applied rewrites 77.9%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. pow2 N/A

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

       17. lift-*.f64 51.6

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

   12. Applied rewrites 51.6%

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

   if 5.7e-111 < t < 2.3000000000000002e106

   1. Initial program 80.4%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 67.4

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

   5. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. associate-/r* N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lower-/.f64 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-pow.f64 78.4

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

   7. Applied rewrites 78.4%

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

   if 2.3000000000000002e106 < t

   1. Initial program 59.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.8%

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

   7. Applied rewrites 79.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 88.1

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

   9. Applied rewrites 88.1%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 86.5

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

   12. Applied rewrites 86.5%

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

Alternative 16: 65.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\frac{t}{\ell}}{\ell}\\ \mathbf{if}\;t \leq 5.7 \cdot 10^{-111}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot t\_1\right) \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_1 \cdot \left({\left(k\_m \cdot t\right)}^{2} \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (/ t l) l)))
   (if (<= t 5.7e-111)
     (/ 2.0 (* (* (* k_m k_m) t_1) (* k_m k_m)))
     (if (<= t 2.3e+106)
       (/ (* (/ l k_m) (/ l k_m)) (pow t 3.0))
       (/ 2.0 (* t_1 (* (pow (* k_m t) 2.0) 2.0)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (t / l) / l;
	double tmp;
	if (t <= 5.7e-111) {
		tmp = 2.0 / (((k_m * k_m) * t_1) * (k_m * k_m));
	} else if (t <= 2.3e+106) {
		tmp = ((l / k_m) * (l / k_m)) / pow(t, 3.0);
	} else {
		tmp = 2.0 / (t_1 * (pow((k_m * t), 2.0) * 2.0));
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t / l) / l
    if (t <= 5.7d-111) then
        tmp = 2.0d0 / (((k_m * k_m) * t_1) * (k_m * k_m))
    else if (t <= 2.3d+106) then
        tmp = ((l / k_m) * (l / k_m)) / (t ** 3.0d0)
    else
        tmp = 2.0d0 / (t_1 * (((k_m * t) ** 2.0d0) * 2.0d0))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = (t / l) / l;
	double tmp;
	if (t <= 5.7e-111) {
		tmp = 2.0 / (((k_m * k_m) * t_1) * (k_m * k_m));
	} else if (t <= 2.3e+106) {
		tmp = ((l / k_m) * (l / k_m)) / Math.pow(t, 3.0);
	} else {
		tmp = 2.0 / (t_1 * (Math.pow((k_m * t), 2.0) * 2.0));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (t / l) / l
	tmp = 0
	if t <= 5.7e-111:
		tmp = 2.0 / (((k_m * k_m) * t_1) * (k_m * k_m))
	elif t <= 2.3e+106:
		tmp = ((l / k_m) * (l / k_m)) / math.pow(t, 3.0)
	else:
		tmp = 2.0 / (t_1 * (math.pow((k_m * t), 2.0) * 2.0))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(t / l) / l)
	tmp = 0.0
	if (t <= 5.7e-111)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * t_1) * Float64(k_m * k_m)));
	elseif (t <= 2.3e+106)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64((Float64(k_m * t) ^ 2.0) * 2.0)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (t / l) / l;
	tmp = 0.0;
	if (t <= 5.7e-111)
		tmp = 2.0 / (((k_m * k_m) * t_1) * (k_m * k_m));
	elseif (t <= 2.3e+106)
		tmp = ((l / k_m) * (l / k_m)) / (t ^ 3.0);
	else
		tmp = 2.0 / (t_1 * (((k_m * t) ^ 2.0) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[t, 5.7e-111], N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.3e+106], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 5.7e-111

   1. Initial program 51.7%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + {k}^{2} \cdot \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{{\ell}^{2}} + \frac{{t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{{\ell}^{2}}\right)\right)}} \]

   5. Applied rewrites 36.5%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.6666666666666666 + {t}^{-2}, -0.16666666666666666, {t}^{-2} \cdot 0.3333333333333333\right) + 0.2833333333333333\right) \cdot {t}^{3}, k \cdot k, \mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right)\right)}{\ell \cdot \ell}, k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{6} \cdot {k}^{2} + 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, {k}^{2}, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      12. lift-*.f64 56.9

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 56.9%

      \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(\color{blue}{k} \cdot k\right)} \]

   9. Taylor expanded in k around 0

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

       1. associate-/l* N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. pow2 N/A

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

       6. associate-/l/ N/A

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

       7. lift-/.f64 N/A

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

       8. lift-/.f64 59.6

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

   11. Applied rewrites 59.6%

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

   if 5.7e-111 < t < 2.3000000000000002e106

   1. Initial program 80.4%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 67.4

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

   5. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. associate-/r* N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lower-/.f64 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-pow.f64 78.4

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

   7. Applied rewrites 78.4%

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

   if 2.3000000000000002e106 < t

   1. Initial program 59.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.8%

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

   7. Applied rewrites 79.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 88.1

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

   9. Applied rewrites 88.1%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 86.5

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

   12. Applied rewrites 86.5%

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

Alternative 17: 54.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) + t \cdot t, k\_m \cdot k\_m, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot {t}^{3}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 9.5e+153)
   (/
    2.0
    (*
     (/ (/ t l) l)
     (*
      (fma
       (+ (fma -0.6666666666666666 (* t t) 1.0) (* t t))
       (* k_m k_m)
       (* (* t t) 2.0))
      (* k_m k_m))))
   (* l (/ l (* (* k_m k_m) (pow t 3.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 9.5e+153) {
		tmp = 2.0 / (((t / l) / l) * (fma((fma(-0.6666666666666666, (t * t), 1.0) + (t * t)), (k_m * k_m), ((t * t) * 2.0)) * (k_m * k_m)));
	} else {
		tmp = l * (l / ((k_m * k_m) * pow(t, 3.0)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 9.5e+153)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64(fma(Float64(fma(-0.6666666666666666, Float64(t * t), 1.0) + Float64(t * t)), Float64(k_m * k_m), Float64(Float64(t * t) * 2.0)) * Float64(k_m * k_m))));
	else
		tmp = Float64(l * Float64(l / Float64(Float64(k_m * k_m) * (t ^ 3.0))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 9.5e+153], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(-0.6666666666666666 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] + N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.4999999999999995e153

   1. Initial program 56.0%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 75.2%

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

   7. Applied rewrites 75.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 79.1

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

   9. Applied rewrites 79.1%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   12. Applied rewrites 53.2%

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

   if 9.4999999999999995e153 < t

   1. Initial program 64.4%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 52.3

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

   5. Applied rewrites 52.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 57.5

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

   7. Applied rewrites 57.5%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) + t \cdot t, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 53.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.18 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) + t \cdot t, k\_m \cdot k\_m, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \mathsf{fma}\left(k\_m \cdot k\_m, 0.16666666666666666, 1\right)}{\ell}}{\ell} \cdot \left(k\_m \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.18e+154)
   (/
    2.0
    (*
     (/ (/ t l) l)
     (*
      (fma
       (+ (fma -0.6666666666666666 (* t t) 1.0) (* t t))
       (* k_m k_m)
       (* (* t t) 2.0))
      (* k_m k_m))))
   (/
    2.0
    (*
     (/
      (/ (* (* (* k_m k_m) t) (fma (* k_m k_m) 0.16666666666666666 1.0)) l)
      l)
     (* k_m k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.18e+154) {
		tmp = 2.0 / (((t / l) / l) * (fma((fma(-0.6666666666666666, (t * t), 1.0) + (t * t)), (k_m * k_m), ((t * t) * 2.0)) * (k_m * k_m)));
	} else {
		tmp = 2.0 / ((((((k_m * k_m) * t) * fma((k_m * k_m), 0.16666666666666666, 1.0)) / l) / l) * (k_m * k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.18e+154)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64(fma(Float64(fma(-0.6666666666666666, Float64(t * t), 1.0) + Float64(t * t)), Float64(k_m * k_m), Float64(Float64(t * t) * 2.0)) * Float64(k_m * k_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(k_m * k_m) * t) * fma(Float64(k_m * k_m), 0.16666666666666666, 1.0)) / l) / l) * Float64(k_m * k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.18e+154], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(-0.6666666666666666 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] + N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(N[(k$95$m * k$95$m), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.18000000000000004e154

   1. Initial program 56.0%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 75.2%

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

   7. Applied rewrites 75.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 79.1

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

   9. Applied rewrites 79.1%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   12. Applied rewrites 53.2%

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

   if 1.18000000000000004e154 < t

   1. Initial program 64.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + {k}^{2} \cdot \left(\frac{{k}^{2} \cdot \left({t}^{3} \cdot \left(\frac{17}{60} + \left(\frac{-1}{6} \cdot \left(\frac{2}{3} + \frac{1}{{t}^{2}}\right) + \frac{1}{3} \cdot \frac{1}{{t}^{2}}\right)\right)\right)}{{\ell}^{2}} + \frac{{t}^{3} \cdot \left(\frac{1}{3} + \frac{1}{{t}^{2}}\right)}{{\ell}^{2}}\right)\right)}} \]

   5. Applied rewrites 52.3%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.6666666666666666 + {t}^{-2}, -0.16666666666666666, {t}^{-2} \cdot 0.3333333333333333\right) + 0.2833333333333333\right) \cdot {t}^{3}, k \cdot k, \mathsf{fma}\left(0.3333333333333333, {t}^{3}, t\right)\right)}{\ell \cdot \ell}, k \cdot k, \frac{2 \cdot {t}^{3}}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(1 + \frac{1}{6} \cdot {k}^{2}\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{6} \cdot {k}^{2} + 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, {k}^{2}, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      9. pow2 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{{\ell}^{2}} \cdot \left(k \cdot k\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      12. lift-*.f64 46.5

          \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

   8. Applied rewrites 46.5%

      \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(0.16666666666666666, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(\color{blue}{k} \cdot k\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(\frac{1}{6}, k \cdot k, 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      7. lift-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right)}{\ell \cdot \ell} \cdot \left(k \cdot k\right)} \]

      8. associate-/r* N/A

         \[\leadsto \frac{2}{\frac{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right)}{\ell}}{\ell} \cdot \left(k \cdot k\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(\frac{1}{6} \cdot \left(k \cdot k\right) + 1\right)}{\ell}}{\ell} \cdot \left(k \cdot k\right)} \]

   10. Applied rewrites 51.4%

       \[\leadsto \frac{2}{\frac{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(k \cdot k, 0.16666666666666666, 1\right)}{\ell}}{\ell} \cdot \left(k \cdot k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.18 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right) + t \cdot t, k \cdot k, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{fma}\left(k \cdot k, 0.16666666666666666, 1\right)}{\ell}}{\ell} \cdot \left(k \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 50.8% accurate, 12.5× speedup

Math

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

FPCore

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

C

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

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = (l * l) / ((k_m * k_m) * ((t * t) * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.4%

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

3. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-pow.f64 50.3

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

5. Applied rewrites 50.3%

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

   1. lift-pow.f64 N/A

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

   2. unpow3 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 50.3

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

7. Applied rewrites 50.3%

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

Reproduce

herbie shell --seed 2025073 
(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))))