Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.4% → 90.5%

Time: 16.0s

Alternatives: 22

Speedup: 7.8×

• 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: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 1.5 \cdot 10^{+142}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \mathsf{fma}\left({\left(\sin k\_m \cdot t\right)}^{2}, 2, {\left(\sin k\_m \cdot k\_m\right)}^{2}\right)}{\ell \cdot \cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \cos k\_m}{{\sin k\_m}^{2} \cdot t} \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.5e+142)
   (/
    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))))
   (* (/ (* (pow (/ l k_m) 2.0) (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 <= 1.5e+142) {
		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 = ((pow((l / k_m), 2.0) * cos(k_m)) / (pow(sin(k_m), 2.0) * t)) * 2.0;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.5e+142)
		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(l / k_m) ^ 2.0) * cos(k_m)) / Float64((sin(k_m) ^ 2.0) * t)) * 2.0);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.5e+142], 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[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.49999999999999987e142

   1. Initial program 54.0%

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

   3. Taylor expanded in t around 0

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

      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 72.9%

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

   7. Applied rewrites 74.0%

      \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \color{blue}{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k}}} \]
   8. 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 81.4

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

   9. Applied rewrites 81.4%

      \[\leadsto \frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{\color{blue}{\mathsf{fma}\left({\left(\sin k \cdot t\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right)}}{\cos k}} \]
   10. 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 85.2%

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

   if 1.49999999999999987e142 < k

   1. Initial program 44.6%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 52.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-cos.f64 N/A

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

      18. lift-sin.f64 N/A

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

      19. lift-pow.f64 N/A

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

      20. lift-*.f64 95.9

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

   7. Applied rewrites 95.9%

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

Alternative 2: 68.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)} \leq 5 \cdot 10^{+237}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(k\_m \cdot t\right) \cdot \left(k\_m \cdot t\right)}{\ell \cdot \ell} \cdot 2\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k\_m}{\ell}\right)}^{2} \cdot k\_m\right) \cdot k\_m\right) \cdot t}\\ \end{array} \end{array} \]

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+237)
   (/ 2.0 (* (* (/ (* (* k_m t) (* k_m t)) (* l l)) 2.0) t))
   (/ 2.0 (* (* (* (pow (/ k_m l) 2.0) k_m) k_m) t))))

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+237) {
		tmp = 2.0 / (((((k_m * t) * (k_m * t)) / (l * l)) * 2.0) * t);
	} else {
		tmp = 2.0 / (((pow((k_m / l), 2.0) * k_m) * k_m) * t);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0d0 + ((k_m / t) ** 2.0d0)) + 1.0d0))) <= 5d+237) then
        tmp = 2.0d0 / (((((k_m * t) * (k_m * t)) / (l * l)) * 2.0d0) * t)
    else
        tmp = 2.0d0 / (((((k_m / l) ** 2.0d0) * k_m) * k_m) * t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((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+237) {
		tmp = 2.0 / (((((k_m * t) * (k_m * t)) / (l * l)) * 2.0) * t);
	} else {
		tmp = 2.0 / (((Math.pow((k_m / l), 2.0) * k_m) * k_m) * t);
	}
	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+237:
		tmp = 2.0 / (((((k_m * t) * (k_m * t)) / (l * l)) * 2.0) * t)
	else:
		tmp = 2.0 / (((math.pow((k_m / l), 2.0) * k_m) * k_m) * t)
	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+237)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m * t) * Float64(k_m * t)) / Float64(l * l)) * 2.0) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m / l) ^ 2.0) * k_m) * k_m) * t));
	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+237)
		tmp = 2.0 / (((((k_m * t) * (k_m * t)) / (l * l)) * 2.0) * t);
	else
		tmp = 2.0 / (((((k_m / l) ^ 2.0) * k_m) * k_m) * t);
	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+237], N[(2.0 / N[(N[(N[(N[(N[(k$95$m * t), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * t), $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)))) < 5.0000000000000002e237

   1. Initial program 78.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.4%

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

   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 80.9

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

   8. Applied rewrites 80.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 80.9

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

   10. Applied rewrites 80.9%

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

   if 5.0000000000000002e237 < (/.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 23.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 51.5%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 36.2%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 43.4

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

   11. Applied rewrites 43.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

   13. Applied rewrites 48.1%

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

Alternative 3: 67.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.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))) < +inf.0

   1. Initial program 81.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 89.2%

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

   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 82.6

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

   8. Applied rewrites 82.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 82.6

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

   10. Applied rewrites 82.6%

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

   if +inf.0 < (*.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 0.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 36.1%

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

   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 24.7

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

   8. Applied rewrites 24.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-/.f64 37.9

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

   10. Applied rewrites 37.9%

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

Alternative 4: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3.6 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t}\\ \mathbf{elif}\;k\_m \leq 2.15 \cdot 10^{+86}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell \cdot \ell} \cdot \frac{\mathsf{fma}\left({\left(\sin k\_m \cdot t\right)}^{2}, 2, {\left(\sin k\_m \cdot k\_m\right)}^{2}\right)}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m}{\ell}\right)}^{2} \cdot \frac{{\sin k\_m}^{2} \cdot t}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.6e-113)
   (/ 2.0 (* (* (/ (/ (pow (* k_m t) 2.0) l) l) 2.0) t))
   (if (<= k_m 2.15e+86)
     (/
      2.0
      (*
       (/ t (* l l))
       (/
        (fma (pow (* (sin k_m) t) 2.0) 2.0 (pow (* (sin k_m) k_m) 2.0))
        (cos k_m))))
     (/ 2.0 (* (pow (/ k_m l) 2.0) (/ (* (pow (sin k_m) 2.0) t) (cos k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.6e-113) {
		tmp = 2.0 / ((((pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	} else if (k_m <= 2.15e+86) {
		tmp = 2.0 / ((t / (l * l)) * (fma(pow((sin(k_m) * t), 2.0), 2.0, pow((sin(k_m) * k_m), 2.0)) / cos(k_m)));
	} else {
		tmp = 2.0 / (pow((k_m / l), 2.0) * ((pow(sin(k_m), 2.0) * t) / cos(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.6e-113)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) / l) * 2.0) * t));
	elseif (k_m <= 2.15e+86)
		tmp = Float64(2.0 / Float64(Float64(t / Float64(l * l)) * Float64(fma((Float64(sin(k_m) * t) ^ 2.0), 2.0, (Float64(sin(k_m) * k_m) ^ 2.0)) / cos(k_m))));
	else
		tmp = Float64(2.0 / Float64((Float64(k_m / l) ^ 2.0) * Float64(Float64((sin(k_m) ^ 2.0) * t) / cos(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3.6e-113], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 2.15e+86], N[(2.0 / N[(N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.59999999999999975e-113

   1. Initial program 55.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 71.9%

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

   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 64.9

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

   8. Applied rewrites 64.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 73.3

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

   10. Applied rewrites 73.3%

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

   if 3.59999999999999975e-113 < k < 2.1500000000000001e86

   1. Initial program 49.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 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 79.3%

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

   if 2.1500000000000001e86 < k

   1. Initial program 46.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 60.2%

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

   7. Applied rewrites 59.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 59.8

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

   9. Applied rewrites 59.8%

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

   10. Taylor expanded in t around 0

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

       1. times-frac N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. times-frac N/A

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

       6. pow2 N/A

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

       7. lower-pow.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-/.f64 N/A

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

       11. lift-sin.f64 N/A

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

       12. lift-pow.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. lift-cos.f64 94.4

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

   12. Applied rewrites 94.4%

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

Alternative 5: 68.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)} \leq 5 \cdot 10^{+237}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(k\_m \cdot t\right) \cdot \left(k\_m \cdot t\right)}{\ell \cdot \ell} \cdot 2\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{k\_m \cdot k\_m}{\ell}}{\ell} \cdot \left(k\_m \cdot k\_m\right)\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       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+237)
   (/ 2.0 (* (* (/ (* (* k_m t) (* k_m t)) (* l l)) 2.0) t))
   (/ 2.0 (* (* (/ (/ (* k_m k_m) l) l) (* k_m k_m)) t))))

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+237) {
		tmp = 2.0 / (((((k_m * t) * (k_m * t)) / (l * l)) * 2.0) * t);
	} else {
		tmp = 2.0 / (((((k_m * k_m) / l) / l) * (k_m * k_m)) * t);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0d0 + ((k_m / t) ** 2.0d0)) + 1.0d0))) <= 5d+237) then
        tmp = 2.0d0 / (((((k_m * t) * (k_m * t)) / (l * l)) * 2.0d0) * t)
    else
        tmp = 2.0d0 / (((((k_m * k_m) / l) / l) * (k_m * k_m)) * t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((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+237) {
		tmp = 2.0 / (((((k_m * t) * (k_m * t)) / (l * l)) * 2.0) * t);
	} else {
		tmp = 2.0 / (((((k_m * k_m) / l) / l) * (k_m * k_m)) * t);
	}
	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+237:
		tmp = 2.0 / (((((k_m * t) * (k_m * t)) / (l * l)) * 2.0) * t)
	else:
		tmp = 2.0 / (((((k_m * k_m) / l) / l) * (k_m * k_m)) * t)
	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+237)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m * t) * Float64(k_m * t)) / Float64(l * l)) * 2.0) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m * k_m) / l) / l) * Float64(k_m * k_m)) * t));
	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+237)
		tmp = 2.0 / (((((k_m * t) * (k_m * t)) / (l * l)) * 2.0) * t);
	else
		tmp = 2.0 / (((((k_m * k_m) / l) / l) * (k_m * k_m)) * t);
	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+237], N[(2.0 / N[(N[(N[(N[(N[(k$95$m * t), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t), $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)))) < 5.0000000000000002e237

   1. Initial program 78.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.4%

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

   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 80.9

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

   8. Applied rewrites 80.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 80.9

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

   10. Applied rewrites 80.9%

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

   if 5.0000000000000002e237 < (/.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 23.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 51.5%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 36.2%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 43.4

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

   11. Applied rewrites 43.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. pow2 N/A

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

       5. associate-/r* N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 47.8

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

   13. Applied rewrites 47.8%

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

Alternative 6: 68.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)} \leq 5 \cdot 10^{+237}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(k\_m \cdot t\right) \cdot \left(k\_m \cdot t\right)}{\ell \cdot \ell} \cdot 2\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{k\_m}{\ell} \cdot \frac{k\_m}{\ell}\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       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+237)
   (/ 2.0 (* (* (/ (* (* k_m t) (* k_m t)) (* l l)) 2.0) t))
   (/ 2.0 (* (* (* (/ k_m l) (/ k_m l)) (* k_m k_m)) t))))

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+237) {
		tmp = 2.0 / (((((k_m * t) * (k_m * t)) / (l * l)) * 2.0) * t);
	} else {
		tmp = 2.0 / ((((k_m / l) * (k_m / l)) * (k_m * k_m)) * t);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0d0 + ((k_m / t) ** 2.0d0)) + 1.0d0))) <= 5d+237) then
        tmp = 2.0d0 / (((((k_m * t) * (k_m * t)) / (l * l)) * 2.0d0) * t)
    else
        tmp = 2.0d0 / ((((k_m / l) * (k_m / l)) * (k_m * k_m)) * t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((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+237) {
		tmp = 2.0 / (((((k_m * t) * (k_m * t)) / (l * l)) * 2.0) * t);
	} else {
		tmp = 2.0 / ((((k_m / l) * (k_m / l)) * (k_m * k_m)) * t);
	}
	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+237:
		tmp = 2.0 / (((((k_m * t) * (k_m * t)) / (l * l)) * 2.0) * t)
	else:
		tmp = 2.0 / ((((k_m / l) * (k_m / l)) * (k_m * k_m)) * t)
	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+237)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m * t) * Float64(k_m * t)) / Float64(l * l)) * 2.0) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k_m / l) * Float64(k_m / l)) * Float64(k_m * k_m)) * t));
	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+237)
		tmp = 2.0 / (((((k_m * t) * (k_m * t)) / (l * l)) * 2.0) * t);
	else
		tmp = 2.0 / ((((k_m / l) * (k_m / l)) * (k_m * k_m)) * t);
	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+237], N[(2.0 / N[(N[(N[(N[(N[(k$95$m * t), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t), $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)))) < 5.0000000000000002e237

   1. Initial program 78.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.4%

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

   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 80.9

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

   8. Applied rewrites 80.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 80.9

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

   10. Applied rewrites 80.9%

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

   if 5.0000000000000002e237 < (/.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 23.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 51.5%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 36.2%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 43.4

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

   11. Applied rewrites 43.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 47.8

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

   13. Applied rewrites 47.8%

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

Alternative 7: 58.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)} \leq 5 \cdot 10^{+237}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(k\_m \cdot k\_m\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot \frac{k\_m}{\ell \cdot \ell}\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       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+237)
   (/ (* l l) (* (* k_m k_m) (* (* t t) t)))
   (/ 2.0 (* (* (* k_m (/ k_m (* l l))) (* k_m k_m)) t))))

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+237) {
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	} else {
		tmp = 2.0 / (((k_m * (k_m / (l * l))) * (k_m * k_m)) * t);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0d0 + ((k_m / t) ** 2.0d0)) + 1.0d0))) <= 5d+237) then
        tmp = (l * l) / ((k_m * k_m) * ((t * t) * t))
    else
        tmp = 2.0d0 / (((k_m * (k_m / (l * l))) * (k_m * k_m)) * t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((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+237) {
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	} else {
		tmp = 2.0 / (((k_m * (k_m / (l * l))) * (k_m * k_m)) * t);
	}
	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+237:
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t))
	else:
		tmp = 2.0 / (((k_m * (k_m / (l * l))) * (k_m * k_m)) * t)
	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+237)
		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 * Float64(k_m / Float64(l * l))) * Float64(k_m * k_m)) * t));
	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+237)
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	else
		tmp = 2.0 / (((k_m * (k_m / (l * l))) * (k_m * k_m)) * t);
	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+237], 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 * N[(k$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t), $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)))) < 5.0000000000000002e237

   1. Initial program 78.5%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 68.5

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

   5. Applied rewrites 68.5%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 68.5

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

   7. Applied rewrites 68.5%

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

   if 5.0000000000000002e237 < (/.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 23.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 51.5%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 36.2%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 43.4

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

   11. Applied rewrites 43.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. pow2 N/A

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

       5. associate-/l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 43.4

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

   13. Applied rewrites 43.4%

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

Alternative 8: 82.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t}\\ \mathbf{elif}\;k\_m \leq 1.95 \cdot 10^{+86}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(\sin k\_m \cdot k\_m\right)}^{2}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m}{\ell}\right)}^{2} \cdot \frac{{\sin k\_m}^{2} \cdot t}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3e-45)
   (/ 2.0 (* (* (/ (/ (pow (* k_m t) 2.0) l) l) 2.0) t))
   (if (<= k_m 1.95e+86)
     (/ 2.0 (* (/ (/ t l) l) (/ (pow (* (sin k_m) k_m) 2.0) (cos k_m))))
     (/ 2.0 (* (pow (/ k_m l) 2.0) (/ (* (pow (sin k_m) 2.0) t) (cos k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3e-45) {
		tmp = 2.0 / ((((pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	} else if (k_m <= 1.95e+86) {
		tmp = 2.0 / (((t / l) / l) * (pow((sin(k_m) * k_m), 2.0) / cos(k_m)));
	} else {
		tmp = 2.0 / (pow((k_m / l), 2.0) * ((pow(sin(k_m), 2.0) * t) / cos(k_m)));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3d-45) then
        tmp = 2.0d0 / ((((((k_m * t) ** 2.0d0) / l) / l) * 2.0d0) * t)
    else if (k_m <= 1.95d+86) then
        tmp = 2.0d0 / (((t / l) / l) * (((sin(k_m) * k_m) ** 2.0d0) / cos(k_m)))
    else
        tmp = 2.0d0 / (((k_m / l) ** 2.0d0) * (((sin(k_m) ** 2.0d0) * t) / cos(k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3e-45) {
		tmp = 2.0 / ((((Math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	} else if (k_m <= 1.95e+86) {
		tmp = 2.0 / (((t / l) / l) * (Math.pow((Math.sin(k_m) * k_m), 2.0) / Math.cos(k_m)));
	} else {
		tmp = 2.0 / (Math.pow((k_m / l), 2.0) * ((Math.pow(Math.sin(k_m), 2.0) * t) / Math.cos(k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 3e-45:
		tmp = 2.0 / ((((math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t)
	elif k_m <= 1.95e+86:
		tmp = 2.0 / (((t / l) / l) * (math.pow((math.sin(k_m) * k_m), 2.0) / math.cos(k_m)))
	else:
		tmp = 2.0 / (math.pow((k_m / l), 2.0) * ((math.pow(math.sin(k_m), 2.0) * t) / math.cos(k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3e-45)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) / l) * 2.0) * t));
	elseif (k_m <= 1.95e+86)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64((Float64(sin(k_m) * k_m) ^ 2.0) / cos(k_m))));
	else
		tmp = Float64(2.0 / Float64((Float64(k_m / l) ^ 2.0) * Float64(Float64((sin(k_m) ^ 2.0) * t) / cos(k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 3e-45)
		tmp = 2.0 / ((((((k_m * t) ^ 2.0) / l) / l) * 2.0) * t);
	elseif (k_m <= 1.95e+86)
		tmp = 2.0 / (((t / l) / l) * (((sin(k_m) * k_m) ^ 2.0) / cos(k_m)));
	else
		tmp = 2.0 / (((k_m / l) ^ 2.0) * (((sin(k_m) ^ 2.0) * t) / cos(k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3e-45], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.95e+86], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.00000000000000011e-45

   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 73.0%

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

   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 66.1

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

   8. Applied rewrites 66.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 73.8

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

   10. Applied rewrites 73.8%

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

   if 3.00000000000000011e-45 < k < 1.9500000000000001e86

   1. Initial program 40.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 69.0%

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

   7. Applied rewrites 73.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 87.1

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

   9. Applied rewrites 87.1%

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

   10. Taylor expanded in t around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-pow.f64 83.2

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

   12. Applied rewrites 83.2%

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

   if 1.9500000000000001e86 < k

   1. Initial program 46.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 60.2%

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

   7. Applied rewrites 59.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 59.8

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

   9. Applied rewrites 59.8%

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

   10. Taylor expanded in t around 0

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

       1. times-frac N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. times-frac N/A

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

       6. pow2 N/A

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

       7. lower-pow.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-/.f64 N/A

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

       11. lift-sin.f64 N/A

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

       12. lift-pow.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. lift-cos.f64 94.4

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

   12. Applied rewrites 94.4%

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

Alternative 9: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t}\\ \mathbf{elif}\;k\_m \leq 1.95 \cdot 10^{+86}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(\sin k\_m \cdot k\_m\right)}^{2}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \cos k\_m}{{\sin k\_m}^{2} \cdot t} \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3e-45)
   (/ 2.0 (* (* (/ (/ (pow (* k_m t) 2.0) l) l) 2.0) t))
   (if (<= k_m 1.95e+86)
     (/ 2.0 (* (/ (/ t l) l) (/ (pow (* (sin k_m) k_m) 2.0) (cos k_m))))
     (* (/ (* (pow (/ l k_m) 2.0) (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 <= 3e-45) {
		tmp = 2.0 / ((((pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	} else if (k_m <= 1.95e+86) {
		tmp = 2.0 / (((t / l) / l) * (pow((sin(k_m) * k_m), 2.0) / cos(k_m)));
	} else {
		tmp = ((pow((l / k_m), 2.0) * 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 <= 3d-45) then
        tmp = 2.0d0 / ((((((k_m * t) ** 2.0d0) / l) / l) * 2.0d0) * t)
    else if (k_m <= 1.95d+86) then
        tmp = 2.0d0 / (((t / l) / l) * (((sin(k_m) * k_m) ** 2.0d0) / cos(k_m)))
    else
        tmp = ((((l / k_m) ** 2.0d0) * 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 <= 3e-45) {
		tmp = 2.0 / ((((Math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	} else if (k_m <= 1.95e+86) {
		tmp = 2.0 / (((t / l) / l) * (Math.pow((Math.sin(k_m) * k_m), 2.0) / Math.cos(k_m)));
	} else {
		tmp = ((Math.pow((l / k_m), 2.0) * 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 <= 3e-45:
		tmp = 2.0 / ((((math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t)
	elif k_m <= 1.95e+86:
		tmp = 2.0 / (((t / l) / l) * (math.pow((math.sin(k_m) * k_m), 2.0) / math.cos(k_m)))
	else:
		tmp = ((math.pow((l / k_m), 2.0) * 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 <= 3e-45)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) / l) * 2.0) * t));
	elseif (k_m <= 1.95e+86)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64((Float64(sin(k_m) * k_m) ^ 2.0) / cos(k_m))));
	else
		tmp = Float64(Float64(Float64((Float64(l / k_m) ^ 2.0) * 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 <= 3e-45)
		tmp = 2.0 / ((((((k_m * t) ^ 2.0) / l) / l) * 2.0) * t);
	elseif (k_m <= 1.95e+86)
		tmp = 2.0 / (((t / l) / l) * (((sin(k_m) * k_m) ^ 2.0) / cos(k_m)));
	else
		tmp = ((((l / k_m) ^ 2.0) * 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, 3e-45], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.95e+86], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.00000000000000011e-45

   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 73.0%

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

   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 66.1

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

   8. Applied rewrites 66.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 73.8

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

   10. Applied rewrites 73.8%

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

   if 3.00000000000000011e-45 < k < 1.9500000000000001e86

   1. Initial program 40.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 69.0%

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

   7. Applied rewrites 73.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 87.1

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

   9. Applied rewrites 87.1%

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

   10. Taylor expanded in t around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-pow.f64 83.2

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

   12. Applied rewrites 83.2%

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

   if 1.9500000000000001e86 < k

   1. Initial program 46.1%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 60.2%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]
   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. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. times-frac N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-cos.f64 N/A

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

      18. lift-sin.f64 N/A

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

      19. lift-pow.f64 N/A

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

      20. lift-*.f64 94.5

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

   7. Applied rewrites 94.5%

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

Alternative 10: 82.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 3 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t}\\ \mathbf{elif}\;k\_m \leq 1.95 \cdot 10^{+86}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{{\left(\sin k\_m \cdot k\_m\right)}^{2}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\cos k\_m}{{\sin k\_m}^{2} \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3e-45)
   (/ 2.0 (* (* (/ (/ (pow (* k_m t) 2.0) l) l) 2.0) t))
   (if (<= k_m 1.95e+86)
     (/ 2.0 (* (/ (/ t l) l) (/ (pow (* (sin k_m) k_m) 2.0) (cos k_m))))
     (*
      (* (* (/ 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 <= 3e-45) {
		tmp = 2.0 / ((((pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	} else if (k_m <= 1.95e+86) {
		tmp = 2.0 / (((t / l) / l) * (pow((sin(k_m) * k_m), 2.0) / cos(k_m)));
	} 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 <= 3d-45) then
        tmp = 2.0d0 / ((((((k_m * t) ** 2.0d0) / l) / l) * 2.0d0) * t)
    else if (k_m <= 1.95d+86) then
        tmp = 2.0d0 / (((t / l) / l) * (((sin(k_m) * k_m) ** 2.0d0) / cos(k_m)))
    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 <= 3e-45) {
		tmp = 2.0 / ((((Math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	} else if (k_m <= 1.95e+86) {
		tmp = 2.0 / (((t / l) / l) * (Math.pow((Math.sin(k_m) * k_m), 2.0) / Math.cos(k_m)));
	} 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 <= 3e-45:
		tmp = 2.0 / ((((math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t)
	elif k_m <= 1.95e+86:
		tmp = 2.0 / (((t / l) / l) * (math.pow((math.sin(k_m) * k_m), 2.0) / math.cos(k_m)))
	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 <= 3e-45)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) / l) * 2.0) * t));
	elseif (k_m <= 1.95e+86)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64((Float64(sin(k_m) * k_m) ^ 2.0) / cos(k_m))));
	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 <= 3e-45)
		tmp = 2.0 / ((((((k_m * t) ^ 2.0) / l) / l) * 2.0) * t);
	elseif (k_m <= 1.95e+86)
		tmp = 2.0 / (((t / l) / l) * (((sin(k_m) * k_m) ^ 2.0) / cos(k_m)));
	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, 3e-45], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.95e+86], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $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 3 regimes

2. if k < 3.00000000000000011e-45

   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 73.0%

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

   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 66.1

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

   8. Applied rewrites 66.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 73.8

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

   10. Applied rewrites 73.8%

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

   if 3.00000000000000011e-45 < k < 1.9500000000000001e86

   1. Initial program 40.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 69.0%

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

   7. Applied rewrites 73.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 87.1

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

   9. Applied rewrites 87.1%

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

   10. Taylor expanded in t around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-pow.f64 83.2

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

   12. Applied rewrites 83.2%

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

   if 1.9500000000000001e86 < k

   1. Initial program 46.1%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 60.2%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]
   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 94.3

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

   7. Applied rewrites 94.3%

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

Alternative 11: 81.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.1 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t}\\ \mathbf{elif}\;k\_m \leq 3 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\cos k\_m}{{\sin k\_m}^{2} \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.1e-115)
   (/ 2.0 (* (* (/ (/ (pow (* k_m t) 2.0) l) l) 2.0) t))
   (if (<= k_m 3e+46)
     (/ 2.0 (* (* (* (* (/ (* t t) l) (/ t l)) (sin k_m)) (tan k_m)) 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 <= 5.1e-115) {
		tmp = 2.0 / ((((pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	} else if (k_m <= 3e+46) {
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * sin(k_m)) * tan(k_m)) * 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 <= 5.1d-115) then
        tmp = 2.0d0 / ((((((k_m * t) ** 2.0d0) / l) / l) * 2.0d0) * t)
    else if (k_m <= 3d+46) then
        tmp = 2.0d0 / ((((((t * t) / l) * (t / l)) * sin(k_m)) * tan(k_m)) * 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 <= 5.1e-115) {
		tmp = 2.0 / ((((Math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	} else if (k_m <= 3e+46) {
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * Math.sin(k_m)) * Math.tan(k_m)) * 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 <= 5.1e-115:
		tmp = 2.0 / ((((math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t)
	elif k_m <= 3e+46:
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * math.sin(k_m)) * math.tan(k_m)) * 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 <= 5.1e-115)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) / l) * 2.0) * t));
	elseif (k_m <= 3e+46)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t * t) / l) * Float64(t / l)) * sin(k_m)) * tan(k_m)) * 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 <= 5.1e-115)
		tmp = 2.0 / ((((((k_m * t) ^ 2.0) / l) / l) * 2.0) * t);
	elseif (k_m <= 3e+46)
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * sin(k_m)) * tan(k_m)) * 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, 5.1e-115], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3e+46], N[(2.0 / N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * 2.0), $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 3 regimes

2. if k < 5.0999999999999997e-115

   1. Initial program 55.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 71.9%

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

   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 64.9

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

   8. Applied rewrites 64.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 73.3

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

   10. Applied rewrites 73.3%

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

   if 5.0999999999999997e-115 < k < 3.00000000000000023e46

   1. Initial program 58.9%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 75.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 81.3

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

   7. Applied rewrites 81.3%

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

   if 3.00000000000000023e46 < k

   1. Initial program 40.6%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 60.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. 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 87.3

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

   7. Applied rewrites 87.3%

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

Alternative 12: 72.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.1 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t}\\ \mathbf{elif}\;k\_m \leq 3 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k\_m \cdot k\_m\right)}^{2}}{\left(\cos k\_m \cdot \ell\right) \cdot \ell} \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.1e-115)
   (/ 2.0 (* (* (/ (/ (pow (* k_m t) 2.0) l) l) 2.0) t))
   (if (<= k_m 3e+46)
     (/ 2.0 (* (* (* (* (/ (* t t) l) (/ t l)) (sin k_m)) (tan k_m)) 2.0))
     (/ 2.0 (* (/ (pow (* (sin k_m) k_m) 2.0) (* (* (cos k_m) l) l)) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.1e-115) {
		tmp = 2.0 / ((((pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	} else if (k_m <= 3e+46) {
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * sin(k_m)) * tan(k_m)) * 2.0);
	} else {
		tmp = 2.0 / ((pow((sin(k_m) * k_m), 2.0) / ((cos(k_m) * l) * l)) * t);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5.1d-115) then
        tmp = 2.0d0 / ((((((k_m * t) ** 2.0d0) / l) / l) * 2.0d0) * t)
    else if (k_m <= 3d+46) then
        tmp = 2.0d0 / ((((((t * t) / l) * (t / l)) * sin(k_m)) * tan(k_m)) * 2.0d0)
    else
        tmp = 2.0d0 / ((((sin(k_m) * k_m) ** 2.0d0) / ((cos(k_m) * l) * l)) * t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.1e-115) {
		tmp = 2.0 / ((((Math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	} else if (k_m <= 3e+46) {
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * Math.sin(k_m)) * Math.tan(k_m)) * 2.0);
	} else {
		tmp = 2.0 / ((Math.pow((Math.sin(k_m) * k_m), 2.0) / ((Math.cos(k_m) * l) * l)) * t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 5.1e-115:
		tmp = 2.0 / ((((math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t)
	elif k_m <= 3e+46:
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * math.sin(k_m)) * math.tan(k_m)) * 2.0)
	else:
		tmp = 2.0 / ((math.pow((math.sin(k_m) * k_m), 2.0) / ((math.cos(k_m) * l) * l)) * t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.1e-115)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) / l) * 2.0) * t));
	elseif (k_m <= 3e+46)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t * t) / l) * Float64(t / l)) * sin(k_m)) * tan(k_m)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k_m) * k_m) ^ 2.0) / Float64(Float64(cos(k_m) * l) * l)) * t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 5.1e-115)
		tmp = 2.0 / ((((((k_m * t) ^ 2.0) / l) / l) * 2.0) * t);
	elseif (k_m <= 3e+46)
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * sin(k_m)) * tan(k_m)) * 2.0);
	else
		tmp = 2.0 / ((((sin(k_m) * k_m) ^ 2.0) / ((cos(k_m) * l) * l)) * t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5.1e-115], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3e+46], N[(2.0 / N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.0999999999999997e-115

   1. Initial program 55.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 71.9%

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

   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 64.9

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

   8. Applied rewrites 64.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 73.3

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

   10. Applied rewrites 73.3%

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

   if 5.0999999999999997e-115 < k < 3.00000000000000023e46

   1. Initial program 58.9%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 75.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 81.3

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

   7. Applied rewrites 81.3%

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

   if 3.00000000000000023e46 < k

   1. Initial program 40.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 60.4%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 18.6%

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

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} \cdot t} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow-prod-down N/A

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

       4. lift-sin.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-pow.f64 N/A

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

       7. *-commutative N/A

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

       8. pow2 N/A

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

       9. associate-*r* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lift-cos.f64 60.5

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

   11. Applied rewrites 60.5%

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

Alternative 13: 74.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.1 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t}\\ \mathbf{elif}\;k\_m \leq 3 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\ell \cdot \ell\right) \cdot \frac{\cos k\_m}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot {\sin k\_m}^{2}}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.1e-115)
   (/ 2.0 (* (* (/ (/ (pow (* k_m t) 2.0) l) l) 2.0) t))
   (if (<= k_m 3e+46)
     (/ 2.0 (* (* (* (* (/ (* t t) l) (/ t l)) (sin k_m)) (tan k_m)) 2.0))
     (*
      (* (* l l) (/ (cos k_m) (* (* (* k_m k_m) t) (pow (sin k_m) 2.0))))
      2.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.1e-115) {
		tmp = 2.0 / ((((pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	} else if (k_m <= 3e+46) {
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * sin(k_m)) * tan(k_m)) * 2.0);
	} else {
		tmp = ((l * l) * (cos(k_m) / (((k_m * k_m) * t) * pow(sin(k_m), 2.0)))) * 2.0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5.1d-115) then
        tmp = 2.0d0 / ((((((k_m * t) ** 2.0d0) / l) / l) * 2.0d0) * t)
    else if (k_m <= 3d+46) then
        tmp = 2.0d0 / ((((((t * t) / l) * (t / l)) * sin(k_m)) * tan(k_m)) * 2.0d0)
    else
        tmp = ((l * l) * (cos(k_m) / (((k_m * k_m) * t) * (sin(k_m) ** 2.0d0)))) * 2.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.1e-115) {
		tmp = 2.0 / ((((Math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	} else if (k_m <= 3e+46) {
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * Math.sin(k_m)) * Math.tan(k_m)) * 2.0);
	} else {
		tmp = ((l * l) * (Math.cos(k_m) / (((k_m * k_m) * t) * Math.pow(Math.sin(k_m), 2.0)))) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 5.1e-115:
		tmp = 2.0 / ((((math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t)
	elif k_m <= 3e+46:
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * math.sin(k_m)) * math.tan(k_m)) * 2.0)
	else:
		tmp = ((l * l) * (math.cos(k_m) / (((k_m * k_m) * t) * math.pow(math.sin(k_m), 2.0)))) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.1e-115)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) / l) * 2.0) * t));
	elseif (k_m <= 3e+46)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t * t) / l) * Float64(t / l)) * sin(k_m)) * tan(k_m)) * 2.0));
	else
		tmp = Float64(Float64(Float64(l * l) * Float64(cos(k_m) / Float64(Float64(Float64(k_m * k_m) * t) * (sin(k_m) ^ 2.0)))) * 2.0);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 5.1e-115)
		tmp = 2.0 / ((((((k_m * t) ^ 2.0) / l) / l) * 2.0) * t);
	elseif (k_m <= 3e+46)
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * sin(k_m)) * tan(k_m)) * 2.0);
	else
		tmp = ((l * l) * (cos(k_m) / (((k_m * k_m) * t) * (sin(k_m) ^ 2.0)))) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5.1e-115], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3e+46], N[(2.0 / N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.0999999999999997e-115

   1. Initial program 55.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 71.9%

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

   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 64.9

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

   8. Applied rewrites 64.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 73.3

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

   10. Applied rewrites 73.3%

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

   if 5.0999999999999997e-115 < k < 3.00000000000000023e46

   1. Initial program 58.9%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 75.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 81.3

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

   7. Applied rewrites 81.3%

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

   if 3.00000000000000023e46 < k

   1. Initial program 40.6%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 60.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. frac-times N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

   7. Applied rewrites 58.3%

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

Alternative 14: 70.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-50}:\\ \;\;\;\;\left(\left(\ell \cdot \frac{\ell}{k\_m \cdot k\_m}\right) \cdot \frac{\cos k\_m}{{\sin k\_m}^{2} \cdot t}\right) \cdot 2\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+67}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{{t}^{3}}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+127}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.4e-50)
   (* (* (* l (/ l (* k_m k_m))) (/ (cos k_m) (* (pow (sin k_m) 2.0) t))) 2.0)
   (if (<= t 4.4e+67)
     (/ (pow (/ l k_m) 2.0) (pow t 3.0))
     (if (<= t 3.1e+127)
       (/ 2.0 (* (* (* (* (/ (* t t) l) (/ t l)) (sin k_m)) (tan k_m)) 2.0))
       (/ 2.0 (* (* (/ (/ (pow (* k_m t) 2.0) l) l) 2.0) t))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.4e-50) {
		tmp = ((l * (l / (k_m * k_m))) * (cos(k_m) / (pow(sin(k_m), 2.0) * t))) * 2.0;
	} else if (t <= 4.4e+67) {
		tmp = pow((l / k_m), 2.0) / pow(t, 3.0);
	} else if (t <= 3.1e+127) {
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * sin(k_m)) * tan(k_m)) * 2.0);
	} else {
		tmp = 2.0 / ((((pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 2.4d-50) then
        tmp = ((l * (l / (k_m * k_m))) * (cos(k_m) / ((sin(k_m) ** 2.0d0) * t))) * 2.0d0
    else if (t <= 4.4d+67) then
        tmp = ((l / k_m) ** 2.0d0) / (t ** 3.0d0)
    else if (t <= 3.1d+127) then
        tmp = 2.0d0 / ((((((t * t) / l) * (t / l)) * sin(k_m)) * tan(k_m)) * 2.0d0)
    else
        tmp = 2.0d0 / ((((((k_m * t) ** 2.0d0) / l) / l) * 2.0d0) * t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.4e-50) {
		tmp = ((l * (l / (k_m * k_m))) * (Math.cos(k_m) / (Math.pow(Math.sin(k_m), 2.0) * t))) * 2.0;
	} else if (t <= 4.4e+67) {
		tmp = Math.pow((l / k_m), 2.0) / Math.pow(t, 3.0);
	} else if (t <= 3.1e+127) {
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * Math.sin(k_m)) * Math.tan(k_m)) * 2.0);
	} else {
		tmp = 2.0 / ((((Math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 2.4e-50:
		tmp = ((l * (l / (k_m * k_m))) * (math.cos(k_m) / (math.pow(math.sin(k_m), 2.0) * t))) * 2.0
	elif t <= 4.4e+67:
		tmp = math.pow((l / k_m), 2.0) / math.pow(t, 3.0)
	elif t <= 3.1e+127:
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * math.sin(k_m)) * math.tan(k_m)) * 2.0)
	else:
		tmp = 2.0 / ((((math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.4e-50)
		tmp = Float64(Float64(Float64(l * Float64(l / Float64(k_m * k_m))) * Float64(cos(k_m) / Float64((sin(k_m) ^ 2.0) * t))) * 2.0);
	elseif (t <= 4.4e+67)
		tmp = Float64((Float64(l / k_m) ^ 2.0) / (t ^ 3.0));
	elseif (t <= 3.1e+127)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t * t) / l) * Float64(t / l)) * sin(k_m)) * tan(k_m)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) / l) * 2.0) * t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 2.4e-50)
		tmp = ((l * (l / (k_m * k_m))) * (cos(k_m) / ((sin(k_m) ^ 2.0) * t))) * 2.0;
	elseif (t <= 4.4e+67)
		tmp = ((l / k_m) ^ 2.0) / (t ^ 3.0);
	elseif (t <= 3.1e+127)
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * sin(k_m)) * tan(k_m)) * 2.0);
	else
		tmp = 2.0 / ((((((k_m * t) ^ 2.0) / l) / l) * 2.0) * t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.4e-50], N[(N[(N[(l * N[(l / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t, 4.4e+67], N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.1e+127], N[(2.0 / N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < 2.40000000000000002e-50

   1. Initial program 49.7%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 62.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 65.9

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

   7. Applied rewrites 65.9%

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

   if 2.40000000000000002e-50 < t < 4.4e67

   1. Initial program 70.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 70.0%

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

   7. Applied rewrites 69.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 70.1

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

   9. Applied rewrites 70.1%

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

   10. Taylor expanded in k around 0

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

       1. associate-/r* N/A

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

       2. pow2 N/A

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

       3. pow2 N/A

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

       4. lower-/.f64 N/A

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

       5. times-frac N/A

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

       6. pow2 N/A

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

       7. lower-pow.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-pow.f64 82.5

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

   12. Applied rewrites 82.5%

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

   if 4.4e67 < t < 3.1000000000000002e127

   1. Initial program 74.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 67.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 93.4

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

   7. Applied rewrites 93.4%

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

   if 3.1000000000000002e127 < t

   1. Initial program 49.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 71.8%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(2 \cdot \frac{{k}^{2} \cdot {t}^{2}}{{\ell}^{2}}\right) \cdot t} \]
   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 66.5

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

   8. Applied rewrites 66.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 83.7

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

   10. Applied rewrites 83.7%

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

Alternative 15: 72.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.1 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t}\\ \mathbf{elif}\;k\_m \leq 3 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k\_m \cdot k\_m} \cdot \frac{\cos k\_m}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right) \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.1e-115)
   (/ 2.0 (* (* (/ (/ (pow (* k_m t) 2.0) l) l) 2.0) t))
   (if (<= k_m 3e+46)
     (/ 2.0 (* (* (* (* (/ (* t t) l) (/ t l)) (sin k_m)) (tan k_m)) 2.0))
     (*
      (*
       (/ (* l l) (* k_m k_m))
       (/ (cos k_m) (* (- 0.5 (* 0.5 (cos (* 2.0 k_m)))) t)))
      2.0))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.1e-115) {
		tmp = 2.0 / ((((pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	} else if (k_m <= 3e+46) {
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * sin(k_m)) * tan(k_m)) * 2.0);
	} else {
		tmp = (((l * l) / (k_m * k_m)) * (cos(k_m) / ((0.5 - (0.5 * cos((2.0 * 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 (k_m <= 5.1d-115) then
        tmp = 2.0d0 / ((((((k_m * t) ** 2.0d0) / l) / l) * 2.0d0) * t)
    else if (k_m <= 3d+46) then
        tmp = 2.0d0 / ((((((t * t) / l) * (t / l)) * sin(k_m)) * tan(k_m)) * 2.0d0)
    else
        tmp = (((l * l) / (k_m * k_m)) * (cos(k_m) / ((0.5d0 - (0.5d0 * cos((2.0d0 * 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 (k_m <= 5.1e-115) {
		tmp = 2.0 / ((((Math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	} else if (k_m <= 3e+46) {
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * Math.sin(k_m)) * Math.tan(k_m)) * 2.0);
	} else {
		tmp = (((l * l) / (k_m * k_m)) * (Math.cos(k_m) / ((0.5 - (0.5 * Math.cos((2.0 * k_m)))) * t))) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 5.1e-115:
		tmp = 2.0 / ((((math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t)
	elif k_m <= 3e+46:
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * math.sin(k_m)) * math.tan(k_m)) * 2.0)
	else:
		tmp = (((l * l) / (k_m * k_m)) * (math.cos(k_m) / ((0.5 - (0.5 * math.cos((2.0 * k_m)))) * t))) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.1e-115)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) / l) * 2.0) * t));
	elseif (k_m <= 3e+46)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t * t) / l) * Float64(t / l)) * sin(k_m)) * tan(k_m)) * 2.0));
	else
		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);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 5.1e-115)
		tmp = 2.0 / ((((((k_m * t) ^ 2.0) / l) / l) * 2.0) * t);
	elseif (k_m <= 3e+46)
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * sin(k_m)) * tan(k_m)) * 2.0);
	else
		tmp = (((l * l) / (k_m * k_m)) * (cos(k_m) / ((0.5 - (0.5 * cos((2.0 * 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[k$95$m, 5.1e-115], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3e+46], N[(2.0 / N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], 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]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.0999999999999997e-115

   1. Initial program 55.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 71.9%

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

   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 64.9

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

   8. Applied rewrites 64.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 73.3

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

   10. Applied rewrites 73.3%

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

   if 5.0999999999999997e-115 < k < 3.00000000000000023e46

   1. Initial program 58.9%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 75.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 81.3

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

   7. Applied rewrites 81.3%

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

   if 3.00000000000000023e46 < k

   1. Initial program 40.6%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   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 60.6%

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

      1. lift-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 60.5

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

   7. Applied rewrites 60.5%

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

Alternative 16: 70.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.1 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t}\\ \mathbf{elif}\;k\_m \leq 1.28 \cdot 10^{+123}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k\_m}{\ell}\right)}^{2} \cdot k\_m\right) \cdot k\_m\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.1e-115)
   (/ 2.0 (* (* (/ (/ (pow (* k_m t) 2.0) l) l) 2.0) t))
   (if (<= k_m 1.28e+123)
     (/ 2.0 (* (* (* (* (/ (* t t) l) (/ t l)) (sin k_m)) (tan k_m)) 2.0))
     (/ 2.0 (* (* (* (pow (/ k_m l) 2.0) k_m) k_m) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.1e-115) {
		tmp = 2.0 / ((((pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	} else if (k_m <= 1.28e+123) {
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * sin(k_m)) * tan(k_m)) * 2.0);
	} else {
		tmp = 2.0 / (((pow((k_m / l), 2.0) * k_m) * k_m) * t);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5.1d-115) then
        tmp = 2.0d0 / ((((((k_m * t) ** 2.0d0) / l) / l) * 2.0d0) * t)
    else if (k_m <= 1.28d+123) then
        tmp = 2.0d0 / ((((((t * t) / l) * (t / l)) * sin(k_m)) * tan(k_m)) * 2.0d0)
    else
        tmp = 2.0d0 / (((((k_m / l) ** 2.0d0) * k_m) * k_m) * t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.1e-115) {
		tmp = 2.0 / ((((Math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	} else if (k_m <= 1.28e+123) {
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * Math.sin(k_m)) * Math.tan(k_m)) * 2.0);
	} else {
		tmp = 2.0 / (((Math.pow((k_m / l), 2.0) * k_m) * k_m) * t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 5.1e-115:
		tmp = 2.0 / ((((math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t)
	elif k_m <= 1.28e+123:
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * math.sin(k_m)) * math.tan(k_m)) * 2.0)
	else:
		tmp = 2.0 / (((math.pow((k_m / l), 2.0) * k_m) * k_m) * t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.1e-115)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) / l) * 2.0) * t));
	elseif (k_m <= 1.28e+123)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t * t) / l) * Float64(t / l)) * sin(k_m)) * tan(k_m)) * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m / l) ^ 2.0) * k_m) * k_m) * t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 5.1e-115)
		tmp = 2.0 / ((((((k_m * t) ^ 2.0) / l) / l) * 2.0) * t);
	elseif (k_m <= 1.28e+123)
		tmp = 2.0 / ((((((t * t) / l) * (t / l)) * sin(k_m)) * tan(k_m)) * 2.0);
	else
		tmp = 2.0 / (((((k_m / l) ^ 2.0) * k_m) * k_m) * t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5.1e-115], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.28e+123], N[(2.0 / N[(N[(N[(N[(N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.0999999999999997e-115

   1. Initial program 55.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 71.9%

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

   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 64.9

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

   8. Applied rewrites 64.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 73.3

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

   10. Applied rewrites 73.3%

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

   if 5.0999999999999997e-115 < k < 1.28000000000000005e123

   1. Initial program 50.5%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 64.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow3 N/A

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

      5. pow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 74.8

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

   7. Applied rewrites 74.8%

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

   if 1.28000000000000005e123 < k

   1. Initial program 44.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 57.0%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 17.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 48.2

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

   11. Applied rewrites 48.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

   13. Applied rewrites 48.7%

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

Alternative 17: 59.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.38 \cdot 10^{-160}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{\ell} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006349206349206349, t \cdot t, 0.044444444444444446\right), k\_m \cdot k\_m, 0.08888888888888889 \cdot \left(t \cdot t\right)\right) - 0.3333333333333333, k\_m \cdot k\_m, \left(t \cdot t\right) \cdot -0.6666666666666666\right) + 1, k\_m \cdot k\_m, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)}{\cos k\_m}}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+65}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.38e-160)
   (/
    2.0
    (*
     (/ (/ t l) l)
     (/
      (*
       (fma
        (+
         (fma
          (-
           (fma
            (fma -0.006349206349206349 (* t t) 0.044444444444444446)
            (* k_m k_m)
            (* 0.08888888888888889 (* t t)))
           0.3333333333333333)
          (* k_m k_m)
          (* (* t t) -0.6666666666666666))
         1.0)
        (* k_m k_m)
        (* (* t t) 2.0))
       (* k_m k_m))
      (cos k_m))))
   (if (<= t 1.9e+65)
     (/ (pow (/ l k_m) 2.0) (pow t 3.0))
     (/ 2.0 (* (* (/ (/ (pow (* k_m t) 2.0) l) l) 2.0) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.38e-160) {
		tmp = 2.0 / (((t / l) / l) * ((fma((fma((fma(fma(-0.006349206349206349, (t * t), 0.044444444444444446), (k_m * k_m), (0.08888888888888889 * (t * t))) - 0.3333333333333333), (k_m * k_m), ((t * t) * -0.6666666666666666)) + 1.0), (k_m * k_m), ((t * t) * 2.0)) * (k_m * k_m)) / cos(k_m)));
	} else if (t <= 1.9e+65) {
		tmp = pow((l / k_m), 2.0) / pow(t, 3.0);
	} else {
		tmp = 2.0 / ((((pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.38e-160)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) / l) * Float64(Float64(fma(Float64(fma(Float64(fma(fma(-0.006349206349206349, Float64(t * t), 0.044444444444444446), Float64(k_m * k_m), Float64(0.08888888888888889 * Float64(t * t))) - 0.3333333333333333), Float64(k_m * k_m), Float64(Float64(t * t) * -0.6666666666666666)) + 1.0), Float64(k_m * k_m), Float64(Float64(t * t) * 2.0)) * Float64(k_m * k_m)) / cos(k_m))));
	elseif (t <= 1.9e+65)
		tmp = Float64((Float64(l / k_m) ^ 2.0) / (t ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) / l) * 2.0) * t));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.38e-160], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.006349206349206349 * N[(t * t), $MachinePrecision] + 0.044444444444444446), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(0.08888888888888889 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $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, 1.9e+65], N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.38e-160

   1. Initial program 51.5%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 72.6%

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

   7. Applied rewrites 73.3%

      \[\leadsto \frac{2}{\frac{t}{\ell \cdot \ell} \cdot \color{blue}{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k}}} \]
   8. 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 80.3

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

   9. Applied rewrites 80.3%

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

   10. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   12. Applied rewrites 52.6%

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

   if 1.38e-160 < t < 1.90000000000000006e65

   1. Initial program 53.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 60.1%

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

   7. Applied rewrites 61.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 61.8

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

   9. Applied rewrites 61.8%

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

   10. Taylor expanded in k around 0

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

       1. associate-/r* N/A

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

       2. pow2 N/A

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

       3. pow2 N/A

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

       4. lower-/.f64 N/A

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

       5. times-frac N/A

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

       6. pow2 N/A

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

       7. lower-pow.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-pow.f64 58.8

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

   12. Applied rewrites 58.8%

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

   if 1.90000000000000006e65 < t

   1. Initial program 56.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 75.3%

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

   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 71.3

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

   8. Applied rewrites 71.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 83.4

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

   10. Applied rewrites 83.4%

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

Alternative 18: 64.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{-116}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k\_m}{\ell}\right)}^{2} \cdot k\_m\right) \cdot k\_m\right) \cdot t}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+65}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.45e-116)
   (/ 2.0 (* (* (* (pow (/ k_m l) 2.0) k_m) k_m) t))
   (if (<= t 1.9e+65)
     (/ (pow (/ l k_m) 2.0) (pow t 3.0))
     (/ 2.0 (* (* (/ (/ (pow (* k_m t) 2.0) l) l) 2.0) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.45e-116) {
		tmp = 2.0 / (((pow((k_m / l), 2.0) * k_m) * k_m) * t);
	} else if (t <= 1.9e+65) {
		tmp = pow((l / k_m), 2.0) / pow(t, 3.0);
	} else {
		tmp = 2.0 / ((((pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 1.45d-116) then
        tmp = 2.0d0 / (((((k_m / l) ** 2.0d0) * k_m) * k_m) * t)
    else if (t <= 1.9d+65) then
        tmp = ((l / k_m) ** 2.0d0) / (t ** 3.0d0)
    else
        tmp = 2.0d0 / ((((((k_m * t) ** 2.0d0) / l) / l) * 2.0d0) * t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.45e-116) {
		tmp = 2.0 / (((Math.pow((k_m / l), 2.0) * k_m) * k_m) * t);
	} else if (t <= 1.9e+65) {
		tmp = Math.pow((l / k_m), 2.0) / Math.pow(t, 3.0);
	} else {
		tmp = 2.0 / ((((Math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 1.45e-116:
		tmp = 2.0 / (((math.pow((k_m / l), 2.0) * k_m) * k_m) * t)
	elif t <= 1.9e+65:
		tmp = math.pow((l / k_m), 2.0) / math.pow(t, 3.0)
	else:
		tmp = 2.0 / ((((math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.45e-116)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m / l) ^ 2.0) * k_m) * k_m) * t));
	elseif (t <= 1.9e+65)
		tmp = Float64((Float64(l / k_m) ^ 2.0) / (t ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) / l) * 2.0) * t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 1.45e-116)
		tmp = 2.0 / (((((k_m / l) ^ 2.0) * k_m) * k_m) * t);
	elseif (t <= 1.9e+65)
		tmp = ((l / k_m) ^ 2.0) / (t ^ 3.0);
	else
		tmp = 2.0 / ((((((k_m * t) ^ 2.0) / l) / l) * 2.0) * t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.45e-116], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.9e+65], N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.4499999999999999e-116

   1. Initial program 51.1%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 71.0%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 38.7%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 58.8

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

   11. Applied rewrites 58.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

   13. Applied rewrites 61.0%

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

   if 1.4499999999999999e-116 < t < 1.90000000000000006e65

   1. Initial program 56.3%

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

   3. Taylor expanded in t around 0

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

      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 63.8%

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

   7. Applied rewrites 63.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 63.7

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

   9. Applied rewrites 63.7%

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

   10. Taylor expanded in k around 0

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

       1. associate-/r* N/A

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

       2. pow2 N/A

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

       3. pow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{{t}^{3}} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{\color{blue}{{t}^{3}}} \]

       5. times-frac N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\color{blue}{t}}^{3}} \]

       6. pow2 N/A

          \[\leadsto \frac{{\left(\frac{\ell}{k}\right)}^{2}}{{\color{blue}{t}}^{3}} \]

       7. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(\frac{\ell}{k}\right)}^{2}}{{\color{blue}{t}}^{3}} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t}^{3}} \]

       9. lift-pow.f64 62.4

          \[\leadsto \frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t}^{\color{blue}{3}}} \]

   12. Applied rewrites 62.4%

       \[\leadsto \color{blue}{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t}^{3}}} \]

   if 1.90000000000000006e65 < t

   1. Initial program 56.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   5. Applied rewrites 75.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   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 71.3

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

   8. Applied rewrites 71.3%

      \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot {t}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

      6. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\frac{\frac{{k}^{2} \cdot {t}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\frac{{k}^{2} \cdot {t}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\frac{{k}^{2} \cdot {t}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t} \]

      9. unpow-prod-down N/A

         \[\leadsto \frac{2}{\left(\frac{\frac{{\left(k \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\frac{{\left(k \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t} \]

      11. lift-*.f64 83.4

          \[\leadsto \frac{2}{\left(\frac{\frac{{\left(k \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t} \]

   10. Applied rewrites 83.4%

       \[\leadsto \frac{2}{\left(\frac{\frac{{\left(k \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 63.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.08 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\frac{k\_m}{\ell}\right)}^{2} \cdot k\_m\right) \cdot k\_m\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.08e-45)
   (/ 2.0 (* (* (* (pow (/ k_m l) 2.0) k_m) k_m) t))
   (/ 2.0 (* (* (/ (/ (pow (* k_m t) 2.0) l) l) 2.0) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.08e-45) {
		tmp = 2.0 / (((pow((k_m / l), 2.0) * k_m) * k_m) * t);
	} else {
		tmp = 2.0 / ((((pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 1.08d-45) then
        tmp = 2.0d0 / (((((k_m / l) ** 2.0d0) * k_m) * k_m) * t)
    else
        tmp = 2.0d0 / ((((((k_m * t) ** 2.0d0) / l) / l) * 2.0d0) * t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.08e-45) {
		tmp = 2.0 / (((Math.pow((k_m / l), 2.0) * k_m) * k_m) * t);
	} else {
		tmp = 2.0 / ((((Math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 1.08e-45:
		tmp = 2.0 / (((math.pow((k_m / l), 2.0) * k_m) * k_m) * t)
	else:
		tmp = 2.0 / ((((math.pow((k_m * t), 2.0) / l) / l) * 2.0) * t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.08e-45)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m / l) ^ 2.0) * k_m) * k_m) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) / l) * 2.0) * t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 1.08e-45)
		tmp = 2.0 / (((((k_m / l) ^ 2.0) * k_m) * k_m) * t);
	else
		tmp = 2.0 / ((((((k_m * t) ^ 2.0) / l) / l) * 2.0) * t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.08e-45], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.08e-45

   1. Initial program 49.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   5. Applied rewrites 69.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \left(\left(\frac{-2}{3} \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) - -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right) \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \left(\left(\frac{-2}{3} \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) - -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right) \cdot {k}^{2}\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \left(\left(\frac{-2}{3} \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) - -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right) \cdot {k}^{2}\right) \cdot t} \]

   8. Applied rewrites 37.7%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       2. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       4. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       5. lift-*.f64 56.1

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]

   11. Applied rewrites 56.1%

       \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]

   13. Applied rewrites 58.2%

       \[\leadsto \frac{2}{\left(\left({\left(\frac{k}{\ell}\right)}^{2} \cdot k\right) \cdot k\right) \cdot t} \]

   if 1.08e-45 < t

   1. Initial program 62.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 74.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   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 69.8

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

   8. Applied rewrites 69.8%

      \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot {t}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

      6. associate-/r* N/A

         \[\leadsto \frac{2}{\left(\frac{\frac{{k}^{2} \cdot {t}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\frac{{k}^{2} \cdot {t}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\frac{{k}^{2} \cdot {t}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t} \]

      9. unpow-prod-down N/A

         \[\leadsto \frac{2}{\left(\frac{\frac{{\left(k \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{\frac{{\left(k \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t} \]

      11. lift-*.f64 80.9

          \[\leadsto \frac{2}{\left(\frac{\frac{{\left(k \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t} \]

   10. Applied rewrites 80.9%

       \[\leadsto \frac{2}{\left(\frac{\frac{{\left(k \cdot t\right)}^{2}}{\ell}}{\ell} \cdot 2\right) \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 59.4% accurate, 7.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 3.9 \cdot 10^{-52}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot \frac{k\_m}{\ell \cdot \ell}\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{\left(k\_m \cdot t\right) \cdot \left(k\_m \cdot t\right)}{\ell \cdot \ell} \cdot 2\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 3.9e-52)
   (/ 2.0 (* (* (* k_m (/ k_m (* l l))) (* k_m k_m)) t))
   (/ 2.0 (* (* (/ (* (* k_m t) (* k_m t)) (* l l)) 2.0) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3.9e-52) {
		tmp = 2.0 / (((k_m * (k_m / (l * l))) * (k_m * k_m)) * t);
	} else {
		tmp = 2.0 / (((((k_m * t) * (k_m * t)) / (l * l)) * 2.0) * t);
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 3.9d-52) then
        tmp = 2.0d0 / (((k_m * (k_m / (l * l))) * (k_m * k_m)) * t)
    else
        tmp = 2.0d0 / (((((k_m * t) * (k_m * t)) / (l * l)) * 2.0d0) * t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3.9e-52) {
		tmp = 2.0 / (((k_m * (k_m / (l * l))) * (k_m * k_m)) * t);
	} else {
		tmp = 2.0 / (((((k_m * t) * (k_m * t)) / (l * l)) * 2.0) * t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 3.9e-52:
		tmp = 2.0 / (((k_m * (k_m / (l * l))) * (k_m * k_m)) * t)
	else:
		tmp = 2.0 / (((((k_m * t) * (k_m * t)) / (l * l)) * 2.0) * t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 3.9e-52)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * Float64(k_m / Float64(l * l))) * Float64(k_m * k_m)) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(k_m * t) * Float64(k_m * t)) / Float64(l * l)) * 2.0) * t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 3.9e-52)
		tmp = 2.0 / (((k_m * (k_m / (l * l))) * (k_m * k_m)) * t);
	else
		tmp = 2.0 / (((((k_m * t) * (k_m * t)) / (l * l)) * 2.0) * t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 3.9e-52], N[(2.0 / N[(N[(N[(k$95$m * N[(k$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(k$95$m * t), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.90000000000000018e-52

   1. Initial program 49.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 69.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \left(\left(\frac{-2}{3} \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) - -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right) \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \left(\left(\frac{-2}{3} \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) - -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right) \cdot {k}^{2}\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \left(\left(\frac{-2}{3} \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) - -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right) \cdot {k}^{2}\right) \cdot t} \]

   8. Applied rewrites 37.9%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       2. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       4. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       5. lift-*.f64 56.4

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]

   11. Applied rewrites 56.4%

       \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       4. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       5. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot \frac{k}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot \frac{k}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot \frac{k}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

       8. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

       9. lift-*.f64 56.4

          \[\leadsto \frac{2}{\left(\left(k \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   13. Applied rewrites 56.4%

       \[\leadsto \frac{2}{\left(\left(k \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   if 3.90000000000000018e-52 < t

   1. Initial program 61.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   5. Applied rewrites 73.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   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 68.9

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

   8. Applied rewrites 68.9%

      \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

      3. unpow2 N/A

         \[\leadsto \frac{2}{\left(\frac{\left(k \cdot t\right) \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\left(k \cdot t\right) \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{\left(k \cdot t\right) \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

      6. lift-*.f64 68.9

         \[\leadsto \frac{2}{\left(\frac{\left(k \cdot t\right) \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]

   10. Applied rewrites 68.9%

       \[\leadsto \frac{2}{\left(\frac{\left(k \cdot t\right) \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot 2\right) \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 56.8% accurate, 7.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{2}{\left(\left(k\_m \cdot \frac{k\_m}{\ell \cdot \ell}\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2e-50)
   (/ 2.0 (* (* (* k_m (/ k_m (* l l))) (* k_m k_m)) t))
   (/ 2.0 (* (* (* (* t (/ t (* l l))) 2.0) (* k_m k_m)) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2e-50) {
		tmp = 2.0 / (((k_m * (k_m / (l * l))) * (k_m * k_m)) * t);
	} else {
		tmp = 2.0 / ((((t * (t / (l * l))) * 2.0) * (k_m * k_m)) * t);
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 2d-50) then
        tmp = 2.0d0 / (((k_m * (k_m / (l * l))) * (k_m * k_m)) * t)
    else
        tmp = 2.0d0 / ((((t * (t / (l * l))) * 2.0d0) * (k_m * k_m)) * t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2e-50) {
		tmp = 2.0 / (((k_m * (k_m / (l * l))) * (k_m * k_m)) * t);
	} else {
		tmp = 2.0 / ((((t * (t / (l * l))) * 2.0) * (k_m * k_m)) * t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 2e-50:
		tmp = 2.0 / (((k_m * (k_m / (l * l))) * (k_m * k_m)) * t)
	else:
		tmp = 2.0 / ((((t * (t / (l * l))) * 2.0) * (k_m * k_m)) * t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2e-50)
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m * Float64(k_m / Float64(l * l))) * Float64(k_m * k_m)) * t));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t * Float64(t / Float64(l * l))) * 2.0) * Float64(k_m * k_m)) * t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 2e-50)
		tmp = 2.0 / (((k_m * (k_m / (l * l))) * (k_m * k_m)) * t);
	else
		tmp = 2.0 / ((((t * (t / (l * l))) * 2.0) * (k_m * k_m)) * t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2e-50], N[(2.0 / N[(N[(N[(k$95$m * N[(k$95$m / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t * N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.00000000000000002e-50

   1. Initial program 49.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   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 69.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \left(\left(\frac{-2}{3} \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) - -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right) \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \left(\left(\frac{-2}{3} \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) - -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right) \cdot {k}^{2}\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \left(\left(\frac{-2}{3} \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) - -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right) \cdot {k}^{2}\right) \cdot t} \]

   8. Applied rewrites 37.9%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       2. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       4. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       5. lift-*.f64 56.4

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]

   11. Applied rewrites 56.4%

       \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\ell \cdot \ell} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       4. pow2 N/A

          \[\leadsto \frac{2}{\left(\frac{k \cdot k}{{\ell}^{2}} \cdot \left(k \cdot k\right)\right) \cdot t} \]

       5. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot \frac{k}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot \frac{k}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot \frac{k}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

       8. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(k \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

       9. lift-*.f64 56.4

          \[\leadsto \frac{2}{\left(\left(k \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   13. Applied rewrites 56.4%

       \[\leadsto \frac{2}{\left(\left(k \cdot \frac{k}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   if 2.00000000000000002e-50 < t

   1. Initial program 61.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   5. Applied rewrites 73.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left({k}^{2} \cdot \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \left(\left(\frac{-2}{3} \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) - -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right) \cdot t} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \left(\left(\frac{-2}{3} \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) - -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right) \cdot {k}^{2}\right) \cdot t} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + {k}^{2} \cdot \left(\left(\frac{-2}{3} \cdot \frac{{t}^{2}}{{\ell}^{2}} + \frac{1}{{\ell}^{2}}\right) - -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right) \cdot {k}^{2}\right) \cdot t} \]

   8. Applied rewrites 31.8%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -0.6666666666666666, {\ell}^{-2}\right) - \left(-\frac{t \cdot t}{\ell \cdot \ell}\right), k \cdot k, \frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{{\ell}^{2}} \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{{\ell}^{2}} \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

       3. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{{\ell}^{2}} \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

       4. associate-/l* N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot \frac{t}{{\ell}^{2}}\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

       5. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

       7. lift-/.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

       8. lift-*.f64 63.5

          \[\leadsto \frac{2}{\left(\left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]

   11. Applied rewrites 63.5%

       \[\leadsto \frac{2}{\left(\left(\left(t \cdot \frac{t}{\ell \cdot \ell}\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 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 53.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

   2. pow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   7. lift-pow.f64 50.9

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

5. Applied rewrites 50.9%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   2. pow3 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   4. lift-*.f64 50.9

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

7. Applied rewrites 50.9%

   \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025043 
(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))))