Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.2% → 91.1%

Time: 7.5s

Alternatives: 24

Speedup: 10.7×

• 26.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 2 \cdot 10^{+141}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left({\left(\sin k\_m \cdot t\right)}^{2}, 2, {\left(\sin k\_m \cdot k\_m\right)}^{2}\right)}{\cos k\_m \cdot \ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \left(\frac{\frac{\cos k\_m}{t}}{{\sin k\_m}^{2}} \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 2e+141)
   (/
    2.0
    (*
     (/
      (fma (pow (* (sin k_m) t) 2.0) 2.0 (pow (* (sin k_m) k_m) 2.0))
      (* (cos k_m) l))
     (/ t l)))
   (*
    (* (/ l k_m) (/ l k_m))
    (* (/ (/ (cos k_m) 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 <= 2e+141) {
		tmp = 2.0 / ((fma(pow((sin(k_m) * t), 2.0), 2.0, pow((sin(k_m) * k_m), 2.0)) / (cos(k_m) * l)) * (t / l));
	} else {
		tmp = ((l / k_m) * (l / k_m)) * (((cos(k_m) / t) / pow(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 <= 2e+141)
		tmp = Float64(2.0 / Float64(Float64(fma((Float64(sin(k_m) * t) ^ 2.0), 2.0, (Float64(sin(k_m) * k_m) ^ 2.0)) / Float64(cos(k_m) * l)) * Float64(t / l)));
	else
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * Float64(Float64(Float64(cos(k_m) / t) / (sin(k_m) ^ 2.0)) * 2.0));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 2e+141], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.00000000000000003e141

   1. Initial program 58.3%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 78.6%

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

   6. Applied rewrites 77.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 90.9%

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

   if 2.00000000000000003e141 < k

   1. Initial program 46.8%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. 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-/.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-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 91.6%

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

Alternative 2: 61.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+180}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k\_m \cdot \left(k\_m \cdot {t}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t}}{\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot k\_m\right)} \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k_m)) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))
      2e+180)
   (/ (* l l) (* k_m (* k_m (pow t 3.0))))
   (* (/ (/ (* l l) t) (* (* k_m k_m) (* k_m k_m))) 2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) + 1.0))) <= 2e+180) {
		tmp = (l * l) / (k_m * (k_m * pow(t, 3.0)));
	} else {
		tmp = (((l * l) / t) / ((k_m * k_m) * (k_m * k_m))) * 2.0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0d0 + ((k_m / t) ** 2.0d0)) + 1.0d0))) <= 2d+180) then
        tmp = (l * l) / (k_m * (k_m * (t ** 3.0d0)))
    else
        tmp = (((l * l) / t) / ((k_m * k_m) * (k_m * k_m))) * 2.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((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))) <= 2e+180) {
		tmp = (l * l) / (k_m * (k_m * Math.pow(t, 3.0)));
	} else {
		tmp = (((l * l) / t) / ((k_m * k_m) * (k_m * k_m))) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k_m)) * math.tan(k_m)) * ((1.0 + math.pow((k_m / t), 2.0)) + 1.0))) <= 2e+180:
		tmp = (l * l) / (k_m * (k_m * math.pow(t, 3.0)))
	else:
		tmp = (((l * l) / t) / ((k_m * k_m) * (k_m * k_m))) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + 1.0))) <= 2e+180)
		tmp = Float64(Float64(l * l) / Float64(k_m * Float64(k_m * (t ^ 3.0))));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / t) / Float64(Float64(k_m * k_m) * Float64(k_m * k_m))) * 2.0);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + ((k_m / t) ^ 2.0)) + 1.0))) <= 2e+180)
		tmp = (l * l) / (k_m * (k_m * (t ^ 3.0)));
	else
		tmp = (((l * l) / t) / ((k_m * k_m) * (k_m * k_m))) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+180], N[(N[(l * l), $MachinePrecision] / N[(k$95$m * N[(k$95$m * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 2e180

   1. Initial program 80.7%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. 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 68.8

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

   4. Applied rewrites 68.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 75.2

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

   6. Applied rewrites 75.2%

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

   if 2e180 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 22.7%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 56.3%

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

   5. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 16.1%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}} \cdot 2 \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}} \cdot 2 \]

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 16.1

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

   9. Applied rewrites 16.1%

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

   10. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 44.1

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

   12. Applied rewrites 44.1%

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

Alternative 3: 57.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\_m\right) \cdot \tan k\_m\right) \cdot \left(\left(1 + {\left(\frac{k\_m}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(k\_m \cdot k\_m\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t}}{\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot k\_m\right)} \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k_m)) (tan k_m))
        (+ (+ 1.0 (pow (/ k_m t) 2.0)) 1.0)))
      2e+117)
   (/ (* l l) (* (* k_m k_m) (* (* t t) t)))
   (* (/ (/ (* l l) t) (* (* k_m k_m) (* k_m k_m))) 2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + pow((k_m / t), 2.0)) + 1.0))) <= 2e+117) {
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	} else {
		tmp = (((l * l) / t) / ((k_m * k_m) * (k_m * k_m))) * 2.0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0d0 + ((k_m / t) ** 2.0d0)) + 1.0d0))) <= 2d+117) then
        tmp = (l * l) / ((k_m * k_m) * ((t * t) * t))
    else
        tmp = (((l * l) / t) / ((k_m * k_m) * (k_m * k_m))) * 2.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if ((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))) <= 2e+117) {
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	} else {
		tmp = (((l * l) / t) / ((k_m * k_m) * (k_m * k_m))) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if (2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k_m)) * math.tan(k_m)) * ((1.0 + math.pow((k_m / t), 2.0)) + 1.0))) <= 2e+117:
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t))
	else:
		tmp = (((l * l) / t) / ((k_m * k_m) * (k_m * k_m))) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k_m)) * tan(k_m)) * Float64(Float64(1.0 + (Float64(k_m / t) ^ 2.0)) + 1.0))) <= 2e+117)
		tmp = Float64(Float64(l * l) / Float64(Float64(k_m * k_m) * Float64(Float64(t * t) * t)));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / t) / Float64(Float64(k_m * k_m) * Float64(k_m * k_m))) * 2.0);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k_m)) * tan(k_m)) * ((1.0 + ((k_m / t) ^ 2.0)) + 1.0))) <= 2e+117)
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	else
		tmp = (((l * l) / t) / ((k_m * k_m) * (k_m * k_m))) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+117], N[(N[(l * l), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 2.0000000000000001e117

   1. Initial program 80.6%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. 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 68.9

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

   4. Applied rewrites 68.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 68.9

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

   6. Applied rewrites 68.9%

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

   if 2.0000000000000001e117 < (/.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 23.3%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 56.0%

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

   5. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 16.1%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}} \cdot 2 \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}} \cdot 2 \]

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 16.1

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

   9. Applied rewrites 16.1%

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

   10. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 43.9

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

   12. Applied rewrites 43.9%

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

Alternative 4: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{-24}:\\ \;\;\;\;\left(\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\cos k\_m}{{\sin k\_m}^{2} \cdot t}\right) \cdot 2\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+85}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \sin k\_m}{\ell} \cdot \left(\tan k\_m \cdot \left(\left({\left(\frac{k\_m}{t}\right)}^{2} + 1\right) + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k\_m \cdot t\right)}^{2} \cdot 2}{\cos k\_m \cdot \ell} \cdot \frac{t}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 1.85e-24)
   (* (* (* (/ l k_m) (/ l k_m)) (/ (cos k_m) (* (pow (sin k_m) 2.0) t))) 2.0)
   (if (<= t 1.5e+85)
     (/
      2.0
      (*
       (/ (* (/ (pow t 3.0) l) (sin k_m)) l)
       (* (tan k_m) (+ (+ (pow (/ k_m t) 2.0) 1.0) 1.0))))
     (/
      2.0
      (* (/ (* (pow (* (sin k_m) t) 2.0) 2.0) (* (cos k_m) l)) (/ t l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 1.85e-24) {
		tmp = (((l / k_m) * (l / k_m)) * (cos(k_m) / (pow(sin(k_m), 2.0) * t))) * 2.0;
	} else if (t <= 1.5e+85) {
		tmp = 2.0 / ((((pow(t, 3.0) / l) * sin(k_m)) / l) * (tan(k_m) * ((pow((k_m / t), 2.0) + 1.0) + 1.0)));
	} else {
		tmp = 2.0 / (((pow((sin(k_m) * t), 2.0) * 2.0) / (cos(k_m) * l)) * (t / l));
	}
	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 <= 1.85d-24) then
        tmp = (((l / k_m) * (l / k_m)) * (cos(k_m) / ((sin(k_m) ** 2.0d0) * t))) * 2.0d0
    else if (t <= 1.5d+85) then
        tmp = 2.0d0 / (((((t ** 3.0d0) / l) * sin(k_m)) / l) * (tan(k_m) * ((((k_m / t) ** 2.0d0) + 1.0d0) + 1.0d0)))
    else
        tmp = 2.0d0 / (((((sin(k_m) * t) ** 2.0d0) * 2.0d0) / (cos(k_m) * l)) * (t / l))
    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 <= 1.85e-24) {
		tmp = (((l / k_m) * (l / k_m)) * (Math.cos(k_m) / (Math.pow(Math.sin(k_m), 2.0) * t))) * 2.0;
	} else if (t <= 1.5e+85) {
		tmp = 2.0 / ((((Math.pow(t, 3.0) / l) * Math.sin(k_m)) / l) * (Math.tan(k_m) * ((Math.pow((k_m / t), 2.0) + 1.0) + 1.0)));
	} else {
		tmp = 2.0 / (((Math.pow((Math.sin(k_m) * t), 2.0) * 2.0) / (Math.cos(k_m) * l)) * (t / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 1.85e-24:
		tmp = (((l / k_m) * (l / k_m)) * (math.cos(k_m) / (math.pow(math.sin(k_m), 2.0) * t))) * 2.0
	elif t <= 1.5e+85:
		tmp = 2.0 / ((((math.pow(t, 3.0) / l) * math.sin(k_m)) / l) * (math.tan(k_m) * ((math.pow((k_m / t), 2.0) + 1.0) + 1.0)))
	else:
		tmp = 2.0 / (((math.pow((math.sin(k_m) * t), 2.0) * 2.0) / (math.cos(k_m) * l)) * (t / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 1.85e-24)
		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);
	elseif (t <= 1.5e+85)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / l) * sin(k_m)) / l) * Float64(tan(k_m) * Float64(Float64((Float64(k_m / t) ^ 2.0) + 1.0) + 1.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(sin(k_m) * t) ^ 2.0) * 2.0) / Float64(cos(k_m) * l)) * Float64(t / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 1.85e-24)
		tmp = (((l / k_m) * (l / k_m)) * (cos(k_m) / ((sin(k_m) ^ 2.0) * t))) * 2.0;
	elseif (t <= 1.5e+85)
		tmp = 2.0 / (((((t ^ 3.0) / l) * sin(k_m)) / l) * (tan(k_m) * ((((k_m / t) ^ 2.0) + 1.0) + 1.0)));
	else
		tmp = 2.0 / (((((sin(k_m) * t) ^ 2.0) * 2.0) / (cos(k_m) * l)) * (t / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 1.85e-24], 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], If[LessEqual[t, 1.5e+85], N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[(N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.8499999999999999e-24

   1. Initial program 50.3%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 63.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 75.0

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

   6. Applied rewrites 75.0%

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

   if 1.8499999999999999e-24 < t < 1.5e85

   1. Initial program 76.7%

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

   3. Applied rewrites 80.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-sin.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if 1.5e85 < t

   1. Initial program 63.9%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 77.6%

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

   6. Applied rewrites 75.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 89.8%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. unpow-prod-down N/A

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

       5. lift-sin.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-pow.f64 87.0

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

   11. Applied rewrites 87.0%

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

Alternative 5: 77.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{+45}:\\ \;\;\;\;\left(\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\cos k\_m}{{\sin k\_m}^{2} \cdot t}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k\_m \cdot t\right)}^{2} \cdot 2}{\cos k\_m \cdot \ell} \cdot \frac{t}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 5.8e+45)
   (* (* (* (/ l k_m) (/ l k_m)) (/ (cos k_m) (* (pow (sin k_m) 2.0) t))) 2.0)
   (/ 2.0 (* (/ (* (pow (* (sin k_m) t) 2.0) 2.0) (* (cos k_m) l)) (/ t l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 5.8e+45) {
		tmp = (((l / k_m) * (l / k_m)) * (cos(k_m) / (pow(sin(k_m), 2.0) * t))) * 2.0;
	} else {
		tmp = 2.0 / (((pow((sin(k_m) * t), 2.0) * 2.0) / (cos(k_m) * l)) * (t / l));
	}
	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 <= 5.8d+45) then
        tmp = (((l / k_m) * (l / k_m)) * (cos(k_m) / ((sin(k_m) ** 2.0d0) * t))) * 2.0d0
    else
        tmp = 2.0d0 / (((((sin(k_m) * t) ** 2.0d0) * 2.0d0) / (cos(k_m) * l)) * (t / l))
    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 <= 5.8e+45) {
		tmp = (((l / k_m) * (l / k_m)) * (Math.cos(k_m) / (Math.pow(Math.sin(k_m), 2.0) * t))) * 2.0;
	} else {
		tmp = 2.0 / (((Math.pow((Math.sin(k_m) * t), 2.0) * 2.0) / (Math.cos(k_m) * l)) * (t / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 5.8e+45:
		tmp = (((l / k_m) * (l / k_m)) * (math.cos(k_m) / (math.pow(math.sin(k_m), 2.0) * t))) * 2.0
	else:
		tmp = 2.0 / (((math.pow((math.sin(k_m) * t), 2.0) * 2.0) / (math.cos(k_m) * l)) * (t / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 5.8e+45)
		tmp = Float64(Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * Float64(cos(k_m) / Float64((sin(k_m) ^ 2.0) * t))) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(sin(k_m) * t) ^ 2.0) * 2.0) / Float64(cos(k_m) * l)) * Float64(t / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 5.8e+45)
		tmp = (((l / k_m) * (l / k_m)) * (cos(k_m) / ((sin(k_m) ^ 2.0) * t))) * 2.0;
	else
		tmp = 2.0 / (((((sin(k_m) * t) ^ 2.0) * 2.0) / (cos(k_m) * l)) * (t / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 5.8e+45], N[(N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.7999999999999994e45

   1. Initial program 52.5%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 63.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 75.0

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

   6. Applied rewrites 75.0%

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

   if 5.7999999999999994e45 < t

   1. Initial program 65.2%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 77.0%

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

   6. Applied rewrites 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 89.2%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. unpow-prod-down N/A

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

       5. lift-sin.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-pow.f64 84.8

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

   11. Applied rewrites 84.8%

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

Alternative 6: 77.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{+45}:\\ \;\;\;\;\left(\left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right) \cdot \frac{\cos k\_m}{{\sin k\_m}^{2} \cdot t}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k\_m \cdot t\right)}^{2} \cdot 2}{\cos k\_m \cdot \ell} \cdot \frac{t}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 5.5e+45)
   (* (* (* (/ l k_m) (/ l k_m)) (/ (cos k_m) (* (pow (sin k_m) 2.0) t))) 2.0)
   (/ 2.0 (* (/ (* (pow (* k_m t) 2.0) 2.0) (* (cos k_m) l)) (/ t l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 5.5e+45) {
		tmp = (((l / k_m) * (l / k_m)) * (cos(k_m) / (pow(sin(k_m), 2.0) * t))) * 2.0;
	} else {
		tmp = 2.0 / (((pow((k_m * t), 2.0) * 2.0) / (cos(k_m) * l)) * (t / l));
	}
	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 <= 5.5d+45) then
        tmp = (((l / k_m) * (l / k_m)) * (cos(k_m) / ((sin(k_m) ** 2.0d0) * t))) * 2.0d0
    else
        tmp = 2.0d0 / (((((k_m * t) ** 2.0d0) * 2.0d0) / (cos(k_m) * l)) * (t / l))
    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 <= 5.5e+45) {
		tmp = (((l / k_m) * (l / k_m)) * (Math.cos(k_m) / (Math.pow(Math.sin(k_m), 2.0) * t))) * 2.0;
	} else {
		tmp = 2.0 / (((Math.pow((k_m * t), 2.0) * 2.0) / (Math.cos(k_m) * l)) * (t / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 5.5e+45:
		tmp = (((l / k_m) * (l / k_m)) * (math.cos(k_m) / (math.pow(math.sin(k_m), 2.0) * t))) * 2.0
	else:
		tmp = 2.0 / (((math.pow((k_m * t), 2.0) * 2.0) / (math.cos(k_m) * l)) * (t / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 5.5e+45)
		tmp = Float64(Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) * Float64(cos(k_m) / Float64((sin(k_m) ^ 2.0) * t))) * 2.0);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) * 2.0) / Float64(cos(k_m) * l)) * Float64(t / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 5.5e+45)
		tmp = (((l / k_m) * (l / k_m)) * (cos(k_m) / ((sin(k_m) ^ 2.0) * t))) * 2.0;
	else
		tmp = 2.0 / (((((k_m * t) ^ 2.0) * 2.0) / (cos(k_m) * l)) * (t / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 5.5e+45], N[(N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.5000000000000001e45

   1. Initial program 52.5%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 63.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 75.0

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

   6. Applied rewrites 75.0%

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

   if 5.5000000000000001e45 < t

   1. Initial program 65.2%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 77.0%

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

   6. Applied rewrites 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 89.2%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow-prod-down N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-*.f64 84.6

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

   11. Applied rewrites 84.6%

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

Alternative 7: 72.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot \ell\\ \mathbf{if}\;t \leq 7.6 \cdot 10^{+28}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k\_m \cdot k\_m\right)}^{2}}{t\_1} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k\_m \cdot t\right)}^{2} \cdot 2}{t\_1} \cdot \frac{t}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l)))
   (if (<= t 7.6e+28)
     (/ 2.0 (* (/ (pow (* (sin k_m) k_m) 2.0) t_1) (/ t l)))
     (/ 2.0 (* (/ (* (pow (* k_m t) 2.0) 2.0) t_1) (/ t l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * l;
	double tmp;
	if (t <= 7.6e+28) {
		tmp = 2.0 / ((pow((sin(k_m) * k_m), 2.0) / t_1) * (t / l));
	} else {
		tmp = 2.0 / (((pow((k_m * t), 2.0) * 2.0) / t_1) * (t / l));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(k_m) * l
    if (t <= 7.6d+28) then
        tmp = 2.0d0 / ((((sin(k_m) * k_m) ** 2.0d0) / t_1) * (t / l))
    else
        tmp = 2.0d0 / (((((k_m * t) ** 2.0d0) * 2.0d0) / t_1) * (t / l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.cos(k_m) * l;
	double tmp;
	if (t <= 7.6e+28) {
		tmp = 2.0 / ((Math.pow((Math.sin(k_m) * k_m), 2.0) / t_1) * (t / l));
	} else {
		tmp = 2.0 / (((Math.pow((k_m * t), 2.0) * 2.0) / t_1) * (t / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.cos(k_m) * l
	tmp = 0
	if t <= 7.6e+28:
		tmp = 2.0 / ((math.pow((math.sin(k_m) * k_m), 2.0) / t_1) * (t / l))
	else:
		tmp = 2.0 / (((math.pow((k_m * t), 2.0) * 2.0) / t_1) * (t / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * l)
	tmp = 0.0
	if (t <= 7.6e+28)
		tmp = Float64(2.0 / Float64(Float64((Float64(sin(k_m) * k_m) ^ 2.0) / t_1) * Float64(t / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) * 2.0) / t_1) * Float64(t / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = cos(k_m) * l;
	tmp = 0.0;
	if (t <= 7.6e+28)
		tmp = 2.0 / ((((sin(k_m) * k_m) ^ 2.0) / t_1) * (t / l));
	else
		tmp = 2.0 / (((((k_m * t) ^ 2.0) * 2.0) / t_1) * (t / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[t, 7.6e+28], N[(2.0 / N[(N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$1), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 7.5999999999999998e28

   1. Initial program 52.0%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 73.7%

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

   6. Applied rewrites 73.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 82.9%

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

   9. Taylor expanded in t around 0

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

       1. *-commutative N/A

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

       2. unpow-prod-down N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-pow.f64 69.0

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

   11. Applied rewrites 69.0%

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

   if 7.5999999999999998e28 < t

   1. Initial program 65.8%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 76.9%

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

   6. Applied rewrites 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 88.8%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow-prod-down N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-*.f64 83.8

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

   11. Applied rewrites 83.8%

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

Alternative 8: 71.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot \ell\\ \mathbf{if}\;t \leq 7.6 \cdot 10^{+28}:\\ \;\;\;\;\frac{2 \cdot \left(t\_1 \cdot \ell\right)}{\left(k\_m \cdot \left(k\_m \cdot t\right)\right) \cdot {\sin k\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k\_m \cdot t\right)}^{2} \cdot 2}{t\_1} \cdot \frac{t}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l)))
   (if (<= t 7.6e+28)
     (/ (* 2.0 (* t_1 l)) (* (* k_m (* k_m t)) (pow (sin k_m) 2.0)))
     (/ 2.0 (* (/ (* (pow (* k_m t) 2.0) 2.0) t_1) (/ t l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * l;
	double tmp;
	if (t <= 7.6e+28) {
		tmp = (2.0 * (t_1 * l)) / ((k_m * (k_m * t)) * pow(sin(k_m), 2.0));
	} else {
		tmp = 2.0 / (((pow((k_m * t), 2.0) * 2.0) / t_1) * (t / l));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(k_m) * l
    if (t <= 7.6d+28) then
        tmp = (2.0d0 * (t_1 * l)) / ((k_m * (k_m * t)) * (sin(k_m) ** 2.0d0))
    else
        tmp = 2.0d0 / (((((k_m * t) ** 2.0d0) * 2.0d0) / t_1) * (t / l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.cos(k_m) * l;
	double tmp;
	if (t <= 7.6e+28) {
		tmp = (2.0 * (t_1 * l)) / ((k_m * (k_m * t)) * Math.pow(Math.sin(k_m), 2.0));
	} else {
		tmp = 2.0 / (((Math.pow((k_m * t), 2.0) * 2.0) / t_1) * (t / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.cos(k_m) * l
	tmp = 0
	if t <= 7.6e+28:
		tmp = (2.0 * (t_1 * l)) / ((k_m * (k_m * t)) * math.pow(math.sin(k_m), 2.0))
	else:
		tmp = 2.0 / (((math.pow((k_m * t), 2.0) * 2.0) / t_1) * (t / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * l)
	tmp = 0.0
	if (t <= 7.6e+28)
		tmp = Float64(Float64(2.0 * Float64(t_1 * l)) / Float64(Float64(k_m * Float64(k_m * t)) * (sin(k_m) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) * 2.0) / t_1) * Float64(t / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = cos(k_m) * l;
	tmp = 0.0;
	if (t <= 7.6e+28)
		tmp = (2.0 * (t_1 * l)) / ((k_m * (k_m * t)) * (sin(k_m) ^ 2.0));
	else
		tmp = 2.0 / (((((k_m * t) ^ 2.0) * 2.0) / t_1) * (t / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[t, 7.6e+28], N[(N[(2.0 * N[(t$95$1 * l), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 7.5999999999999998e28

   1. Initial program 52.0%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 63.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. 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-/.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-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. *-commutative N/A

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

   6. Applied rewrites 64.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 67.7

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

   8. Applied rewrites 67.7%

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

   if 7.5999999999999998e28 < t

   1. Initial program 65.8%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 76.9%

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

   6. Applied rewrites 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 88.8%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow-prod-down N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-*.f64 83.8

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

   11. Applied rewrites 83.8%

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

Alternative 9: 68.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{+28}:\\ \;\;\;\;\left(\left(\ell \cdot \ell\right) \cdot \frac{\cos k\_m}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot {\sin k\_m}^{2}}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(k\_m \cdot t\right)}^{2} \cdot 2}{\cos k\_m \cdot \ell} \cdot \frac{t}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 7.6e+28)
   (* (* (* l l) (/ (cos k_m) (* (* (* k_m k_m) t) (pow (sin k_m) 2.0)))) 2.0)
   (/ 2.0 (* (/ (* (pow (* k_m t) 2.0) 2.0) (* (cos k_m) l)) (/ t l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 7.6e+28) {
		tmp = ((l * l) * (cos(k_m) / (((k_m * k_m) * t) * pow(sin(k_m), 2.0)))) * 2.0;
	} else {
		tmp = 2.0 / (((pow((k_m * t), 2.0) * 2.0) / (cos(k_m) * l)) * (t / l));
	}
	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 <= 7.6d+28) then
        tmp = ((l * l) * (cos(k_m) / (((k_m * k_m) * t) * (sin(k_m) ** 2.0d0)))) * 2.0d0
    else
        tmp = 2.0d0 / (((((k_m * t) ** 2.0d0) * 2.0d0) / (cos(k_m) * l)) * (t / l))
    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 <= 7.6e+28) {
		tmp = ((l * l) * (Math.cos(k_m) / (((k_m * k_m) * t) * Math.pow(Math.sin(k_m), 2.0)))) * 2.0;
	} else {
		tmp = 2.0 / (((Math.pow((k_m * t), 2.0) * 2.0) / (Math.cos(k_m) * l)) * (t / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 7.6e+28:
		tmp = ((l * l) * (math.cos(k_m) / (((k_m * k_m) * t) * math.pow(math.sin(k_m), 2.0)))) * 2.0
	else:
		tmp = 2.0 / (((math.pow((k_m * t), 2.0) * 2.0) / (math.cos(k_m) * l)) * (t / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 7.6e+28)
		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);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) * 2.0) / Float64(cos(k_m) * l)) * Float64(t / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 7.6e+28)
		tmp = ((l * l) * (cos(k_m) / (((k_m * k_m) * t) * (sin(k_m) ^ 2.0)))) * 2.0;
	else
		tmp = 2.0 / (((((k_m * t) ^ 2.0) * 2.0) / (cos(k_m) * l)) * (t / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 7.6e+28], 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], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.5999999999999998e28

   1. Initial program 52.0%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]
   5. 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

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

   6. Applied rewrites 64.2%

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

   if 7.5999999999999998e28 < t

   1. Initial program 65.8%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 76.9%

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

   6. Applied rewrites 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 88.8%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow-prod-down N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-*.f64 83.8

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

   11. Applied rewrites 83.8%

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

Alternative 10: 77.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \cos k\_m \cdot \ell\\ \mathbf{if}\;k\_m \leq 9.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\frac{{\left(k\_m \cdot t\right)}^{2} \cdot 2}{t\_1} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(t\_1 \cdot \ell\right)}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (cos k_m) l)))
   (if (<= k_m 9.4e-5)
     (/ 2.0 (* (/ (* (pow (* k_m t) 2.0) 2.0) t_1) (/ t l)))
     (/
      (* 2.0 (* t_1 l))
      (* (* (* k_m k_m) t) (- 0.5 (* 0.5 (cos (* 2.0 k_m)))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = cos(k_m) * l;
	double tmp;
	if (k_m <= 9.4e-5) {
		tmp = 2.0 / (((pow((k_m * t), 2.0) * 2.0) / t_1) * (t / l));
	} else {
		tmp = (2.0 * (t_1 * l)) / (((k_m * k_m) * t) * (0.5 - (0.5 * cos((2.0 * k_m)))));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(k_m) * l
    if (k_m <= 9.4d-5) then
        tmp = 2.0d0 / (((((k_m * t) ** 2.0d0) * 2.0d0) / t_1) * (t / l))
    else
        tmp = (2.0d0 * (t_1 * l)) / (((k_m * k_m) * t) * (0.5d0 - (0.5d0 * cos((2.0d0 * k_m)))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.cos(k_m) * l;
	double tmp;
	if (k_m <= 9.4e-5) {
		tmp = 2.0 / (((Math.pow((k_m * t), 2.0) * 2.0) / t_1) * (t / l));
	} else {
		tmp = (2.0 * (t_1 * l)) / (((k_m * k_m) * t) * (0.5 - (0.5 * Math.cos((2.0 * k_m)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = math.cos(k_m) * l
	tmp = 0
	if k_m <= 9.4e-5:
		tmp = 2.0 / (((math.pow((k_m * t), 2.0) * 2.0) / t_1) * (t / l))
	else:
		tmp = (2.0 * (t_1 * l)) / (((k_m * k_m) * t) * (0.5 - (0.5 * math.cos((2.0 * k_m)))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(cos(k_m) * l)
	tmp = 0.0
	if (k_m <= 9.4e-5)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) * 2.0) / t_1) * Float64(t / l)));
	else
		tmp = Float64(Float64(2.0 * Float64(t_1 * l)) / Float64(Float64(Float64(k_m * k_m) * t) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = cos(k_m) * l;
	tmp = 0.0;
	if (k_m <= 9.4e-5)
		tmp = 2.0 / (((((k_m * t) ^ 2.0) * 2.0) / t_1) * (t / l));
	else
		tmp = (2.0 * (t_1 * l)) / (((k_m * k_m) * t) * (0.5 - (0.5 * cos((2.0 * k_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]}, If[LessEqual[k$95$m, 9.4e-5], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(t$95$1 * l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 9.39999999999999945e-5

   1. Initial program 62.1%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 76.8%

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

   6. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 89.9%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow-prod-down N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-*.f64 85.0

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

   11. Applied rewrites 85.0%

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

   if 9.39999999999999945e-5 < k

   1. Initial program 48.5%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. 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-/.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-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. *-commutative N/A

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

   6. Applied rewrites 70.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 70.5

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

   8. Applied rewrites 70.5%

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

Alternative 11: 77.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 9.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\cos k\_m \cdot \ell\right) \cdot \ell\right)}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 9.4e-5)
   (/ 2.0 (* (* (/ (pow (* k_m t) 2.0) l) 2.0) (/ t l)))
   (/
    (* 2.0 (* (* (cos k_m) l) l))
    (* (* (* k_m k_m) t) (- 0.5 (* 0.5 (cos (* 2.0 k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9.4e-5) {
		tmp = 2.0 / (((pow((k_m * t), 2.0) / l) * 2.0) * (t / l));
	} else {
		tmp = (2.0 * ((cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (0.5 - (0.5 * cos((2.0 * k_m)))));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 9.4d-5) then
        tmp = 2.0d0 / (((((k_m * t) ** 2.0d0) / l) * 2.0d0) * (t / l))
    else
        tmp = (2.0d0 * ((cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (0.5d0 - (0.5d0 * cos((2.0d0 * k_m)))))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 9.4e-5) {
		tmp = 2.0 / (((Math.pow((k_m * t), 2.0) / l) * 2.0) * (t / l));
	} else {
		tmp = (2.0 * ((Math.cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (0.5 - (0.5 * Math.cos((2.0 * k_m)))));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 9.4e-5:
		tmp = 2.0 / (((math.pow((k_m * t), 2.0) / l) * 2.0) * (t / l))
	else:
		tmp = (2.0 * ((math.cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (0.5 - (0.5 * math.cos((2.0 * k_m)))))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 9.4e-5)
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) * 2.0) * Float64(t / l)));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k_m) * l) * l)) / Float64(Float64(Float64(k_m * k_m) * t) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * k_m))))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 9.4e-5)
		tmp = 2.0 / (((((k_m * t) ^ 2.0) / l) * 2.0) * (t / l));
	else
		tmp = (2.0 * ((cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (0.5 - (0.5 * cos((2.0 * k_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 9.4e-5], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * k$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.39999999999999945e-5

   1. Initial program 62.1%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 76.8%

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

   6. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 89.9%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow-prod-down N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-*.f64 85.0

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

   11. Applied rewrites 85.0%

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

   if 9.39999999999999945e-5 < k

   1. Initial program 48.5%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. 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-/.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-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. *-commutative N/A

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

   6. Applied rewrites 70.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 70.5

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

   8. Applied rewrites 70.5%

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

Alternative 12: 61.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{t \cdot t}{\ell}\\ \mathbf{if}\;t \leq 3 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_1, -0.6666666666666666, {\ell}^{-1}\right) - \left(-t\_1\right), k\_m \cdot k\_m, t\_1 \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* t t) l)))
   (if (<= t 3e-14)
     (/
      2.0
      (*
       (*
        (fma
         (- (fma t_1 -0.6666666666666666 (pow l -1.0)) (- t_1))
         (* k_m k_m)
         (* t_1 2.0))
        (* k_m k_m))
       (/ t l)))
     (if (<= t 4.2e+60)
       (/ (* (/ l k_m) (/ l k_m)) (pow t 3.0))
       (/ 2.0 (* (* (/ (pow (* k_m t) 2.0) l) 2.0) (/ t l)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (t * t) / l;
	double tmp;
	if (t <= 3e-14) {
		tmp = 2.0 / ((fma((fma(t_1, -0.6666666666666666, pow(l, -1.0)) - -t_1), (k_m * k_m), (t_1 * 2.0)) * (k_m * k_m)) * (t / l));
	} else if (t <= 4.2e+60) {
		tmp = ((l / k_m) * (l / k_m)) / pow(t, 3.0);
	} else {
		tmp = 2.0 / (((pow((k_m * t), 2.0) / l) * 2.0) * (t / l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(t * t) / l)
	tmp = 0.0
	if (t <= 3e-14)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(fma(t_1, -0.6666666666666666, (l ^ -1.0)) - Float64(-t_1)), Float64(k_m * k_m), Float64(t_1 * 2.0)) * Float64(k_m * k_m)) * Float64(t / l)));
	elseif (t <= 4.2e+60)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) * 2.0) * Float64(t / l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[t, 3e-14], N[(2.0 / N[(N[(N[(N[(N[(t$95$1 * -0.6666666666666666 + N[Power[l, -1.0], $MachinePrecision]), $MachinePrecision] - (-t$95$1)), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(t$95$1 * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e+60], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.9999999999999998e-14

   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. Taylor expanded in t around 0

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

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

   4. Applied rewrites 73.4%

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

   6. Applied rewrites 73.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 82.8%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 54.0%

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

   if 2.9999999999999998e-14 < t < 4.2000000000000002e60

   1. Initial program 79.3%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. 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 66.4

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

   4. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. associate-/r* N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lower-/.f64 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-pow.f64 77.7

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

   6. Applied rewrites 77.7%

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

   if 4.2000000000000002e60 < t

   1. Initial program 64.6%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 77.4%

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

   6. Applied rewrites 75.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow-prod-down N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-*.f64 85.2

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

   11. Applied rewrites 85.2%

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

Alternative 13: 63.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k\_m \cdot k\_m, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)}{\cos k\_m \cdot \ell} \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 3e-14)
   (/
    2.0
    (*
     (/
      (*
       (fma (fma -0.6666666666666666 (* t t) 1.0) (* k_m k_m) (* (* t t) 2.0))
       (* k_m k_m))
      (* (cos k_m) l))
     (/ t l)))
   (if (<= t 4.2e+60)
     (/ (* (/ l k_m) (/ l k_m)) (pow t 3.0))
     (/ 2.0 (* (* (/ (pow (* k_m t) 2.0) l) 2.0) (/ t l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3e-14) {
		tmp = 2.0 / (((fma(fma(-0.6666666666666666, (t * t), 1.0), (k_m * k_m), ((t * t) * 2.0)) * (k_m * k_m)) / (cos(k_m) * l)) * (t / l));
	} else if (t <= 4.2e+60) {
		tmp = ((l / k_m) * (l / k_m)) / pow(t, 3.0);
	} else {
		tmp = 2.0 / (((pow((k_m * t), 2.0) / l) * 2.0) * (t / l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 3e-14)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(-0.6666666666666666, Float64(t * t), 1.0), Float64(k_m * k_m), Float64(Float64(t * t) * 2.0)) * Float64(k_m * k_m)) / Float64(cos(k_m) * l)) * Float64(t / l)));
	elseif (t <= 4.2e+60)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) * 2.0) * Float64(t / l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 3e-14], N[(2.0 / N[(N[(N[(N[(N[(-0.6666666666666666 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e+60], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.9999999999999998e-14

   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. Taylor expanded in t around 0

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

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

   4. Applied rewrites 73.4%

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

   6. Applied rewrites 73.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 82.8%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. pow2 N/A

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

       15. lift-*.f64 N/A

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

       16. pow2 N/A

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

       17. lift-*.f64 56.2

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

   11. Applied rewrites 56.2%

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

   if 2.9999999999999998e-14 < t < 4.2000000000000002e60

   1. Initial program 79.3%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. 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 66.4

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

   4. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. associate-/r* N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lower-/.f64 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-pow.f64 77.7

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

   6. Applied rewrites 77.7%

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

   if 4.2000000000000002e60 < t

   1. Initial program 64.6%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 77.4%

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

   6. Applied rewrites 75.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow-prod-down N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-*.f64 85.2

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

   11. Applied rewrites 85.2%

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

Alternative 14: 59.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.6666666666666666, t \cdot t, 1\right), k\_m \cdot k\_m, \left(t \cdot t\right) \cdot 2\right) \cdot \left(k\_m \cdot k\_m\right)}{\cos k\_m \cdot \left(\ell \cdot \ell\right)} \cdot t}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 3e-14)
   (/
    2.0
    (*
     (/
      (*
       (fma (fma -0.6666666666666666 (* t t) 1.0) (* k_m k_m) (* (* t t) 2.0))
       (* k_m k_m))
      (* (cos k_m) (* l l)))
     t))
   (if (<= t 4.2e+60)
     (/ (* (/ l k_m) (/ l k_m)) (pow t 3.0))
     (/ 2.0 (* (* (/ (pow (* k_m t) 2.0) l) 2.0) (/ t l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3e-14) {
		tmp = 2.0 / (((fma(fma(-0.6666666666666666, (t * t), 1.0), (k_m * k_m), ((t * t) * 2.0)) * (k_m * k_m)) / (cos(k_m) * (l * l))) * t);
	} else if (t <= 4.2e+60) {
		tmp = ((l / k_m) * (l / k_m)) / pow(t, 3.0);
	} else {
		tmp = 2.0 / (((pow((k_m * t), 2.0) / l) * 2.0) * (t / l));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 3e-14)
		tmp = Float64(2.0 / Float64(Float64(Float64(fma(fma(-0.6666666666666666, Float64(t * t), 1.0), Float64(k_m * k_m), Float64(Float64(t * t) * 2.0)) * Float64(k_m * k_m)) / Float64(cos(k_m) * Float64(l * l))) * t));
	elseif (t <= 4.2e+60)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) * 2.0) * Float64(t / l)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 3e-14], N[(2.0 / N[(N[(N[(N[(N[(-0.6666666666666666 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e+60], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.9999999999999998e-14

   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. Taylor expanded in t around 0

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

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

   4. Applied rewrites 73.4%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 51.0

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

   7. Applied rewrites 51.0%

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

   if 2.9999999999999998e-14 < t < 4.2000000000000002e60

   1. Initial program 79.3%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. 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 66.4

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

   4. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. associate-/r* N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lower-/.f64 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-pow.f64 77.7

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

   6. Applied rewrites 77.7%

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

   if 4.2000000000000002e60 < t

   1. Initial program 64.6%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 77.4%

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

   6. Applied rewrites 75.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow-prod-down N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-*.f64 85.2

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

   11. Applied rewrites 85.2%

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

Alternative 15: 64.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-14}:\\ \;\;\;\;\left(\frac{\ell \cdot \ell}{k\_m \cdot k\_m} \cdot \frac{\cos k\_m}{\left(k\_m \cdot k\_m\right) \cdot t}\right) \cdot 2\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 3e-14)
   (* (* (/ (* l l) (* k_m k_m)) (/ (cos k_m) (* (* k_m k_m) t))) 2.0)
   (if (<= t 4.2e+60)
     (/ (* (/ l k_m) (/ l k_m)) (pow t 3.0))
     (/ 2.0 (* (* (/ (pow (* k_m t) 2.0) l) 2.0) (/ t l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3e-14) {
		tmp = (((l * l) / (k_m * k_m)) * (cos(k_m) / ((k_m * k_m) * t))) * 2.0;
	} else if (t <= 4.2e+60) {
		tmp = ((l / k_m) * (l / k_m)) / pow(t, 3.0);
	} else {
		tmp = 2.0 / (((pow((k_m * t), 2.0) / l) * 2.0) * (t / l));
	}
	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 <= 3d-14) then
        tmp = (((l * l) / (k_m * k_m)) * (cos(k_m) / ((k_m * k_m) * t))) * 2.0d0
    else if (t <= 4.2d+60) then
        tmp = ((l / k_m) * (l / k_m)) / (t ** 3.0d0)
    else
        tmp = 2.0d0 / (((((k_m * t) ** 2.0d0) / l) * 2.0d0) * (t / l))
    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 <= 3e-14) {
		tmp = (((l * l) / (k_m * k_m)) * (Math.cos(k_m) / ((k_m * k_m) * t))) * 2.0;
	} else if (t <= 4.2e+60) {
		tmp = ((l / k_m) * (l / k_m)) / Math.pow(t, 3.0);
	} else {
		tmp = 2.0 / (((Math.pow((k_m * t), 2.0) / l) * 2.0) * (t / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 3e-14:
		tmp = (((l * l) / (k_m * k_m)) * (math.cos(k_m) / ((k_m * k_m) * t))) * 2.0
	elif t <= 4.2e+60:
		tmp = ((l / k_m) * (l / k_m)) / math.pow(t, 3.0)
	else:
		tmp = 2.0 / (((math.pow((k_m * t), 2.0) / l) * 2.0) * (t / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 3e-14)
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(k_m * k_m)) * Float64(cos(k_m) / Float64(Float64(k_m * k_m) * t))) * 2.0);
	elseif (t <= 4.2e+60)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) * 2.0) * Float64(t / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 3e-14)
		tmp = (((l * l) / (k_m * k_m)) * (cos(k_m) / ((k_m * k_m) * t))) * 2.0;
	elseif (t <= 4.2e+60)
		tmp = ((l / k_m) * (l / k_m)) / (t ^ 3.0);
	else
		tmp = 2.0 / (((((k_m * t) ^ 2.0) / l) * 2.0) * (t / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 3e-14], 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[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t, 4.2e+60], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.9999999999999998e-14

   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. Taylor expanded in t around 0

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

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

   4. Applied rewrites 63.6%

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

   5. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 58.0

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

   7. Applied rewrites 58.0%

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

   if 2.9999999999999998e-14 < t < 4.2000000000000002e60

   1. Initial program 79.3%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. 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 66.4

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

   4. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. associate-/r* N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lower-/.f64 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-pow.f64 77.7

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

   6. Applied rewrites 77.7%

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

   if 4.2000000000000002e60 < t

   1. Initial program 64.6%

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

   2. Taylor expanded in t around 0

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

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

   4. Applied rewrites 77.4%

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

   6. Applied rewrites 75.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. times-frac N/A

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow-prod-down N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-*.f64 85.2

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

   11. Applied rewrites 85.2%

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

Alternative 16: 63.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-14}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\cos k\_m \cdot \ell\right) \cdot \ell\right)}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(k\_m \cdot t\right)}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 3e-14)
   (/ (* 2.0 (* (* (cos k_m) l) l)) (* (* (* k_m k_m) t) (* k_m k_m)))
   (if (<= t 4.2e+60)
     (/ (* (/ l k_m) (/ l k_m)) (pow t 3.0))
     (/ 2.0 (* (* (/ (pow (* k_m t) 2.0) l) 2.0) (/ t l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3e-14) {
		tmp = (2.0 * ((cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (k_m * k_m));
	} else if (t <= 4.2e+60) {
		tmp = ((l / k_m) * (l / k_m)) / pow(t, 3.0);
	} else {
		tmp = 2.0 / (((pow((k_m * t), 2.0) / l) * 2.0) * (t / l));
	}
	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 <= 3d-14) then
        tmp = (2.0d0 * ((cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (k_m * k_m))
    else if (t <= 4.2d+60) then
        tmp = ((l / k_m) * (l / k_m)) / (t ** 3.0d0)
    else
        tmp = 2.0d0 / (((((k_m * t) ** 2.0d0) / l) * 2.0d0) * (t / l))
    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 <= 3e-14) {
		tmp = (2.0 * ((Math.cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (k_m * k_m));
	} else if (t <= 4.2e+60) {
		tmp = ((l / k_m) * (l / k_m)) / Math.pow(t, 3.0);
	} else {
		tmp = 2.0 / (((Math.pow((k_m * t), 2.0) / l) * 2.0) * (t / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 3e-14:
		tmp = (2.0 * ((math.cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (k_m * k_m))
	elif t <= 4.2e+60:
		tmp = ((l / k_m) * (l / k_m)) / math.pow(t, 3.0)
	else:
		tmp = 2.0 / (((math.pow((k_m * t), 2.0) / l) * 2.0) * (t / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 3e-14)
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k_m) * l) * l)) / Float64(Float64(Float64(k_m * k_m) * t) * Float64(k_m * k_m)));
	elseif (t <= 4.2e+60)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((Float64(k_m * t) ^ 2.0) / l) * 2.0) * Float64(t / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 3e-14)
		tmp = (2.0 * ((cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (k_m * k_m));
	elseif (t <= 4.2e+60)
		tmp = ((l / k_m) * (l / k_m)) / (t ^ 3.0);
	else
		tmp = 2.0 / (((((k_m * t) ^ 2.0) / l) * 2.0) * (t / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 3e-14], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e+60], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.9999999999999998e-14

   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. Taylor expanded in t around 0

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

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

   4. Applied rewrites 63.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. 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-/.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-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. *-commutative N/A

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

   6. Applied rewrites 64.2%

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

   7. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 56.5

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

   9. Applied rewrites 56.5%

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

   if 2.9999999999999998e-14 < t < 4.2000000000000002e60

   1. Initial program 79.3%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. 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 66.4

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

   4. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      5. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      7. pow2 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{\color{blue}{{t}^{3}}} \]

      9. pow2 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{{t}^{3}} \]

      10. pow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{{t}^{3}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{\color{blue}{{t}^{3}}} \]

      12. times-frac N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\color{blue}{t}}^{3}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\color{blue}{t}}^{3}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}} \]

      16. lift-pow.f64 77.7

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{\color{blue}{3}}} \]

   6. Applied rewrites 77.7%

      \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{{t}^{3}}} \]

   if 4.2000000000000002e60 < t

   1. Initial program 64.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t \cdot \left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right)}} \]
   3. 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

         \[\leadsto \frac{2}{\left(2 \cdot \frac{{t}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k} + \frac{{k}^{2} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}\right) \cdot \color{blue}{t}} \]

   4. Applied rewrites 77.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{\mathsf{fma}\left(2, {\left(\sin k \cdot t\right)}^{2}, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot t}} \]

   6. Applied rewrites 75.8%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right) \cdot t}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right) \cdot t}{\color{blue}{\left(\cos k \cdot \ell\right) \cdot \ell}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right) \cdot t}{\color{blue}{\left(\cos k \cdot \ell\right)} \cdot \ell}} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({\left(\sin k \cdot t\right)}^{2} \cdot 2 + {\left(\sin k \cdot k\right)}^{2}\right) \cdot t}{\left(\color{blue}{\cos k} \cdot \ell\right) \cdot \ell}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({\left(\sin k \cdot t\right)}^{2} \cdot 2 + {\left(\sin k \cdot k\right)}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({\left(\sin k \cdot t\right)}^{2} \cdot 2 + {\left(\sin k \cdot k\right)}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({\left(\sin k \cdot t\right)}^{2} \cdot 2 + {\left(\sin k \cdot k\right)}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({\left(\sin k \cdot t\right)}^{2} \cdot 2 + {\left(\sin k \cdot k\right)}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({\left(\sin k \cdot t\right)}^{2} \cdot 2 + {\left(\sin k \cdot k\right)}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      9. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\frac{\left({\left(\sin k \cdot t\right)}^{2} \cdot 2 + {\left(\sin k \cdot k\right)}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left({\left(\sin k \cdot t\right)}^{2} \cdot 2 + {\left(\sin k \cdot k\right)}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \color{blue}{\ell}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\frac{\left({\left(\sin k \cdot t\right)}^{2} \cdot 2 + {\left(\sin k \cdot k\right)}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      12. lift-cos.f64 N/A

          \[\leadsto \frac{2}{\frac{\left({\left(\sin k \cdot t\right)}^{2} \cdot 2 + {\left(\sin k \cdot k\right)}^{2}\right) \cdot t}{\left(\cos k \cdot \ell\right) \cdot \ell}} \]

      13. times-frac N/A

          \[\leadsto \frac{2}{\frac{{\left(\sin k \cdot t\right)}^{2} \cdot 2 + {\left(\sin k \cdot k\right)}^{2}}{\cos k \cdot \ell} \cdot \color{blue}{\frac{t}{\ell}}} \]

   8. Applied rewrites 89.5%

      \[\leadsto \frac{2}{\frac{\mathsf{fma}\left({\left(\sin k \cdot t\right)}^{2}, 2, {\left(\sin k \cdot k\right)}^{2}\right)}{\cos k \cdot \ell} \cdot \color{blue}{\frac{t}{\ell}}} \]

   9. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(2 \cdot \frac{{k}^{2} \cdot {t}^{2}}{\ell}\right) \cdot \frac{\color{blue}{t}}{\ell}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot {t}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot {t}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot {t}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}} \]

       4. pow-prod-down N/A

          \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}} \]

       5. lower-pow.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}} \]

       6. lower-*.f64 85.2

          \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell} \cdot 2\right) \cdot \frac{t}{\ell}} \]

   11. Applied rewrites 85.2%

       \[\leadsto \frac{2}{\left(\frac{{\left(k \cdot t\right)}^{2}}{\ell} \cdot 2\right) \cdot \frac{\color{blue}{t}}{\ell}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 61.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-14}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\cos k\_m \cdot \ell\right) \cdot \ell\right)}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(k\_m \cdot t\right)}^{2} \cdot t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 3e-14)
   (/ (* 2.0 (* (* (cos k_m) l) l)) (* (* (* k_m k_m) t) (* k_m k_m)))
   (if (<= t 2.4e+60)
     (/ (* (/ l k_m) (/ l k_m)) (pow t 3.0))
     (/ (* l l) (* (pow (* k_m t) 2.0) t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3e-14) {
		tmp = (2.0 * ((cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (k_m * k_m));
	} else if (t <= 2.4e+60) {
		tmp = ((l / k_m) * (l / k_m)) / pow(t, 3.0);
	} else {
		tmp = (l * l) / (pow((k_m * t), 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 <= 3d-14) then
        tmp = (2.0d0 * ((cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (k_m * k_m))
    else if (t <= 2.4d+60) then
        tmp = ((l / k_m) * (l / k_m)) / (t ** 3.0d0)
    else
        tmp = (l * l) / (((k_m * t) ** 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 <= 3e-14) {
		tmp = (2.0 * ((Math.cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (k_m * k_m));
	} else if (t <= 2.4e+60) {
		tmp = ((l / k_m) * (l / k_m)) / Math.pow(t, 3.0);
	} else {
		tmp = (l * l) / (Math.pow((k_m * t), 2.0) * t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 3e-14:
		tmp = (2.0 * ((math.cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (k_m * k_m))
	elif t <= 2.4e+60:
		tmp = ((l / k_m) * (l / k_m)) / math.pow(t, 3.0)
	else:
		tmp = (l * l) / (math.pow((k_m * t), 2.0) * t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 3e-14)
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k_m) * l) * l)) / Float64(Float64(Float64(k_m * k_m) * t) * Float64(k_m * k_m)));
	elseif (t <= 2.4e+60)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t ^ 3.0));
	else
		tmp = Float64(Float64(l * l) / Float64((Float64(k_m * t) ^ 2.0) * t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 3e-14)
		tmp = (2.0 * ((cos(k_m) * l) * l)) / (((k_m * k_m) * t) * (k_m * k_m));
	elseif (t <= 2.4e+60)
		tmp = ((l / k_m) * (l / k_m)) / (t ^ 3.0);
	else
		tmp = (l * l) / (((k_m * t) ^ 2.0) * t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 3e-14], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.4e+60], N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.9999999999999998e-14

   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. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. 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

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

   4. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot \color{blue}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      5. 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-/.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-cos.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      10. lift-sin.f64 N/A

          \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      11. pow2 N/A

          \[\leadsto \left(\frac{{\ell}^{2}}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      12. pow2 N/A

          \[\leadsto \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      13. frac-times N/A

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \cdot 2 \]

      14. *-commutative N/A

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2 \]

   6. Applied rewrites 64.2%

      \[\leadsto \frac{2 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]
   8. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 56.5

         \[\leadsto \frac{2 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   9. Applied rewrites 56.5%

      \[\leadsto \frac{2 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

   if 2.9999999999999998e-14 < t < 2.4e60

   1. Initial program 79.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. 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 66.3

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 66.3%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      5. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      7. pow2 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\frac{{\ell}^{2}}{{k}^{2}}}{\color{blue}{{t}^{3}}} \]

      9. pow2 N/A

         \[\leadsto \frac{\frac{\ell \cdot \ell}{{k}^{2}}}{{t}^{3}} \]

      10. pow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{{t}^{3}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{k \cdot k}}{\color{blue}{{t}^{3}}} \]

      12. times-frac N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\color{blue}{t}}^{3}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\color{blue}{t}}^{3}} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}} \]

      16. lift-pow.f64 77.7

          \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{\color{blue}{3}}} \]

   6. Applied rewrites 77.7%

      \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{{t}^{3}}} \]

   if 2.4e60 < t

   1. Initial program 64.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. 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 55.2

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 55.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. unpow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. lower-*.f64 55.2

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   6. Applied rewrites 55.2%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. pow-prod-down N/A

         \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      11. lower-*.f64 75.0

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

   8. Applied rewrites 75.0%

      \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot \color{blue}{t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 65.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(k\_m \cdot t\right)}^{2} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t} \cdot -0.16666666666666666\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.5e+15)
   (/ (* l l) (* (pow (* k_m t) 2.0) t))
   (* (* (/ (pow (/ l k_m) 2.0) t) -0.16666666666666666) 2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.5e+15) {
		tmp = (l * l) / (pow((k_m * t), 2.0) * t);
	} else {
		tmp = ((pow((l / k_m), 2.0) / t) * -0.16666666666666666) * 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 <= 5.5d+15) then
        tmp = (l * l) / (((k_m * t) ** 2.0d0) * t)
    else
        tmp = ((((l / k_m) ** 2.0d0) / t) * (-0.16666666666666666d0)) * 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 <= 5.5e+15) {
		tmp = (l * l) / (Math.pow((k_m * t), 2.0) * t);
	} else {
		tmp = ((Math.pow((l / k_m), 2.0) / t) * -0.16666666666666666) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 5.5e+15:
		tmp = (l * l) / (math.pow((k_m * t), 2.0) * t)
	else:
		tmp = ((math.pow((l / k_m), 2.0) / t) * -0.16666666666666666) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.5e+15)
		tmp = Float64(Float64(l * l) / Float64((Float64(k_m * t) ^ 2.0) * t));
	else
		tmp = Float64(Float64(Float64((Float64(l / k_m) ^ 2.0) / t) * -0.16666666666666666) * 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 <= 5.5e+15)
		tmp = (l * l) / (((k_m * t) ^ 2.0) * t);
	else
		tmp = ((((l / k_m) ^ 2.0) / t) * -0.16666666666666666) * 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, 5.5e+15], N[(N[(l * l), $MachinePrecision] / N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.5e15

   1. Initial program 61.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 56.5

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 56.5%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. unpow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. lower-*.f64 56.5

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   6. Applied rewrites 56.5%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. pow-prod-down N/A

         \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      11. lower-*.f64 72.0

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

   8. Applied rewrites 72.0%

      \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot \color{blue}{t}} \]

   if 5.5e15 < k

   1. Initial program 47.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. 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

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

   4. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) + \frac{{\ell}^{2}}{t}}{{k}^{4}} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) + \frac{{\ell}^{2}}{t}}{{k}^{4}} \cdot 2 \]

   7. Applied rewrites 20.2%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}} \cdot 2 \]

   8. Taylor expanded in k around inf

      \[\leadsto \left(\frac{-1}{6} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \cdot 2 \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{6}\right) \cdot 2 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{6}\right) \cdot 2 \]

      3. associate-/r* N/A

         \[\leadsto \left(\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{6}\right) \cdot 2 \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t} \cdot \frac{-1}{6}\right) \cdot 2 \]

      5. pow2 N/A

         \[\leadsto \left(\frac{\frac{\ell \cdot \ell}{{k}^{2}}}{t} \cdot \frac{-1}{6}\right) \cdot 2 \]

      6. pow2 N/A

         \[\leadsto \left(\frac{\frac{\ell \cdot \ell}{k \cdot k}}{t} \cdot \frac{-1}{6}\right) \cdot 2 \]

      7. times-frac N/A

         \[\leadsto \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{-1}{6}\right) \cdot 2 \]

      8. pow2 N/A

         \[\leadsto \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{-1}{6}\right) \cdot 2 \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{-1}{6}\right) \cdot 2 \]

      10. lower-/.f64 59.0

          \[\leadsto \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.16666666666666666\right) \cdot 2 \]

   10. Applied rewrites 59.0%

       \[\leadsto \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.16666666666666666\right) \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 65.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(k\_m \cdot t\right)}^{2} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \left(\ell \cdot \ell\right)}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.5e+15)
   (/ (* l l) (* (pow (* k_m t) 2.0) t))
   (* (/ (* -0.16666666666666666 (* l l)) (* (* k_m k_m) t)) 2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.5e+15) {
		tmp = (l * l) / (pow((k_m * t), 2.0) * t);
	} else {
		tmp = ((-0.16666666666666666 * (l * l)) / ((k_m * k_m) * t)) * 2.0;
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5.5d+15) then
        tmp = (l * l) / (((k_m * t) ** 2.0d0) * t)
    else
        tmp = (((-0.16666666666666666d0) * (l * l)) / ((k_m * k_m) * t)) * 2.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.5e+15) {
		tmp = (l * l) / (Math.pow((k_m * t), 2.0) * t);
	} else {
		tmp = ((-0.16666666666666666 * (l * l)) / ((k_m * k_m) * t)) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 5.5e+15:
		tmp = (l * l) / (math.pow((k_m * t), 2.0) * t)
	else:
		tmp = ((-0.16666666666666666 * (l * l)) / ((k_m * k_m) * t)) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.5e+15)
		tmp = Float64(Float64(l * l) / Float64((Float64(k_m * t) ^ 2.0) * t));
	else
		tmp = Float64(Float64(Float64(-0.16666666666666666 * Float64(l * l)) / Float64(Float64(k_m * k_m) * t)) * 2.0);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 5.5e+15)
		tmp = (l * l) / (((k_m * t) ^ 2.0) * t);
	else
		tmp = ((-0.16666666666666666 * (l * l)) / ((k_m * k_m) * t)) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5.5e+15], N[(N[(l * l), $MachinePrecision] / N[(N[Power[N[(k$95$m * t), $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.5e15

   1. Initial program 61.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 56.5

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 56.5%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. unpow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. lower-*.f64 56.5

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   6. Applied rewrites 56.5%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot t\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \left({t}^{2} \cdot t\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left({k}^{2} \cdot {t}^{2}\right) \cdot \color{blue}{t}} \]

      9. pow-prod-down N/A

         \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

      11. lower-*.f64 72.0

          \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot t} \]

   8. Applied rewrites 72.0%

      \[\leadsto \frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2} \cdot \color{blue}{t}} \]

   if 5.5e15 < k

   1. Initial program 47.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. 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

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

   4. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) + \frac{{\ell}^{2}}{t}}{{k}^{4}} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) + \frac{{\ell}^{2}}{t}}{{k}^{4}} \cdot 2 \]

   7. Applied rewrites 20.2%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}} \cdot 2 \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}} \cdot 2 \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{\left(2 + 2\right)}} \cdot 2 \]

      3. pow-prod-up N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{2} \cdot {k}^{2}} \cdot 2 \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{2} \cdot {k}^{2}} \cdot 2 \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot 2 \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot 2 \]

      7. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

      8. lift-*.f64 20.2

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

   9. Applied rewrites 20.2%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

   10. Taylor expanded in k around inf

       \[\leadsto \left(\frac{-1}{6} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \cdot 2 \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \cdot 2 \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \cdot 2 \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \cdot 2 \]

       4. pow2 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot t} \cdot 2 \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot t} \cdot 2 \]

       6. pow2 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \cdot 2 \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \cdot 2 \]

       8. lift-*.f64 57.8

          \[\leadsto \frac{-0.16666666666666666 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \cdot 2 \]

   12. Applied rewrites 57.8%

       \[\leadsto \frac{-0.16666666666666666 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 56.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t}}{\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot k\_m\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(k\_m \cdot k\_m\right) \cdot {t}^{3}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.1e-8)
   (* (/ (/ (* l l) t) (* (* k_m k_m) (* k_m k_m))) 2.0)
   (* l (/ l (* (* k_m k_m) (pow t 3.0))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.1e-8) {
		tmp = (((l * l) / t) / ((k_m * k_m) * (k_m * k_m))) * 2.0;
	} else {
		tmp = l * (l / ((k_m * k_m) * pow(t, 3.0)));
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 2.1d-8) then
        tmp = (((l * l) / t) / ((k_m * k_m) * (k_m * k_m))) * 2.0d0
    else
        tmp = l * (l / ((k_m * k_m) * (t ** 3.0d0)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.1e-8) {
		tmp = (((l * l) / t) / ((k_m * k_m) * (k_m * k_m))) * 2.0;
	} else {
		tmp = l * (l / ((k_m * k_m) * Math.pow(t, 3.0)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 2.1e-8:
		tmp = (((l * l) / t) / ((k_m * k_m) * (k_m * k_m))) * 2.0
	else:
		tmp = l * (l / ((k_m * k_m) * math.pow(t, 3.0)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.1e-8)
		tmp = Float64(Float64(Float64(Float64(l * l) / t) / Float64(Float64(k_m * k_m) * Float64(k_m * k_m))) * 2.0);
	else
		tmp = Float64(l * Float64(l / Float64(Float64(k_m * k_m) * (t ^ 3.0))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 2.1e-8)
		tmp = (((l * l) / t) / ((k_m * k_m) * (k_m * k_m))) * 2.0;
	else
		tmp = l * (l / ((k_m * k_m) * (t ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.1e-8], N[(N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(l * N[(l / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.09999999999999994e-8

   1. Initial program 50.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. 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

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

   4. Applied rewrites 63.7%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) + \frac{{\ell}^{2}}{t}}{{k}^{4}} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) + \frac{{\ell}^{2}}{t}}{{k}^{4}} \cdot 2 \]

   7. Applied rewrites 23.1%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}} \cdot 2 \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}} \cdot 2 \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{\left(2 + 2\right)}} \cdot 2 \]

      3. pow-prod-up N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{2} \cdot {k}^{2}} \cdot 2 \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{2} \cdot {k}^{2}} \cdot 2 \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot 2 \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot 2 \]

      7. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

      8. lift-*.f64 23.1

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

   9. Applied rewrites 23.1%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{\frac{{\ell}^{2}}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]
   11. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

       3. lift-*.f64 53.9

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

   12. Applied rewrites 53.9%

       \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

   if 2.09999999999999994e-8 < t

   1. Initial program 67.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. 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 57.7

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 57.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      3. associate-/l* N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lift-*.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      7. pow2 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {\color{blue}{t}}^{3}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{{k}^{2} \cdot {t}^{\color{blue}{3}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \ell \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      10. pow2 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      11. lift-pow.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      12. lift-*.f64 N/A

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \color{blue}{{t}^{3}}} \]

      13. lift-*.f64 61.9

          \[\leadsto \ell \cdot \frac{\ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

   6. Applied rewrites 61.9%

      \[\leadsto \ell \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 45.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{t}\\ \mathbf{if}\;\ell \leq 1.35 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(t\_1 \cdot -0.16666666666666666\right) \cdot k\_m, k\_m, t\_1\right)}{k\_m \cdot k\_m}}{k\_m \cdot k\_m} \cdot 2\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+145}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(k\_m \cdot k\_m\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\left(\mathsf{fma}\left(-0.5, k\_m \cdot k\_m, 1\right) \cdot \ell\right) \cdot \ell\right)}{\left(\left(k\_m \cdot k\_m\right) \cdot t\right) \cdot \left(k\_m \cdot k\_m\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (/ (* l l) t)))
   (if (<= l 1.35e-112)
     (*
      (/
       (/ (fma (* (* t_1 -0.16666666666666666) k_m) k_m t_1) (* k_m k_m))
       (* k_m k_m))
      2.0)
     (if (<= l 6.8e+145)
       (/ (* l l) (* (* k_m k_m) (* (* t t) t)))
       (/
        (* 2.0 (* (* (fma -0.5 (* k_m k_m) 1.0) l) l))
        (* (* (* k_m k_m) t) (* k_m k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (l * l) / t;
	double tmp;
	if (l <= 1.35e-112) {
		tmp = ((fma(((t_1 * -0.16666666666666666) * k_m), k_m, t_1) / (k_m * k_m)) / (k_m * k_m)) * 2.0;
	} else if (l <= 6.8e+145) {
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	} else {
		tmp = (2.0 * ((fma(-0.5, (k_m * k_m), 1.0) * l) * l)) / (((k_m * k_m) * t) * (k_m * k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(l * l) / t)
	tmp = 0.0
	if (l <= 1.35e-112)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(t_1 * -0.16666666666666666) * k_m), k_m, t_1) / Float64(k_m * k_m)) / Float64(k_m * k_m)) * 2.0);
	elseif (l <= 6.8e+145)
		tmp = Float64(Float64(l * l) / Float64(Float64(k_m * k_m) * Float64(Float64(t * t) * t)));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(fma(-0.5, Float64(k_m * k_m), 1.0) * l) * l)) / Float64(Float64(Float64(k_m * k_m) * t) * Float64(k_m * k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[l, 1.35e-112], N[(N[(N[(N[(N[(N[(t$95$1 * -0.16666666666666666), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m + t$95$1), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] / N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[l, 6.8e+145], N[(N[(l * l), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[(-0.5 * N[(k$95$m * k$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.35e-112

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. 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

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

   4. Applied rewrites 60.3%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) + \frac{{\ell}^{2}}{t}}{{k}^{4}} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) + \frac{{\ell}^{2}}{t}}{{k}^{4}} \cdot 2 \]

   7. Applied rewrites 25.3%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}} \cdot 2 \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}} \cdot 2 \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{\left(2 + 2\right)}} \cdot 2 \]

      3. pow-prod-up N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{2} \cdot {k}^{2}} \cdot 2 \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{2} \cdot {k}^{2}} \cdot 2 \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot 2 \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot 2 \]

      7. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

      8. lift-*.f64 25.3

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

   9. Applied rewrites 25.3%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

       2. lift-fma.f64 N/A

          \[\leadsto \frac{\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}\right) \cdot \left(k \cdot k\right) + \frac{\ell \cdot \ell}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}\right) \cdot \left(k \cdot k\right) + \frac{\ell \cdot \ell}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}\right) \cdot \left(k \cdot k\right) + \frac{\ell \cdot \ell}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

       5. lift-/.f64 N/A

          \[\leadsto \frac{\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}\right) \cdot \left(k \cdot k\right) + \frac{\ell \cdot \ell}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}\right) \cdot \left(k \cdot k\right) + \frac{\ell \cdot \ell}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}\right) \cdot \left(k \cdot k\right) + \frac{\ell \cdot \ell}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

       8. lift-/.f64 N/A

          \[\leadsto \frac{\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}\right) \cdot \left(k \cdot k\right) + \frac{\ell \cdot \ell}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}\right) \cdot \left(k \cdot k\right) + \frac{\ell \cdot \ell}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}\right) \cdot \left(k \cdot k\right) + \frac{\ell \cdot \ell}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

       11. lift-*.f64 N/A

           \[\leadsto \frac{\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}\right) \cdot \left(k \cdot k\right) + \frac{\ell \cdot \ell}{t}}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

       12. pow2 N/A

           \[\leadsto \frac{\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}\right) \cdot \left(k \cdot k\right) + \frac{\ell \cdot \ell}{t}}{{k}^{2} \cdot \left(k \cdot k\right)} \cdot 2 \]

       13. pow2 N/A

           \[\leadsto \frac{\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}\right) \cdot \left(k \cdot k\right) + \frac{\ell \cdot \ell}{t}}{{k}^{2} \cdot {k}^{2}} \cdot 2 \]

       14. associate-/r* N/A

           \[\leadsto \frac{\frac{\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}\right) \cdot \left(k \cdot k\right) + \frac{\ell \cdot \ell}{t}}{{k}^{2}}}{{k}^{2}} \cdot 2 \]

       15. lower-/.f64 N/A

           \[\leadsto \frac{\frac{\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}\right) \cdot \left(k \cdot k\right) + \frac{\ell \cdot \ell}{t}}{{k}^{2}}}{{k}^{2}} \cdot 2 \]

   11. Applied rewrites 41.3%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666\right) \cdot k, k, \frac{\ell \cdot \ell}{t}\right)}{k \cdot k}}{k \cdot k} \cdot 2 \]

   if 1.35e-112 < l < 6.7999999999999998e145

   1. Initial program 65.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. 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 59.7

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. unpow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. lower-*.f64 59.7

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   6. Applied rewrites 59.7%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   if 6.7999999999999998e145 < l

   1. Initial program 35.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. 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

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

   4. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot \color{blue}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      5. 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-/.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-cos.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      10. lift-sin.f64 N/A

          \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      11. pow2 N/A

          \[\leadsto \left(\frac{{\ell}^{2}}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      12. pow2 N/A

          \[\leadsto \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      13. frac-times N/A

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \cdot 2 \]

      14. *-commutative N/A

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2 \]

   6. Applied rewrites 52.4%

      \[\leadsto \frac{2 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]
   8. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 49.5

         \[\leadsto \frac{2 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   9. Applied rewrites 49.5%

      \[\leadsto \frac{2 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2 \cdot \left(\left(\left(1 + \frac{-1}{2} \cdot {k}^{2}\right) \cdot \ell\right) \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{2 \cdot \left(\left(\left(\frac{-1}{2} \cdot {k}^{2} + 1\right) \cdot \ell\right) \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{2 \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{2}, {k}^{2}, 1\right) \cdot \ell\right) \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       3. pow2 N/A

          \[\leadsto \frac{2 \cdot \left(\left(\mathsf{fma}\left(\frac{-1}{2}, k \cdot k, 1\right) \cdot \ell\right) \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

       4. lift-*.f64 44.1

          \[\leadsto \frac{2 \cdot \left(\left(\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \ell\right) \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   12. Applied rewrites 44.1%

       \[\leadsto \frac{2 \cdot \left(\left(\mathsf{fma}\left(-0.5, k \cdot k, 1\right) \cdot \ell\right) \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 58.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := \left(k\_m \cdot k\_m\right) \cdot t\\ \mathbf{if}\;k\_m \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{t\_1 \cdot \left(k\_m \cdot k\_m\right)}\\ \mathbf{elif}\;k\_m \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(k\_m \cdot k\_m\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \left(\ell \cdot \ell\right)}{t\_1} \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (* (* k_m k_m) t)))
   (if (<= k_m 1.55e-162)
     (/ (* 2.0 (* l l)) (* t_1 (* k_m k_m)))
     (if (<= k_m 5.5e+15)
       (/ (* l l) (* (* k_m k_m) (* (* t t) t)))
       (* (/ (* -0.16666666666666666 (* l l)) t_1) 2.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = (k_m * k_m) * t;
	double tmp;
	if (k_m <= 1.55e-162) {
		tmp = (2.0 * (l * l)) / (t_1 * (k_m * k_m));
	} else if (k_m <= 5.5e+15) {
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	} else {
		tmp = ((-0.16666666666666666 * (l * l)) / t_1) * 2.0;
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k_m * k_m) * t
    if (k_m <= 1.55d-162) then
        tmp = (2.0d0 * (l * l)) / (t_1 * (k_m * k_m))
    else if (k_m <= 5.5d+15) then
        tmp = (l * l) / ((k_m * k_m) * ((t * t) * t))
    else
        tmp = (((-0.16666666666666666d0) * (l * l)) / t_1) * 2.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = (k_m * k_m) * t;
	double tmp;
	if (k_m <= 1.55e-162) {
		tmp = (2.0 * (l * l)) / (t_1 * (k_m * k_m));
	} else if (k_m <= 5.5e+15) {
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	} else {
		tmp = ((-0.16666666666666666 * (l * l)) / t_1) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	t_1 = (k_m * k_m) * t
	tmp = 0
	if k_m <= 1.55e-162:
		tmp = (2.0 * (l * l)) / (t_1 * (k_m * k_m))
	elif k_m <= 5.5e+15:
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t))
	else:
		tmp = ((-0.16666666666666666 * (l * l)) / t_1) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = Float64(Float64(k_m * k_m) * t)
	tmp = 0.0
	if (k_m <= 1.55e-162)
		tmp = Float64(Float64(2.0 * Float64(l * l)) / Float64(t_1 * Float64(k_m * k_m)));
	elseif (k_m <= 5.5e+15)
		tmp = Float64(Float64(l * l) / Float64(Float64(k_m * k_m) * Float64(Float64(t * t) * t)));
	else
		tmp = Float64(Float64(Float64(-0.16666666666666666 * Float64(l * l)) / t_1) * 2.0);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	t_1 = (k_m * k_m) * t;
	tmp = 0.0;
	if (k_m <= 1.55e-162)
		tmp = (2.0 * (l * l)) / (t_1 * (k_m * k_m));
	elseif (k_m <= 5.5e+15)
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	else
		tmp = ((-0.16666666666666666 * (l * l)) / t_1) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[k$95$m, 1.55e-162], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 5.5e+15], N[(N[(l * l), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * N[(l * l), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.5499999999999999e-162

   1. Initial program 62.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. 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

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

   4. Applied rewrites 51.3%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot \color{blue}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      5. 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-/.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-cos.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      9. lift-pow.f64 N/A

         \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      10. lift-sin.f64 N/A

          \[\leadsto \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      11. pow2 N/A

          \[\leadsto \left(\frac{{\ell}^{2}}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      12. pow2 N/A

          \[\leadsto \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]

      13. frac-times N/A

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \cdot 2 \]

      14. *-commutative N/A

          \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot 2 \]

   6. Applied rewrites 51.3%

      \[\leadsto \frac{2 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {\sin k}^{2}}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \frac{2 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot {k}^{\color{blue}{2}}} \]
   8. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

      2. lift-*.f64 51.3

         \[\leadsto \frac{2 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   9. Applied rewrites 51.3%

      \[\leadsto \frac{2 \cdot \left(\left(\cos k \cdot \ell\right) \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot \color{blue}{k}\right)} \]

   10. Taylor expanded in k around 0

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 51.3%

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(k \cdot k\right)} \]

   if 1.5499999999999999e-162 < k < 5.5e15

   1. Initial program 61.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. 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 65.4

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 65.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. unpow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. lower-*.f64 65.4

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   6. Applied rewrites 65.4%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   if 5.5e15 < k

   1. Initial program 47.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. 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

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

   4. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) + \frac{{\ell}^{2}}{t}}{{k}^{4}} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) + \frac{{\ell}^{2}}{t}}{{k}^{4}} \cdot 2 \]

   7. Applied rewrites 20.2%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}} \cdot 2 \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}} \cdot 2 \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{\left(2 + 2\right)}} \cdot 2 \]

      3. pow-prod-up N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{2} \cdot {k}^{2}} \cdot 2 \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{2} \cdot {k}^{2}} \cdot 2 \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot 2 \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot 2 \]

      7. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

      8. lift-*.f64 20.2

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

   9. Applied rewrites 20.2%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

   10. Taylor expanded in k around inf

       \[\leadsto \left(\frac{-1}{6} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \cdot 2 \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \cdot 2 \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \cdot 2 \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \cdot 2 \]

       4. pow2 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot t} \cdot 2 \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot t} \cdot 2 \]

       6. pow2 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \cdot 2 \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \cdot 2 \]

       8. lift-*.f64 57.8

          \[\leadsto \frac{-0.16666666666666666 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \cdot 2 \]

   12. Applied rewrites 57.8%

       \[\leadsto \frac{-0.16666666666666666 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \cdot 2 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 57.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\left(k\_m \cdot k\_m\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \left(\ell \cdot \ell\right)}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot 2\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.5e+15)
   (/ (* l l) (* (* k_m k_m) (* (* t t) t)))
   (* (/ (* -0.16666666666666666 (* l l)) (* (* k_m k_m) t)) 2.0)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.5e+15) {
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	} else {
		tmp = ((-0.16666666666666666 * (l * l)) / ((k_m * k_m) * t)) * 2.0;
	}
	return tmp;
}

Fortran

k_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5.5d+15) then
        tmp = (l * l) / ((k_m * k_m) * ((t * t) * t))
    else
        tmp = (((-0.16666666666666666d0) * (l * l)) / ((k_m * k_m) * t)) * 2.0d0
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.5e+15) {
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	} else {
		tmp = ((-0.16666666666666666 * (l * l)) / ((k_m * k_m) * t)) * 2.0;
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 5.5e+15:
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t))
	else:
		tmp = ((-0.16666666666666666 * (l * l)) / ((k_m * k_m) * t)) * 2.0
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.5e+15)
		tmp = Float64(Float64(l * l) / Float64(Float64(k_m * k_m) * Float64(Float64(t * t) * t)));
	else
		tmp = Float64(Float64(Float64(-0.16666666666666666 * Float64(l * l)) / Float64(Float64(k_m * k_m) * t)) * 2.0);
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 5.5e+15)
		tmp = (l * l) / ((k_m * k_m) * ((t * t) * t));
	else
		tmp = ((-0.16666666666666666 * (l * l)) / ((k_m * k_m) * t)) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5.5e+15], N[(N[(l * l), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.5e15

   1. Initial program 61.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot {t}^{3}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{k}^{2}} \cdot {t}^{3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot \color{blue}{{t}^{3}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {\color{blue}{t}}^{3}} \]

      7. lift-pow.f64 56.5

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

   4. Applied rewrites 56.5%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{\color{blue}{3}}} \]

      2. unpow3 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot t\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left({t}^{2} \cdot \color{blue}{t}\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

      6. lower-*.f64 56.5

         \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)} \]

   6. Applied rewrites 56.5%

      \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{t}\right)} \]

   if 5.5e15 < k

   1. Initial program 47.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   3. 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

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

   4. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) + \frac{{\ell}^{2}}{t}}{{k}^{4}} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) + \frac{{\ell}^{2}}{t}}{{k}^{4}} \cdot 2 \]

   7. Applied rewrites 20.2%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}} \cdot 2 \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}} \cdot 2 \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{\left(2 + 2\right)}} \cdot 2 \]

      3. pow-prod-up N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{2} \cdot {k}^{2}} \cdot 2 \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{2} \cdot {k}^{2}} \cdot 2 \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot 2 \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot 2 \]

      7. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

      8. lift-*.f64 20.2

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

   9. Applied rewrites 20.2%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

   10. Taylor expanded in k around inf

       \[\leadsto \left(\frac{-1}{6} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \cdot 2 \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \cdot 2 \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \cdot 2 \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \cdot 2 \]

       4. pow2 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot t} \cdot 2 \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot t} \cdot 2 \]

       6. pow2 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \cdot 2 \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \cdot 2 \]

       8. lift-*.f64 57.8

          \[\leadsto \frac{-0.16666666666666666 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \cdot 2 \]

   12. Applied rewrites 57.8%

       \[\leadsto \frac{-0.16666666666666666 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 32.5% accurate, 12.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{-0.16666666666666666 \cdot \left(\ell \cdot \ell\right)}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot 2 \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (* -0.16666666666666666 (* l l)) (* (* k_m k_m) t)) 2.0))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((-0.16666666666666666 * (l * l)) / ((k_m * k_m) * t)) * 2.0;
}

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 = (((-0.16666666666666666d0) * (l * l)) / ((k_m * k_m) * t)) * 2.0d0
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((-0.16666666666666666 * (l * l)) / ((k_m * k_m) * t)) * 2.0;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((-0.16666666666666666 * (l * l)) / ((k_m * k_m) * t)) * 2.0

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(-0.16666666666666666 * Float64(l * l)) / Float64(Float64(k_m * k_m) * t)) * 2.0)
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((-0.16666666666666666 * (l * l)) / ((k_m * k_m) * t)) * 2.0;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(-0.16666666666666666 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 55.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in t around 0

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
3. 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

      \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \cdot \color{blue}{2} \]

4. Applied rewrites 60.2%

   \[\leadsto \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]

5. Taylor expanded in k around 0

   \[\leadsto \frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) + \frac{{\ell}^{2}}{t}}{{k}^{4}} \cdot 2 \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{k}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\ell}^{2}}{t} - \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right) + \frac{{\ell}^{2}}{t}}{{k}^{4}} \cdot 2 \]

7. Applied rewrites 23.0%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}} \cdot 2 \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{4}} \cdot 2 \]

   2. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{\left(2 + 2\right)}} \cdot 2 \]

   3. pow-prod-up N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{2} \cdot {k}^{2}} \cdot 2 \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{{k}^{2} \cdot {k}^{2}} \cdot 2 \]

   5. pow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot 2 \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot {k}^{2}} \cdot 2 \]

   7. pow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{6}, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

   8. lift-*.f64 23.0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

9. Applied rewrites 23.0%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666, k \cdot k, \frac{\ell \cdot \ell}{t}\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

10. Taylor expanded in k around inf

    \[\leadsto \left(\frac{-1}{6} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) \cdot 2 \]
11. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{-1}{6} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \cdot 2 \]

    2. lower-/.f64 N/A

       \[\leadsto \frac{\frac{-1}{6} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \cdot 2 \]

    3. lower-*.f64 N/A

       \[\leadsto \frac{\frac{-1}{6} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \cdot 2 \]

    4. pow2 N/A

       \[\leadsto \frac{\frac{-1}{6} \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot t} \cdot 2 \]

    5. lift-*.f64 N/A

       \[\leadsto \frac{\frac{-1}{6} \cdot \left(\ell \cdot \ell\right)}{{k}^{2} \cdot t} \cdot 2 \]

    6. pow2 N/A

       \[\leadsto \frac{\frac{-1}{6} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \cdot 2 \]

    7. lift-*.f64 N/A

       \[\leadsto \frac{\frac{-1}{6} \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \cdot 2 \]

    8. lift-*.f64 32.5

       \[\leadsto \frac{-0.16666666666666666 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \cdot 2 \]

12. Applied rewrites 32.5%

    \[\leadsto \frac{-0.16666666666666666 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot t} \cdot 2 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025095 
(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))))