Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.5% → 83.5%

Time: 8.2s

Alternatives: 10

Speedup: 5.7×

• 40.3% 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: 36.5% 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: 83.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0085:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(k\_m \cdot k\_m, -0.3333333333333333, 2\right)}{\left(\left(k\_m \cdot k\_m\right) \cdot k\_m\right) \cdot \left(k\_m \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\cos k\_m \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot k\_m}}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.0085)
   (*
    l
    (*
     l
     (/
      (fma (* k_m k_m) -0.3333333333333333 2.0)
      (* (* (* k_m k_m) k_m) (* k_m t)))))
   (/
    (/
     (* (* (* (cos k_m) l) l) 2.0)
     (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) k_m))
    k_m)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0085) {
		tmp = l * (l * (fma((k_m * k_m), -0.3333333333333333, 2.0) / (((k_m * k_m) * k_m) * (k_m * t))));
	} else {
		tmp = ((((cos(k_m) * l) * l) * 2.0) / (((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m)) / k_m;
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.0085)
		tmp = Float64(l * Float64(l * Float64(fma(Float64(k_m * k_m), -0.3333333333333333, 2.0) / Float64(Float64(Float64(k_m * k_m) * k_m) * Float64(k_m * t)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(cos(k_m) * l) * l) * 2.0) / Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * k_m)) / k_m);
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.0085], N[(l * N[(l * N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * -0.3333333333333333 + 2.0), $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0085000000000000006

   1. Initial program 36.5%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 48.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   8. Applied rewrites 55.2%

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

   if 0.0085000000000000006 < k

   1. Initial program 36.5%

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

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. sqr-sin-a-rev N/A

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

      6. unpow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-sin.f64 74.3

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

   6. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-cos.f64 74.4

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

   8. Applied rewrites 74.4%

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

   10. Applied rewrites 72.2%

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

Alternative 2: 83.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0085:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(k\_m \cdot k\_m, -0.3333333333333333, 2\right)}{\left(\left(k\_m \cdot k\_m\right) \cdot k\_m\right) \cdot \left(k\_m \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot k\_m} \cdot \frac{\left(\cos k\_m \cdot \ell\right) \cdot \ell}{k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.0085)
   (*
    l
    (*
     l
     (/
      (fma (* k_m k_m) -0.3333333333333333 2.0)
      (* (* (* k_m k_m) k_m) (* k_m t)))))
   (*
    (/ 2.0 (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) k_m))
    (/ (* (* (cos k_m) l) l) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0085) {
		tmp = l * (l * (fma((k_m * k_m), -0.3333333333333333, 2.0) / (((k_m * k_m) * k_m) * (k_m * t))));
	} else {
		tmp = (2.0 / (((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m)) * (((cos(k_m) * l) * l) / k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.0085)
		tmp = Float64(l * Float64(l * Float64(fma(Float64(k_m * k_m), -0.3333333333333333, 2.0) / Float64(Float64(Float64(k_m * k_m) * k_m) * Float64(k_m * t)))));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * k_m)) * Float64(Float64(Float64(cos(k_m) * l) * l) / k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.0085], N[(l * N[(l * N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * -0.3333333333333333 + 2.0), $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0085000000000000006

   1. Initial program 36.5%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 48.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   8. Applied rewrites 55.2%

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

   if 0.0085000000000000006 < k

   1. Initial program 36.5%

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

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. sqr-sin-a-rev N/A

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

      6. unpow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-sin.f64 74.3

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

   6. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-cos.f64 74.4

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

   8. Applied rewrites 74.4%

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

   10. Applied rewrites 72.3%

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

Alternative 3: 81.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0085:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(k\_m \cdot k\_m, -0.3333333333333333, 2\right)}{\left(\left(k\_m \cdot k\_m\right) \cdot k\_m\right) \cdot \left(k\_m \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k\_m \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.0085)
   (*
    l
    (*
     l
     (/
      (fma (* k_m k_m) -0.3333333333333333 2.0)
      (* (* (* k_m k_m) k_m) (* k_m t)))))
   (/
    (* 2.0 (* (cos k_m) (* l l)))
    (* (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) k_m) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0085) {
		tmp = l * (l * (fma((k_m * k_m), -0.3333333333333333, 2.0) / (((k_m * k_m) * k_m) * (k_m * t))));
	} else {
		tmp = (2.0 * (cos(k_m) * (l * l))) / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.0085)
		tmp = Float64(l * Float64(l * Float64(fma(Float64(k_m * k_m), -0.3333333333333333, 2.0) / Float64(Float64(Float64(k_m * k_m) * k_m) * Float64(k_m * t)))));
	else
		tmp = Float64(Float64(2.0 * Float64(cos(k_m) * Float64(l * l))) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * k_m) * k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.0085], N[(l * N[(l * N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * -0.3333333333333333 + 2.0), $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0085000000000000006

   1. Initial program 36.5%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 48.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   8. Applied rewrites 55.2%

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

   if 0.0085000000000000006 < k

   1. Initial program 36.5%

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

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 70.6%

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

Alternative 4: 81.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0085:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(k\_m \cdot k\_m, -0.3333333333333333, 2\right)}{\left(\left(k\_m \cdot k\_m\right) \cdot k\_m\right) \cdot \left(k\_m \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\cos k\_m \cdot \ell\right) \cdot \ell\right) \cdot 2}{\left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.0085)
   (*
    l
    (*
     l
     (/
      (fma (* k_m k_m) -0.3333333333333333 2.0)
      (* (* (* k_m k_m) k_m) (* k_m t)))))
   (/
    (* (* (* (cos k_m) l) l) 2.0)
    (* (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) k_m) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0085) {
		tmp = l * (l * (fma((k_m * k_m), -0.3333333333333333, 2.0) / (((k_m * k_m) * k_m) * (k_m * t))));
	} else {
		tmp = (((cos(k_m) * l) * l) * 2.0) / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.0085)
		tmp = Float64(l * Float64(l * Float64(fma(Float64(k_m * k_m), -0.3333333333333333, 2.0) / Float64(Float64(Float64(k_m * k_m) * k_m) * Float64(k_m * t)))));
	else
		tmp = Float64(Float64(Float64(Float64(cos(k_m) * l) * l) * 2.0) / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * k_m) * k_m));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.0085], N[(l * N[(l * N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * -0.3333333333333333 + 2.0), $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0085000000000000006

   1. Initial program 36.5%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 48.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   8. Applied rewrites 55.2%

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

   if 0.0085000000000000006 < k

   1. Initial program 36.5%

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

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.0%

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

   6. Applied rewrites 70.6%

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

Alternative 5: 81.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 0.0085:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\mathsf{fma}\left(k\_m \cdot k\_m, -0.3333333333333333, 2\right)}{\left(\left(k\_m \cdot k\_m\right) \cdot k\_m\right) \cdot \left(k\_m \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\cos k\_m \cdot \ell\right) \cdot \ell\right) \cdot \frac{2}{\left(\left(\left(0.5 - \cos \left(k\_m + k\_m\right) \cdot 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 0.0085)
   (*
    l
    (*
     l
     (/
      (fma (* k_m k_m) -0.3333333333333333 2.0)
      (* (* (* k_m k_m) k_m) (* k_m t)))))
   (*
    (* (* (cos k_m) l) l)
    (/ 2.0 (* (* (* (- 0.5 (* (cos (+ k_m k_m)) 0.5)) t) k_m) k_m)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 0.0085) {
		tmp = l * (l * (fma((k_m * k_m), -0.3333333333333333, 2.0) / (((k_m * k_m) * k_m) * (k_m * t))));
	} else {
		tmp = ((cos(k_m) * l) * l) * (2.0 / ((((0.5 - (cos((k_m + k_m)) * 0.5)) * t) * k_m) * k_m));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 0.0085)
		tmp = Float64(l * Float64(l * Float64(fma(Float64(k_m * k_m), -0.3333333333333333, 2.0) / Float64(Float64(Float64(k_m * k_m) * k_m) * Float64(k_m * t)))));
	else
		tmp = Float64(Float64(Float64(cos(k_m) * l) * l) * Float64(2.0 / Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(k_m + k_m)) * 0.5)) * t) * k_m) * k_m)));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 0.0085], N[(l * N[(l * N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * -0.3333333333333333 + 2.0), $MachinePrecision] / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * l), $MachinePrecision] * l), $MachinePrecision] * N[(2.0 / N[(N[(N[(N[(0.5 - N[(N[Cos[N[(k$95$m + k$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0085000000000000006

   1. Initial program 36.5%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 48.5%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(k \cdot k, -0.3333333333333333, 2\right)}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   8. Applied rewrites 55.2%

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

   if 0.0085000000000000006 < k

   1. Initial program 36.5%

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

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. sqr-sin-a-rev N/A

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

      6. unpow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-sin.f64 74.3

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

   6. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-cos.f64 74.4

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

   8. Applied rewrites 74.4%

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

   10. Applied rewrites 70.6%

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

Alternative 6: 69.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{+216}:\\ \;\;\;\;\left(\ell \cdot \frac{\ell}{\left(\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot t}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k\_m \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(\left(0.5 - 0.5\right) \cdot t\right) \cdot k\_m\right) \cdot k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= l 6e+216)
   (* (* l (/ l (* (* (* k_m k_m) (* k_m k_m)) t))) 2.0)
   (/ (* 2.0 (* (cos k_m) (* l l))) (* (* (* (- 0.5 0.5) t) k_m) k_m))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 6e+216) {
		tmp = (l * (l / (((k_m * k_m) * (k_m * k_m)) * t))) * 2.0;
	} else {
		tmp = (2.0 * (cos(k_m) * (l * l))) / ((((0.5 - 0.5) * t) * k_m) * k_m);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 6d+216) then
        tmp = (l * (l / (((k_m * k_m) * (k_m * k_m)) * t))) * 2.0d0
    else
        tmp = (2.0d0 * (cos(k_m) * (l * l))) / ((((0.5d0 - 0.5d0) * t) * k_m) * k_m)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (l <= 6e+216) {
		tmp = (l * (l / (((k_m * k_m) * (k_m * k_m)) * t))) * 2.0;
	} else {
		tmp = (2.0 * (Math.cos(k_m) * (l * l))) / ((((0.5 - 0.5) * t) * k_m) * k_m);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if l <= 6e+216:
		tmp = (l * (l / (((k_m * k_m) * (k_m * k_m)) * t))) * 2.0
	else:
		tmp = (2.0 * (math.cos(k_m) * (l * l))) / ((((0.5 - 0.5) * t) * k_m) * k_m)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (l <= 6e+216)
		tmp = Float64(Float64(l * Float64(l / Float64(Float64(Float64(k_m * k_m) * Float64(k_m * k_m)) * t))) * 2.0);
	else
		tmp = Float64(Float64(2.0 * Float64(cos(k_m) * Float64(l * l))) / Float64(Float64(Float64(Float64(0.5 - 0.5) * t) * k_m) * k_m));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (l <= 6e+216)
		tmp = (l * (l / (((k_m * k_m) * (k_m * k_m)) * t))) * 2.0;
	else
		tmp = (2.0 * (cos(k_m) * (l * l))) / ((((0.5 - 0.5) * t) * k_m) * k_m);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[l, 6e+216], N[(N[(l * N[(l / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(0.5 - 0.5), $MachinePrecision] * t), $MachinePrecision] * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.9999999999999995e216

   1. Initial program 36.5%

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

   2. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

      4. pow2 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot \color{blue}{t}} \]

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 62.8

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

   4. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. pow-prod-up N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t} \]

      13. associate-*r/ N/A

          \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

      14. *-commutative N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   6. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow3 N/A

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

      10. pow-plus N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow-prod-up N/A

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

      16. unpow-prod-down N/A

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

      17. lift-*.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 68.8

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

   8. Applied rewrites 68.8%

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

   if 5.9999999999999995e216 < l

   1. Initial program 36.5%

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

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t\right) \cdot \left(k \cdot k\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 70.6%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 37.4%

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

Alternative 7: 68.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \left(\ell \cdot \frac{\ell}{\left(\left(k\_m \cdot k\_m\right) \cdot \left(k\_m \cdot k\_m\right)\right) \cdot t}\right) \cdot 2 \end{array} \]

FPCore

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

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return (l * (l / (((k_m * k_m) * (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 = (l * (l / (((k_m * k_m) * (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 (l * (l / (((k_m * k_m) * (k_m * k_m)) * t))) * 2.0;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return (l * (l / (((k_m * k_m) * (k_m * k_m)) * t))) * 2.0

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(l * Float64(l / Float64(Float64(Float64(k_m * k_m) * Float64(k_m * k_m)) * t))) * 2.0)
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = (l * (l / (((k_m * k_m) * (k_m * k_m)) * t))) * 2.0;
end

Wolfram

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

Derivation

1. Initial program 36.5%

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

2. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4} \cdot t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{{k}^{4}} \cdot t} \]

   4. pow2 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{\color{blue}{4}} \cdot t} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot \color{blue}{t}} \]

   7. metadata-eval N/A

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

   8. pow-prod-up N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 62.8

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

4. Applied rewrites 62.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. pow2 N/A

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

   11. pow-prod-up N/A

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

   12. metadata-eval N/A

       \[\leadsto \frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t} \]

   13. associate-*r/ N/A

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   14. *-commutative N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{4} \cdot t} \cdot \color{blue}{2} \]

6. Applied rewrites 62.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. pow3 N/A

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

   10. pow-plus N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. pow-prod-up N/A

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

   16. unpow-prod-down N/A

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

   17. lift-*.f64 N/A

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

   18. pow2 N/A

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

   19. lift-*.f64 68.8

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

8. Applied rewrites 68.8%

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

Alternative 8: 65.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k\_m \leq 5.9 \cdot 10^{+47}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(\left(k\_m \cdot k\_m\right) \cdot k\_m\right) \cdot \left(k\_m \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{k\_m \cdot \left(k\_m \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.9e+47)
   (* (* l l) (/ 2.0 (* (* (* k_m k_m) k_m) (* k_m t))))
   (/ (* (* l l) -0.3333333333333333) (* k_m (* k_m t)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.9e+47) {
		tmp = (l * l) * (2.0 / (((k_m * k_m) * k_m) * (k_m * t)));
	} else {
		tmp = ((l * l) * -0.3333333333333333) / (k_m * (k_m * t));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5.9d+47) then
        tmp = (l * l) * (2.0d0 / (((k_m * k_m) * k_m) * (k_m * t)))
    else
        tmp = ((l * l) * (-0.3333333333333333d0)) / (k_m * (k_m * t))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.9e+47) {
		tmp = (l * l) * (2.0 / (((k_m * k_m) * k_m) * (k_m * t)));
	} else {
		tmp = ((l * l) * -0.3333333333333333) / (k_m * (k_m * t));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 5.9e+47:
		tmp = (l * l) * (2.0 / (((k_m * k_m) * k_m) * (k_m * t)))
	else:
		tmp = ((l * l) * -0.3333333333333333) / (k_m * (k_m * t))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.9e+47)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(Float64(Float64(k_m * k_m) * k_m) * Float64(k_m * t))));
	else
		tmp = Float64(Float64(Float64(l * l) * -0.3333333333333333) / Float64(k_m * Float64(k_m * t)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 5.9e+47)
		tmp = (l * l) * (2.0 / (((k_m * k_m) * k_m) * (k_m * t)));
	else
		tmp = ((l * l) * -0.3333333333333333) / (k_m * (k_m * t));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5.9e+47], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[(N[(k$95$m * k$95$m), $MachinePrecision] * k$95$m), $MachinePrecision] * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.90000000000000034e47

   1. Initial program 36.5%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 48.5%

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

   7. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot \color{blue}{t}} \]

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow-prod-up N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. pow-prod-up N/A

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

      15. metadata-eval N/A

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

      16. unpow3 N/A

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

      17. pow2 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 63.7

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

   9. Applied rewrites 63.7%

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

   if 5.90000000000000034e47 < k

   1. Initial program 36.5%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in k around inf

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

      3. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 29.9

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

   7. Applied rewrites 29.9%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 30.3

         \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot \left(k \cdot t\right)} \]

   9. Applied rewrites 30.3%

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot \left(k \cdot \color{blue}{t}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 30.3% accurate, 7.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{k\_m \cdot \left(k\_m \cdot t\right)} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ (* (* l l) -0.3333333333333333) (* k_m (* k_m t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l * l) * -0.3333333333333333) / (k_m * (k_m * t));
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((l * l) * (-0.3333333333333333d0)) / (k_m * (k_m * t))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l * l) * -0.3333333333333333) / (k_m * (k_m * t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l * l) * -0.3333333333333333) / (k_m * (k_m * t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l * l) * -0.3333333333333333) / Float64(k_m * Float64(k_m * t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l * l) * -0.3333333333333333) / (k_m * (k_m * t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l * l), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(k$95$m * N[(k$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.5%

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

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

4. Applied rewrites 50.1%

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

5. Taylor expanded in k around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   3. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 29.9

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

7. Applied rewrites 29.9%

   \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 30.3

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot \left(k \cdot t\right)} \]

9. Applied rewrites 30.3%

   \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{k \cdot \left(k \cdot \color{blue}{t}\right)} \]
10. Add Preprocessing

Alternative 10: 30.0% accurate, 7.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{\ell \cdot \ell}{\left(k\_m \cdot k\_m\right) \cdot t} \cdot -0.3333333333333333 \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* (/ (* l l) (* (* k_m k_m) t)) -0.3333333333333333))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return ((l * l) / ((k_m * k_m) * t)) * -0.3333333333333333;
}

Fortran

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

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

real(8) function code(t, l, k_m)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = ((l * l) / ((k_m * k_m) * t)) * (-0.3333333333333333d0)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return ((l * l) / ((k_m * k_m) * t)) * -0.3333333333333333;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return ((l * l) / ((k_m * k_m) * t)) * -0.3333333333333333

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(Float64(Float64(l * l) / Float64(Float64(k_m * k_m) * t)) * -0.3333333333333333)
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = ((l * l) / ((k_m * k_m) * t)) * -0.3333333333333333;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(N[(N[(l * l), $MachinePrecision] / N[(N[(k$95$m * k$95$m), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 36.5%

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

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{\color{blue}{{k}^{4}}} \]

4. Applied rewrites 50.1%

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

5. Taylor expanded in k around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{t}} \]

   3. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 29.9

      \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\left(k \cdot k\right) \cdot t} \]

7. Applied rewrites 29.9%

   \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

      \[\leadsto \frac{{\ell}^{2} \cdot \frac{-1}{3}}{{k}^{2} \cdot t} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{{k}^{2} \cdot t} \]

   9. associate-*r/ N/A

      \[\leadsto \frac{-1}{3} \cdot \frac{{\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

   10. *-commutative N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   12. lower-/.f64 N/A

       \[\leadsto \frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   13. pow2 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\ell \cdot \ell}{{k}^{2} \cdot t} \cdot \frac{-1}{3} \]

   15. pow2 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 30.0

       \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]

9. Applied rewrites 30.0%

   \[\leadsto \frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(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))))