Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.3% → 94.0%

Time: 21.6s

Alternatives: 15

Speedup: 15.1×

• 39.1% 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: 36.3% 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: 94.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.4 \cdot 10^{+60}:\\ \;\;\;\;\left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{ratio\_of\_squares}\left(l\_m, \sin k\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(l\_m, k\right) \cdot \cos k}{{\sin k}^{2} \cdot t} \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= l_m 2.4e+60)
   (* (* (/ (cos k) (* k (* k t))) (ratio-of-squares l_m (sin k))) 2.0)
   (* (/ (* (ratio-of-squares l_m k) (cos k)) (* (pow (sin k) 2.0) t)) 2.0)))

Derivation

1. Split input into 2 regimes

2. if l < 2.4e60

   1. Initial program 35.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 86.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 89.8

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

   7. Applied rewrites 89.8%

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

   if 2.4e60 < l

   1. Initial program 42.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 64.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-ratio-of-squares.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 97.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-*.f64 97.3

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

   9. Applied rewrites 97.3%

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

Alternative 2: 93.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 5.2 \cdot 10^{+75}:\\ \;\;\;\;\left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{ratio\_of\_squares}\left(l\_m, \sin k\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{ratio\_of\_squares}\left(l\_m, k\right) \cdot \frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= l_m 5.2e+75)
   (* (* (/ (cos k) (* k (* k t))) (ratio-of-squares l_m (sin k))) 2.0)
   (*
    (*
     (ratio-of-squares l_m k)
     (/ (cos k) (* (- 0.5 (* 0.5 (cos (* 2.0 k)))) t)))
    2.0)))

Derivation

1. Split input into 2 regimes

2. if l < 5.1999999999999997e75

   1. Initial program 35.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 86.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 89.9

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

   7. Applied rewrites 89.9%

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

   if 5.1999999999999997e75 < l

   1. Initial program 42.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-ratio-of-squares.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 97.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. pow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 97.2

         \[\leadsto \left(\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot \frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t}\right) \cdot 2 \]

   9. Applied rewrites 97.2%

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

Alternative 3: 93.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 5.2 \cdot 10^{+75}:\\ \;\;\;\;\left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{ratio\_of\_squares}\left(l\_m, \sin k\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(l\_m, k\right) \cdot \cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t} \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= l_m 5.2e+75)
   (* (* (/ (cos k) (* k (* k t))) (ratio-of-squares l_m (sin k))) 2.0)
   (*
    (/
     (* (ratio-of-squares l_m k) (cos k))
     (* (- 0.5 (* 0.5 (cos (+ k k)))) t))
    2.0)))

Derivation

1. Split input into 2 regimes

2. if l < 5.1999999999999997e75

   1. Initial program 35.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 86.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 89.9

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

   7. Applied rewrites 89.9%

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

   if 5.1999999999999997e75 < l

   1. Initial program 42.4%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 62.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-ratio-of-squares.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 97.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-*.f64 97.3

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

   9. Applied rewrites 97.3%

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

       1. lift-pow.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. pow2 N/A

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

       4. sqr-sin-a N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. cos-2 N/A

          \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \left(\cos k \cdot \cos k - \sin k \cdot \sin k\right)\right) \cdot t} \cdot 2 \]

       8. cos-sum N/A

          \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t} \cdot 2 \]

       9. lower-cos.f64 N/A

          \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot \cos k}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(k + k\right)\right) \cdot t} \cdot 2 \]

       10. lower-+.f64 96.9

           \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot \cos k}{\left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right) \cdot t} \cdot 2 \]

   11. Applied rewrites 96.9%

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

Alternative 4: 83.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 2.75 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{1} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, l\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{ratio\_of\_squares}\left(\left(\frac{l\_m}{k}\right), \sin k\right) \cdot \frac{\cos k}{t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 2.75e-46)
   (/ 2.0 (* (/ (* k (* k t)) 1.0) (ratio-of-squares (sin k) l_m)))
   (* (* (ratio-of-squares (/ l_m k) (sin k)) (/ (cos k) t)) 2.0)))

Derivation

1. Split input into 2 regimes

2. if k < 2.74999999999999992e-46

   1. Initial program 41.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. lower-ratio-of-squares.f64 N/A

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

      13. lift-sin.f64 85.9

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

   5. Applied rewrites 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 90.2

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

   7. Applied rewrites 90.2%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{1} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 83.8%

       \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{1} \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   if 2.74999999999999992e-46 < k

   1. Initial program 25.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 76.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-ratio-of-squares.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-*.f64 94.0

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

   9. Applied rewrites 94.0%

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

   11. Applied rewrites 95.0%

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

Alternative 5: 77.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 10^{-273}:\\ \;\;\;\;\frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\left(\frac{l\_m}{k}\right), k\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{ratio\_of\_squares}\left(l\_m, k\right) \cdot \frac{1}{{\sin k}^{2} \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= (* l_m l_m) 1e-273)
   (/ (* 2.0 (ratio-of-squares (/ l_m k) k)) t)
   (* (* (ratio-of-squares l_m k) (/ 1.0 (* (pow (sin k) 2.0) t))) 2.0)))

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1e-273

   1. Initial program 28.5%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \color{blue}{\left({k}^{2}\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      11. lower-*.f64 44.3

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

   5. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \color{blue}{\mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\color{blue}{\ell}, \left(k \cdot k\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      4. lift-ratio-of-squares.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{t} \]

      9. lift-*.f64 44.3

         \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{t} \]

   7. Applied rewrites 44.3%

      \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{\color{blue}{t}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{t} \]

      2. pow2 N/A

         \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left({k}^{2}\right)\right)}{t} \]

      3. lower-ratio-of-squares.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. times-frac N/A

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

      9. pow2 N/A

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

      10. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), k\right)}{t} \]

      11. lower-/.f64 96.4

          \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), k\right)}{t} \]

   9. Applied rewrites 96.4%

      \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), k\right)}{t} \]

   if 1e-273 < (*.f64 l l)

   1. Initial program 40.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-ratio-of-squares.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 N/A

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

   7. Applied rewrites 96.5%

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

   8. Taylor expanded in k around 0

      \[\leadsto \left(\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot \frac{1}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]
   9. Step-by-step derivation

   10. Applied rewrites 72.3%

       \[\leadsto \left(\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot \frac{1}{{\sin k}^{2} \cdot t}\right) \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 85.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{ratio\_of\_squares}\left(l\_m, \sin k\right)\right) \cdot 2 \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (* (* (/ (cos k) (* k (* k t))) (ratio-of-squares l_m (sin k))) 2.0))

Derivation

1. Initial program 36.9%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 82.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 87.0

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

7. Applied rewrites 87.0%

   \[\leadsto \left(\frac{\cos k}{k \cdot \left(k \cdot t\right)} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \sin k\right)\right) \cdot 2 \]
8. Add Preprocessing

Alternative 7: 76.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 10^{-273}:\\ \;\;\;\;\frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\left(\frac{l\_m}{k}\right), k\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\cos k}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{ratio\_of\_squares}\left(l\_m, k\right)\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= (* l_m l_m) 1e-273)
   (/ (* 2.0 (ratio-of-squares (/ l_m k) k)) t)
   (* (* (/ (cos k) (* (* k k) t)) (ratio-of-squares l_m k)) 2.0)))

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1e-273

   1. Initial program 28.5%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \color{blue}{\left({k}^{2}\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      11. lower-*.f64 44.3

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

   5. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \color{blue}{\mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\color{blue}{\ell}, \left(k \cdot k\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      4. lift-ratio-of-squares.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{t} \]

      9. lift-*.f64 44.3

         \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{t} \]

   7. Applied rewrites 44.3%

      \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{\color{blue}{t}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{t} \]

      2. pow2 N/A

         \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left({k}^{2}\right)\right)}{t} \]

      3. lower-ratio-of-squares.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. times-frac N/A

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

      9. pow2 N/A

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

      10. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), k\right)}{t} \]

      11. lower-/.f64 96.4

          \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), k\right)}{t} \]

   9. Applied rewrites 96.4%

      \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), k\right)}{t} \]

   if 1e-273 < (*.f64 l l)

   1. Initial program 40.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 72.3%

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

Alternative 8: 75.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 2 \cdot 10^{-313}:\\ \;\;\;\;\frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\left(\frac{l\_m}{k}\right), k\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{ratio\_of\_squares}\left(\sin k, l\_m\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= (* l_m l_m) 2e-313)
   (/ (* 2.0 (ratio-of-squares (/ l_m k) k)) t)
   (/ 2.0 (* (* (* k k) t) (ratio-of-squares (sin k) l_m)))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999998e-313

   1. Initial program 25.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \color{blue}{\left({k}^{2}\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      11. lower-*.f64 45.7

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

   5. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \color{blue}{\mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\color{blue}{\ell}, \left(k \cdot k\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      4. lift-ratio-of-squares.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{t} \]

      9. lift-*.f64 45.7

         \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{t} \]

   7. Applied rewrites 45.7%

      \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{\color{blue}{t}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{t} \]

      2. pow2 N/A

         \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left({k}^{2}\right)\right)}{t} \]

      3. lower-ratio-of-squares.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. times-frac N/A

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

      9. pow2 N/A

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

      10. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), k\right)}{t} \]

      11. lower-/.f64 95.9

          \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), k\right)}{t} \]

   9. Applied rewrites 95.9%

      \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), k\right)}{t} \]

   if 1.99999999998e-313 < (*.f64 l l)

   1. Initial program 41.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. unpow2 N/A

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

      11. pow2 N/A

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

      12. lower-ratio-of-squares.f64 N/A

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

      13. lift-sin.f64 80.8

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in k around 0

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

      1. pow2 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

      3. lift-*.f64 71.0

         \[\leadsto \frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \mathsf{ratio\_of\_squares}\left(\sin k, \ell\right)} \]

   8. Applied rewrites 71.0%

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

Alternative 9: 74.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m}{k \cdot k}\\ \mathbf{if}\;k \leq 1800:\\ \;\;\;\;\frac{2}{t} \cdot \left(t\_1 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(l\_m, k\right) \cdot -0.16666666666666666}{t} \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (let* ((t_1 (/ l_m (* k k))))
   (if (<= k 1800.0)
     (* (/ 2.0 t) (* t_1 t_1))
     (* (/ (* (ratio-of-squares l_m k) -0.16666666666666666) t) 2.0))))

Derivation

1. Split input into 2 regimes

2. if k < 1800

   1. Initial program 39.8%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \color{blue}{\left({k}^{2}\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      11. lower-*.f64 32.7

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

   5. Applied rewrites 32.7%

      \[\leadsto \color{blue}{\frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      2. pow2 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left({k}^{\color{blue}{2}}\right)\right) \]

      3. lower-ratio-of-squares.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 81.8

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

   7. Applied rewrites 81.8%

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

   if 1800 < k

   1. Initial program 28.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 27.6%

      \[\leadsto \frac{\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666\right) \cdot \left(k \cdot k\right) + \frac{\ell \cdot \ell}{t}}{{k}^{4}} \cdot 2 \]

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. pow2 N/A

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

       5. pow2 N/A

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

       6. lift-ratio-of-squares.f64 N/A

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

       7. lower-/.f64 64.1

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

   11. Applied rewrites 64.1%

       \[\leadsto \left(\frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot -0.16666666666666666\right) \cdot 2 \]
   12. Step-by-step derivation

   13. Applied rewrites 64.1%

       \[\leadsto \color{blue}{\frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot -0.16666666666666666}{t} \cdot 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 75.1% accurate, 12.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 4 \cdot 10^{-137}:\\ \;\;\;\;\frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\left(\frac{l\_m}{k}\right), k\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(l\_m, k\right) \cdot 2}{\left(k \cdot k\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= l_m 4e-137)
   (/ (* 2.0 (ratio-of-squares (/ l_m k) k)) t)
   (/ (* (ratio-of-squares l_m k) 2.0) (* (* k k) t))))

Derivation

1. Split input into 2 regimes

2. if l < 3.99999999999999991e-137

   1. Initial program 35.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \color{blue}{\left({k}^{2}\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      11. lower-*.f64 35.2

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

   5. Applied rewrites 35.2%

      \[\leadsto \color{blue}{\frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \color{blue}{\mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\color{blue}{\ell}, \left(k \cdot k\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      4. lift-ratio-of-squares.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{t} \]

      9. lift-*.f64 35.2

         \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{t} \]

   7. Applied rewrites 35.2%

      \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{\color{blue}{t}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{t} \]

      2. pow2 N/A

         \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left({k}^{2}\right)\right)}{t} \]

      3. lower-ratio-of-squares.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. times-frac N/A

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

      9. pow2 N/A

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

      10. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), k\right)}{t} \]

      11. lower-/.f64 75.4

          \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), k\right)}{t} \]

   9. Applied rewrites 75.4%

      \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), k\right)}{t} \]

   if 3.99999999999999991e-137 < l

   1. Initial program 40.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \color{blue}{\left({k}^{2}\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      11. lower-*.f64 26.1

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

   5. Applied rewrites 26.1%

      \[\leadsto \color{blue}{\frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \color{blue}{\mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\color{blue}{\ell}, \left(k \cdot k\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      4. lift-ratio-of-squares.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. pow2 N/A

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

      15. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot 2}{{\color{blue}{k}}^{2} \cdot t} \]

      16. pow2 N/A

          \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot 2}{\left(k \cdot k\right) \cdot t} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot 2}{\left(k \cdot k\right) \cdot t} \]

      18. lift-*.f64 67.2

          \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot 2}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

   7. Applied rewrites 67.2%

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

Alternative 11: 73.3% accurate, 13.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 1800:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(l\_m, k\right) \cdot 2}{\left(k \cdot k\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(l\_m, k\right) \cdot -0.16666666666666666}{t} \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 1800.0)
   (/ (* (ratio-of-squares l_m k) 2.0) (* (* k k) t))
   (* (/ (* (ratio-of-squares l_m k) -0.16666666666666666) t) 2.0)))

Derivation

1. Split input into 2 regimes

2. if k < 1800

   1. Initial program 39.8%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \color{blue}{\left({k}^{2}\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      11. lower-*.f64 32.7

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

   5. Applied rewrites 32.7%

      \[\leadsto \color{blue}{\frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \color{blue}{\mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\color{blue}{\ell}, \left(k \cdot k\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      4. lift-ratio-of-squares.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. pow2 N/A

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

      15. lower-ratio-of-squares.f64 N/A

          \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot 2}{{\color{blue}{k}}^{2} \cdot t} \]

      16. pow2 N/A

          \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot 2}{\left(k \cdot k\right) \cdot t} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot 2}{\left(k \cdot k\right) \cdot t} \]

      18. lift-*.f64 81.0

          \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot 2}{\left(k \cdot k\right) \cdot \color{blue}{t}} \]

   7. Applied rewrites 81.0%

      \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot 2}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]

   if 1800 < k

   1. Initial program 28.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 27.6%

      \[\leadsto \frac{\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666\right) \cdot \left(k \cdot k\right) + \frac{\ell \cdot \ell}{t}}{{k}^{4}} \cdot 2 \]

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. pow2 N/A

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

       5. pow2 N/A

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

       6. lift-ratio-of-squares.f64 N/A

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

       7. lower-/.f64 64.1

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

   11. Applied rewrites 64.1%

       \[\leadsto \left(\frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot -0.16666666666666666\right) \cdot 2 \]
   12. Step-by-step derivation

   13. Applied rewrites 64.1%

       \[\leadsto \color{blue}{\frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot -0.16666666666666666}{t} \cdot 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 44.9% accurate, 15.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 1800:\\ \;\;\;\;\frac{2 \cdot \mathsf{ratio\_of\_squares}\left(l\_m, \left(k \cdot k\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(l\_m, k\right) \cdot -0.16666666666666666}{t} \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 1800.0)
   (/ (* 2.0 (ratio-of-squares l_m (* k k))) t)
   (* (/ (* (ratio-of-squares l_m k) -0.16666666666666666) t) 2.0)))

Derivation

1. Split input into 2 regimes

2. if k < 1800

   1. Initial program 39.8%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \color{blue}{\left({k}^{2}\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      11. lower-*.f64 32.7

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

   5. Applied rewrites 32.7%

      \[\leadsto \color{blue}{\frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \color{blue}{\mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\color{blue}{\ell}, \left(k \cdot k\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      4. lift-ratio-of-squares.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{t} \]

      9. lift-*.f64 32.7

         \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{t} \]

   7. Applied rewrites 32.7%

      \[\leadsto \frac{2 \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{\color{blue}{t}} \]

   if 1800 < k

   1. Initial program 28.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 27.6%

      \[\leadsto \frac{\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666\right) \cdot \left(k \cdot k\right) + \frac{\ell \cdot \ell}{t}}{{k}^{4}} \cdot 2 \]

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. pow2 N/A

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

       5. pow2 N/A

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

       6. lift-ratio-of-squares.f64 N/A

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

       7. lower-/.f64 64.1

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

   11. Applied rewrites 64.1%

       \[\leadsto \left(\frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot -0.16666666666666666\right) \cdot 2 \]
   12. Step-by-step derivation

   13. Applied rewrites 64.1%

       \[\leadsto \color{blue}{\frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot -0.16666666666666666}{t} \cdot 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 44.9% accurate, 15.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;k \leq 1800:\\ \;\;\;\;\frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(l\_m, \left(k \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(l\_m, k\right) \cdot -0.16666666666666666}{t} \cdot 2\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (if (<= k 1800.0)
   (* (/ 2.0 t) (ratio-of-squares l_m (* k k)))
   (* (/ (* (ratio-of-squares l_m k) -0.16666666666666666) t) 2.0)))

Derivation

1. Split input into 2 regimes

2. if k < 1800

   1. Initial program 39.8%

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

   3. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-ratio-of-squares.f64 N/A

         \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \color{blue}{\left({k}^{2}\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

      11. lower-*.f64 32.7

          \[\leadsto \frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot \color{blue}{k}\right)\right) \]

   5. Applied rewrites 32.7%

      \[\leadsto \color{blue}{\frac{2}{t} \cdot \mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)} \]

   if 1800 < k

   1. Initial program 28.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 27.6%

      \[\leadsto \frac{\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666\right) \cdot \left(k \cdot k\right) + \frac{\ell \cdot \ell}{t}}{{k}^{4}} \cdot 2 \]

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. pow2 N/A

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

       5. pow2 N/A

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

       6. lift-ratio-of-squares.f64 N/A

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

       7. lower-/.f64 64.1

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

   11. Applied rewrites 64.1%

       \[\leadsto \left(\frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot -0.16666666666666666\right) \cdot 2 \]
   12. Step-by-step derivation

   13. Applied rewrites 64.1%

       \[\leadsto \color{blue}{\frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot -0.16666666666666666}{t} \cdot 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 31.9% accurate, 18.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \frac{\mathsf{ratio\_of\_squares}\left(l\_m, k\right) \cdot -0.16666666666666666}{t} \cdot 2 \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (* (/ (* (ratio-of-squares l_m k) -0.16666666666666666) t) 2.0))

Derivation

1. Initial program 36.9%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 82.8%

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

6. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

8. Applied rewrites 33.8%

   \[\leadsto \frac{\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666\right) \cdot \left(k \cdot k\right) + \frac{\ell \cdot \ell}{t}}{{k}^{4}} \cdot 2 \]

9. Taylor expanded in k around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. associate-/r* N/A

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

    4. pow2 N/A

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

    5. pow2 N/A

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

    6. lift-ratio-of-squares.f64 N/A

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

    7. lower-/.f64 31.2

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

11. Applied rewrites 31.2%

    \[\leadsto \left(\frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot -0.16666666666666666\right) \cdot 2 \]
12. Step-by-step derivation

13. Applied rewrites 31.2%

    \[\leadsto \color{blue}{\frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot -0.16666666666666666}{t} \cdot 2} \]
14. Add Preprocessing

Alternative 15: 31.9% accurate, 18.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \left(\frac{\mathsf{ratio\_of\_squares}\left(l\_m, k\right)}{t} \cdot -0.16666666666666666\right) \cdot 2 \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m k)
 :precision binary64
 (* (* (/ (ratio-of-squares l_m k) t) -0.16666666666666666) 2.0))

Derivation

1. Initial program 36.9%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 82.8%

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

6. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

8. Applied rewrites 33.8%

   \[\leadsto \frac{\left(\frac{\ell \cdot \ell}{t} \cdot -0.16666666666666666\right) \cdot \left(k \cdot k\right) + \frac{\ell \cdot \ell}{t}}{{k}^{4}} \cdot 2 \]

9. Taylor expanded in k around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. associate-/r* N/A

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

    4. pow2 N/A

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

    5. pow2 N/A

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

    6. lift-ratio-of-squares.f64 N/A

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

    7. lower-/.f64 31.2

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

11. Applied rewrites 31.2%

    \[\leadsto \left(\frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{t} \cdot -0.16666666666666666\right) \cdot 2 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025058 
(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))))