Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.6% → 93.3%

Time: 23.4s

Alternatives: 16

Speedup: 18.8×

• 27.1% 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: 54.6% 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: 93.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (+
    (/ (ratio-of-squares (* k (sin k)) l) (cos k))
    (* (/ (ratio-of-squares (* t (sin k)) l) (cos k)) 2.0))
   t)))

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 94.7%

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

Alternative 2: 77.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\\ \mathbf{if}\;t\_1 \leq 0 \lor \neg \left(t\_1 \leq \infty\right):\\ \;\;\;\;\frac{2}{\frac{\mathsf{ratio\_of\_squares}\left(\left(\sin k \cdot k\right), \ell\right)}{\cos k} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), t\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))
   (if (or (<= t_1 0.0) (not (<= t_1 INFINITY)))
     (/ 2.0 (* (/ (ratio-of-squares (* (sin k) k) l) (cos k)) t))
     (/ (ratio-of-squares (/ l k) t) t))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < 0.0 or +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))

   1. Initial program 43.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. pow-prod-down N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

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

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-/.f64 72.0

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

      10. lift-*.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 72.0

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

   8. Applied rewrites 72.0%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 81.6%

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

   3. Taylor expanded in k around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. unpow2 N/A

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

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

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{\color{blue}{t}}^{3}} \]

      6. lift-pow.f64 76.8

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{t}^{\color{blue}{3}}} \]

   5. Applied rewrites 76.8%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{t}^{\color{blue}{3}}} \]

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 76.7

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

   7. Applied rewrites 76.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

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

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

      8. times-frac N/A

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

      9. pow2 N/A

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

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

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

      11. lower-/.f64 82.8

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

   9. Applied rewrites 82.8%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \leq 0 \lor \neg \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \leq \infty\right):\\ \;\;\;\;\frac{2}{\frac{\mathsf{ratio\_of\_squares}\left(\left(\sin k \cdot k\right), \ell\right)}{\cos k} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), t\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), t\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot \frac{\cos k}{\left(k \cdot k\right) \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
        (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
      2e+275)
   (/ (ratio-of-squares (/ l k) t) t)
   (* (* (ratio-of-squares l k) (/ (cos k) (* (* k k) t))) 2.0)))

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999992e275

   1. Initial program 80.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. unpow2 N/A

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

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

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{\color{blue}{t}}^{3}} \]

      6. lift-pow.f64 78.6

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{t}^{\color{blue}{3}}} \]

   5. Applied rewrites 78.6%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{t}^{\color{blue}{3}}} \]

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 78.6

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

   7. Applied rewrites 78.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

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

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

      8. times-frac N/A

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

      9. pow2 N/A

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

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

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

      11. lower-/.f64 82.7

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

   9. Applied rewrites 82.7%

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

   if 1.99999999999999992e275 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 18.5%

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

   3. Taylor expanded in t around 0

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

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. unpow2 N/A

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

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

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-sin.f64 77.1

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

   5. Applied rewrites 77.1%

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

   6. Taylor expanded in k around 0

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

      1. pow2 N/A

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

      2. lower-*.f64 61.4

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

   8. Applied rewrites 61.4%

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

Alternative 4: 67.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), t\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot t\_1}{t} \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (* k k))))
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
        2e+275)
     (/ (ratio-of-squares (/ l k) t) t)
     (* (/ (* t_1 t_1) t) 2.0))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999992e275

   1. Initial program 80.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. unpow2 N/A

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

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

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{\color{blue}{t}}^{3}} \]

      6. lift-pow.f64 78.6

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{t}^{\color{blue}{3}}} \]

   5. Applied rewrites 78.6%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{t}^{\color{blue}{3}}} \]

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 78.6

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

   7. Applied rewrites 78.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

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

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

      8. times-frac N/A

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

      9. pow2 N/A

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

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

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

      11. lower-/.f64 82.7

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

   9. Applied rewrites 82.7%

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

   if 1.99999999999999992e275 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 18.5%

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

   3. Taylor expanded in t around 0

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

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. unpow2 N/A

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

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

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-sin.f64 77.1

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

   5. Applied rewrites 77.1%

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

   6. Taylor expanded in k around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. sqr-pow N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

      8. pow2 N/A

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

      9. lower-*.f64 26.5

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

   8. Applied rewrites 26.5%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

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

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 55.8

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

   10. Applied rewrites 55.8%

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

Alternative 5: 66.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), t\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot 1}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot -0.3333333333333333 + t\right) \cdot \left(k \cdot k\right)} \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
        (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
      2e+275)
   (/ (ratio-of-squares (/ l k) t) t)
   (*
    (/
     (* (ratio-of-squares l k) 1.0)
     (* (+ (* (* (* k k) t) -0.3333333333333333) t) (* k k)))
    2.0)))

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999992e275

   1. Initial program 80.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. unpow2 N/A

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

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

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{\color{blue}{t}}^{3}} \]

      6. lift-pow.f64 78.6

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{t}^{\color{blue}{3}}} \]

   5. Applied rewrites 78.6%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{t}^{\color{blue}{3}}} \]

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 78.6

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

   7. Applied rewrites 78.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

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

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

      8. times-frac N/A

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

      9. pow2 N/A

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

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

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

      11. lower-/.f64 82.7

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

   9. Applied rewrites 82.7%

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

   if 1.99999999999999992e275 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 18.5%

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

   3. Taylor expanded in t around 0

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

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. unpow2 N/A

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

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

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-sin.f64 77.1

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

   5. Applied rewrites 77.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 55.5

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

   10. Applied rewrites 55.5%

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

   11. Taylor expanded in k around 0

       \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot 1}{{k}^{2} \cdot \left(t + \frac{-1}{3} \cdot \left({k}^{2} \cdot t\right)\right)} \cdot 2 \]
   12. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 56.7

           \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot 1}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot -0.3333333333333333 + t\right) \cdot \left(k \cdot k\right)} \cdot 2 \]

   13. Applied rewrites 56.7%

       \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot 1}{\left(\left(\left(k \cdot k\right) \cdot t\right) \cdot -0.3333333333333333 + t\right) \cdot \left(k \cdot k\right)} \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 66.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), t\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot \frac{1}{\left(\left(\left(k \cdot k\right) \cdot -0.3333333333333333 + 1\right) \cdot \left(k \cdot k\right)\right) \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
        (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
      2e+275)
   (/ (ratio-of-squares (/ l k) t) t)
   (*
    (*
     (ratio-of-squares l k)
     (/ 1.0 (* (* (+ (* (* k k) -0.3333333333333333) 1.0) (* k k)) t)))
    2.0)))

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999992e275

   1. Initial program 80.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. unpow2 N/A

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

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

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{\color{blue}{t}}^{3}} \]

      6. lift-pow.f64 78.6

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{t}^{\color{blue}{3}}} \]

   5. Applied rewrites 78.6%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{t}^{\color{blue}{3}}} \]

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 78.6

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

   7. Applied rewrites 78.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

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

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

      8. times-frac N/A

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

      9. pow2 N/A

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

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

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

      11. lower-/.f64 82.7

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

   9. Applied rewrites 82.7%

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

   if 1.99999999999999992e275 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 18.5%

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

   3. Taylor expanded in t around 0

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

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. unpow2 N/A

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

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

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-sin.f64 77.1

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

   5. Applied rewrites 77.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 55.6%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 55.8

           \[\leadsto \left(\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot \frac{1}{\left(\left(\left(k \cdot k\right) \cdot -0.3333333333333333 + 1\right) \cdot \left(k \cdot k\right)\right) \cdot t}\right) \cdot 2 \]

   11. Applied rewrites 55.8%

       \[\leadsto \left(\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot \frac{1}{\left(\left(\left(k \cdot k\right) \cdot -0.3333333333333333 + 1\right) \cdot \left(k \cdot k\right)\right) \cdot t}\right) \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 66.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), t\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot \frac{1}{\left(k \cdot k\right) \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<=
      (/
       2.0
       (*
        (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
        (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0)))
      2e+275)
   (/ (ratio-of-squares (/ l k) t) t)
   (* (* (ratio-of-squares l k) (/ 1.0 (* (* k k) t))) 2.0)))

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 1.99999999999999992e275

   1. Initial program 80.1%

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

   3. Taylor expanded in k around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. unpow2 N/A

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

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

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{\color{blue}{t}}^{3}} \]

      6. lift-pow.f64 78.6

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{t}^{\color{blue}{3}}} \]

   5. Applied rewrites 78.6%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{t}^{\color{blue}{3}}} \]

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 78.6

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

   7. Applied rewrites 78.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

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

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

      8. times-frac N/A

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

      9. pow2 N/A

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

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

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

      11. lower-/.f64 82.7

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

   9. Applied rewrites 82.7%

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

   if 1.99999999999999992e275 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 18.5%

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

   3. Taylor expanded in t around 0

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

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. unpow2 N/A

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

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

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-sin.f64 77.1

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

   5. Applied rewrites 77.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 55.6%

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

   9. Taylor expanded in k around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 54.7

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

   11. Applied rewrites 54.7%

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

Alternative 8: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \leq 0:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{\left(t \cdot t\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), t\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<=
      (*
       (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
       (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))
      0.0)
   (/ (ratio-of-squares l k) (* (* t t) t))
   (/ (ratio-of-squares (/ l k) t) t)))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < 0.0

   1. Initial program 80.0%

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

   3. Taylor expanded in k around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. unpow2 N/A

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

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

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{\color{blue}{t}}^{3}} \]

      6. lift-pow.f64 81.1

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{t}^{\color{blue}{3}}} \]

   5. Applied rewrites 81.1%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{t}^{\color{blue}{3}}} \]

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 81.1

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

   7. Applied rewrites 81.1%

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

   if 0.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))

   1. Initial program 36.6%

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

   3. Taylor expanded in k around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. unpow2 N/A

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

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

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{\color{blue}{t}}^{3}} \]

      6. lift-pow.f64 47.4

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{t}^{\color{blue}{3}}} \]

   5. Applied rewrites 47.4%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{t}^{\color{blue}{3}}} \]

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 47.3

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

   7. Applied rewrites 47.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

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

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

      8. times-frac N/A

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

      9. pow2 N/A

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

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

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

      11. lower-/.f64 60.8

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

   9. Applied rewrites 60.8%

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

Alternative 9: 73.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5600:\\ \;\;\;\;\frac{2}{\left(\mathsf{ratio\_of\_squares}\left(\left(k \cdot k\right), \ell\right) + \frac{\mathsf{ratio\_of\_squares}\left(\left(t \cdot \sin k\right), \ell\right)}{\cos k} \cdot 2\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot \frac{\frac{\cos k}{{\sin k}^{2}}}{t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5600.0)
   (/
    2.0
    (*
     (+
      (ratio-of-squares (* k k) l)
      (* (/ (ratio-of-squares (* t (sin k)) l) (cos k)) 2.0))
     t))
   (* (* (ratio-of-squares l k) (/ (/ (cos k) (pow (sin k) 2.0)) t)) 2.0)))

Derivation

1. Split input into 2 regimes

2. if k < 5600

   1. Initial program 56.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{{k}^{4}}{{\ell}^{2}} + \frac{\mathsf{ratio\_of\_squares}\left(\left(t \cdot \sin k\right), \ell\right)}{\cos k} \cdot 2\right) \cdot t} \]
   7. Step-by-step derivation

      1. sqr-pow N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}{{\ell}^{2}} + \frac{\mathsf{ratio\_of\_squares}\left(\left(t \cdot \sin k\right), \ell\right)}{\cos k} \cdot 2\right) \cdot t} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot {k}^{\left(\frac{4}{2}\right)}}{{\ell}^{2}} + \frac{\mathsf{ratio\_of\_squares}\left(\left(t \cdot \sin k\right), \ell\right)}{\cos k} \cdot 2\right) \cdot t} \]

      3. metadata-eval N/A

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

      4. pow2 N/A

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

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

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

      6. pow2 N/A

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

      7. lower-*.f64 62.4

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

   8. Applied rewrites 62.4%

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

   if 5600 < k

   1. Initial program 44.4%

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

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. unpow2 N/A

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

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

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-sin.f64 81.5

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

   5. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\left(\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2} \]
   6. 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-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-/.f64 81.6

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

   7. Applied rewrites 81.6%

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

Alternative 10: 73.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5600:\\ \;\;\;\;\frac{2}{\left(\mathsf{ratio\_of\_squares}\left(\left(k \cdot k\right), \ell\right) + \frac{\mathsf{ratio\_of\_squares}\left(\left(t \cdot \sin k\right), \ell\right)}{\cos k} \cdot 2\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5600.0)
   (/
    2.0
    (*
     (+
      (ratio-of-squares (* k k) l)
      (* (/ (ratio-of-squares (* t (sin k)) l) (cos k)) 2.0))
     t))
   (* (* (ratio-of-squares l k) (/ (cos k) (* (pow (sin k) 2.0) t))) 2.0)))

Derivation

1. Split input into 2 regimes

2. if k < 5600

   1. Initial program 56.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{{k}^{4}}{{\ell}^{2}} + \frac{\mathsf{ratio\_of\_squares}\left(\left(t \cdot \sin k\right), \ell\right)}{\cos k} \cdot 2\right) \cdot t} \]
   7. Step-by-step derivation

      1. sqr-pow N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}{{\ell}^{2}} + \frac{\mathsf{ratio\_of\_squares}\left(\left(t \cdot \sin k\right), \ell\right)}{\cos k} \cdot 2\right) \cdot t} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot {k}^{\left(\frac{4}{2}\right)}}{{\ell}^{2}} + \frac{\mathsf{ratio\_of\_squares}\left(\left(t \cdot \sin k\right), \ell\right)}{\cos k} \cdot 2\right) \cdot t} \]

      3. metadata-eval N/A

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

      4. pow2 N/A

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

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

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

      6. pow2 N/A

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

      7. lower-*.f64 62.4

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

   8. Applied rewrites 62.4%

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

   if 5600 < k

   1. Initial program 44.4%

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

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. unpow2 N/A

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

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

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-sin.f64 81.5

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

   5. Applied rewrites 81.5%

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

Alternative 11: 73.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5600:\\ \;\;\;\;\frac{2}{\left(\mathsf{ratio\_of\_squares}\left(\left(k \cdot k\right), \ell\right) + \frac{\mathsf{ratio\_of\_squares}\left(\left(t \cdot \sin k\right), \ell\right)}{\cos k} \cdot 2\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot \frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5600.0)
   (/
    2.0
    (*
     (+
      (ratio-of-squares (* k k) l)
      (* (/ (ratio-of-squares (* t (sin k)) l) (cos k)) 2.0))
     t))
   (*
    (*
     (ratio-of-squares l k)
     (/ (cos k) (* (- 0.5 (* 0.5 (cos (* 2.0 k)))) t)))
    2.0)))

Derivation

1. Split input into 2 regimes

2. if k < 5600

   1. Initial program 56.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\left(\frac{{k}^{4}}{{\ell}^{2}} + \frac{\mathsf{ratio\_of\_squares}\left(\left(t \cdot \sin k\right), \ell\right)}{\cos k} \cdot 2\right) \cdot t} \]
   7. Step-by-step derivation

      1. sqr-pow N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}{{\ell}^{2}} + \frac{\mathsf{ratio\_of\_squares}\left(\left(t \cdot \sin k\right), \ell\right)}{\cos k} \cdot 2\right) \cdot t} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\frac{{k}^{2} \cdot {k}^{\left(\frac{4}{2}\right)}}{{\ell}^{2}} + \frac{\mathsf{ratio\_of\_squares}\left(\left(t \cdot \sin k\right), \ell\right)}{\cos k} \cdot 2\right) \cdot t} \]

      3. metadata-eval N/A

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

      4. pow2 N/A

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

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

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

      6. pow2 N/A

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

      7. lower-*.f64 62.4

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

   8. Applied rewrites 62.4%

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

   if 5600 < k

   1. Initial program 44.4%

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

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. unpow2 N/A

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

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

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-sin.f64 81.5

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

   5. Applied rewrites 81.5%

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

      1. lift-pow.f64 N/A

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

      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. unpow2 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 81.4

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

   7. Applied rewrites 81.4%

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

Alternative 12: 77.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 10000:\\ \;\;\;\;\frac{2}{\frac{\mathsf{ratio\_of\_squares}\left(\left(\sin k \cdot t\right), \ell\right) \cdot 2}{\cos k} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{ratio\_of\_squares}\left(\ell, k\right) \cdot \frac{\cos k}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right) \cdot t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 10000.0)
   (/ 2.0 (* (/ (* (ratio-of-squares (* (sin k) t) l) 2.0) (cos k)) t))
   (*
    (*
     (ratio-of-squares l k)
     (/ (cos k) (* (- 0.5 (* 0.5 (cos (* 2.0 k)))) t)))
    2.0)))

Derivation

1. Split input into 2 regimes

2. if k < 1e4

   1. Initial program 56.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. pow-prod-down N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 72.2%

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

   if 1e4 < k

   1. Initial program 44.4%

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

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. unpow2 N/A

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

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

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-sin.f64 81.5

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

   5. Applied rewrites 81.5%

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

      1. lift-pow.f64 N/A

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

      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. unpow2 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 81.4

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

   7. Applied rewrites 81.4%

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

Alternative 13: 70.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-150}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{\left(t \cdot t\right) \cdot t}\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\left(\left({\ell}^{-2} + \left(\mathsf{ratio\_of\_squares}\left(t, \ell\right) \cdot 0.16666666666666666\right) \cdot 2\right) \cdot \left(k \cdot k\right) + \mathsf{ratio\_of\_squares}\left(t, \ell\right) \cdot 2\right) \cdot \left(k \cdot k\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), \sin k\right) \cdot \frac{\cos k}{t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.7e-150)
   (/ (ratio-of-squares l k) (* (* t t) t))
   (if (<= k 2.8e-6)
     (/
      2.0
      (*
       (*
        (+
         (*
          (+
           (pow l -2.0)
           (* (* (ratio-of-squares t l) 0.16666666666666666) 2.0))
          (* k k))
         (* (ratio-of-squares t l) 2.0))
        (* k k))
       t))
     (* (* (ratio-of-squares (/ l k) (sin k)) (/ (cos k) t)) 2.0))))

Derivation

1. Split input into 3 regimes

2. if k < 3.70000000000000001e-150

   1. Initial program 55.6%

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

   3. Taylor expanded in k around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. unpow2 N/A

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

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

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{\color{blue}{t}}^{3}} \]

      6. lift-pow.f64 62.9

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{t}^{\color{blue}{3}}} \]

   5. Applied rewrites 62.9%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{t}^{\color{blue}{3}}} \]

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 62.9

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

   7. Applied rewrites 62.9%

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

   if 3.70000000000000001e-150 < k < 2.79999999999999987e-6

   1. Initial program 66.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.1%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 83.9%

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

   if 2.79999999999999987e-6 < k

   1. Initial program 42.8%

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

   4. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 1\right) + 1\right) \cdot \left(\mathsf{ratio\_of\_squares}\left(\left({t}^{1.5}\right), \ell\right) \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   6. Applied rewrites 27.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\mathsf{ratio\_of\_squares}\left(\left({t}^{1.5}\right), \ell\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 2\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\mathsf{ratio\_of\_squares}\left(\left({t}^{\frac{3}{2}}\right), \ell\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 2\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\mathsf{ratio\_of\_squares}\left(\left({t}^{\frac{3}{2}}\right), \ell\right) \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 2\right)\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\mathsf{ratio\_of\_squares}\left(\color{blue}{\left({t}^{\frac{3}{2}}\right)}, \ell\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 2\right)\right)} \]

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

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{\frac{3}{2}} \cdot {t}^{\frac{3}{2}}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 2\right)\right)} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{\frac{3}{2}} \cdot {t}^{\frac{3}{2}}}{\ell \cdot \ell} \cdot \color{blue}{\sin k}\right) \cdot \left(\tan k \cdot \left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 2\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{\frac{3}{2}} \cdot {t}^{\frac{3}{2}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 2\right)\right)}} \]

      7. lift-tan.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{\frac{3}{2}} \cdot {t}^{\frac{3}{2}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 2\right)\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{\frac{3}{2}} \cdot {t}^{\frac{3}{2}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 2\right)}\right)} \]

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

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

      10. associate-*l* N/A

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

      11. pow-prod-up N/A

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

      12. metadata-eval N/A

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

      13. pow2 N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 27.7%

      \[\leadsto \color{blue}{\frac{2}{\mathsf{ratio\_of\_squares}\left(\left({t}^{1.5}\right), \ell\right) \cdot \left(\sin k \cdot \left(\left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 2\right) \cdot \tan k\right)\right)}} \]

   9. Taylor expanded in t around 0

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

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

       3. associate-*l/ N/A

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

       4. pow2 N/A

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

       5. pow2 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

   11. Applied rewrites 78.5%

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

Alternative 14: 77.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 10000:\\ \;\;\;\;\frac{2}{\frac{\mathsf{ratio\_of\_squares}\left(\left(\sin k \cdot t\right), \ell\right) \cdot 2}{\cos k} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{ratio\_of\_squares}\left(\left(\frac{\ell}{k}\right), \sin k\right) \cdot \frac{\cos k}{t}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 10000.0)
   (/ 2.0 (* (/ (* (ratio-of-squares (* (sin k) t) l) 2.0) (cos k)) t))
   (* (* (ratio-of-squares (/ l k) (sin k)) (/ (cos k) t)) 2.0)))

Derivation

1. Split input into 2 regimes

2. if k < 1e4

   1. Initial program 56.3%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. pow-prod-down N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 72.2%

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

   if 1e4 < k

   1. Initial program 44.4%

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

   4. Applied rewrites 28.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 1\right) + 1\right) \cdot \left(\mathsf{ratio\_of\_squares}\left(\left({t}^{1.5}\right), \ell\right) \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   6. Applied rewrites 28.8%

      \[\leadsto \frac{2}{\color{blue}{\left(\mathsf{ratio\_of\_squares}\left(\left({t}^{1.5}\right), \ell\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 2\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\mathsf{ratio\_of\_squares}\left(\left({t}^{\frac{3}{2}}\right), \ell\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 2\right)\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\mathsf{ratio\_of\_squares}\left(\left({t}^{\frac{3}{2}}\right), \ell\right) \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 2\right)\right)} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{2}{\left(\mathsf{ratio\_of\_squares}\left(\color{blue}{\left({t}^{\frac{3}{2}}\right)}, \ell\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 2\right)\right)} \]

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

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{{t}^{\frac{3}{2}} \cdot {t}^{\frac{3}{2}}}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 2\right)\right)} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{\frac{3}{2}} \cdot {t}^{\frac{3}{2}}}{\ell \cdot \ell} \cdot \color{blue}{\sin k}\right) \cdot \left(\tan k \cdot \left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 2\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{\frac{3}{2}} \cdot {t}^{\frac{3}{2}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 2\right)\right)}} \]

      7. lift-tan.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{\frac{3}{2}} \cdot {t}^{\frac{3}{2}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\tan k} \cdot \left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 2\right)\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{{t}^{\frac{3}{2}} \cdot {t}^{\frac{3}{2}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 2\right)}\right)} \]

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

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

      10. associate-*l* N/A

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

      11. pow-prod-up N/A

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

      12. metadata-eval N/A

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

      13. pow2 N/A

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

      14. lower-*.f64 N/A

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

   8. Applied rewrites 28.8%

      \[\leadsto \color{blue}{\frac{2}{\mathsf{ratio\_of\_squares}\left(\left({t}^{1.5}\right), \ell\right) \cdot \left(\sin k \cdot \left(\left(\mathsf{ratio\_of\_squares}\left(k, t\right) + 2\right) \cdot \tan k\right)\right)}} \]

   9. Taylor expanded in t around 0

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

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

       3. associate-*l/ N/A

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

       4. pow2 N/A

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

       5. pow2 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

   11. Applied rewrites 81.4%

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

Alternative 15: 59.6% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{\left(t \cdot t\right) \cdot t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (/ (ratio-of-squares l k) (* (* t t) t)))

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in k around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. pow2 N/A

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

   4. unpow2 N/A

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

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

      \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{\color{blue}{t}}^{3}} \]

   6. lift-pow.f64 60.8

      \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{t}^{\color{blue}{3}}} \]

5. Applied rewrites 60.8%

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

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, k\right)}{{t}^{\color{blue}{3}}} \]

   2. unpow3 N/A

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

   3. pow2 N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lower-*.f64 60.8

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

7. Applied rewrites 60.8%

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

Alternative 16: 26.1% accurate, 18.8× speedup

Math

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

FPCore

(FPCore (t l k) :precision binary64 (* (/ (ratio-of-squares l (* k k)) t) 2.0))

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in t around 0

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

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. unpow2 N/A

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

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

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

   8. lower-/.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lift-sin.f64 67.6

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

5. Applied rewrites 67.6%

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

6. Taylor expanded in k around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. pow2 N/A

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

   4. sqr-pow N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

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

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

   8. pow2 N/A

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

   9. lower-*.f64 24.3

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

8. Applied rewrites 24.3%

   \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\ell, \left(k \cdot k\right)\right)}{t} \cdot 2 \]
9. 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))))