Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.1% → 97.8%

Time: 41.5s

Alternatives: 14

Speedup: 38.3×

• 40.0% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.1% 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

real(8) function code(t, l, k)
    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: 97.8% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ (* 2.0 l) k) (/ (/ l k) (* (sin k) (* t (tan k))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.0%

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

3. Taylor expanded in 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. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. /-lowering-/.f64 N/A

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

5. Simplified 69.6%

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

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(2 \cdot \ell\right) \cdot \ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   5. /-lowering-/.f64 78.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

7. Applied egg-rr 78.4%

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

   1. associate-*l* N/A

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

   2. times-frac N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \ell}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}}{k}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \ell\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}}}{k}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t}} \cdot \frac{\cos k}{{\sin k}^{2}}}{k}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right), \color{blue}{k}\right)\right) \]

   7. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{1}{\frac{{\sin k}^{2}}{\cos k}}\right), k\right)\right) \]

   8. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\frac{{\sin k}^{2}}{\cos k}}\right), k\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\frac{\sin k \cdot \sin k}{\cos k}}\right), k\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\sin k \cdot \frac{\sin k}{\cos k}}\right), k\right)\right) \]

   11. tan-quot N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\sin k \cdot \tan k}\right), k\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(\sin k \cdot \tan k\right)\right), k\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\sin k \cdot \tan k\right)\right), k\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\sin k, \tan k\right)\right), k\right)\right) \]

   15. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \tan k\right)\right), k\right)\right) \]

   16. tan-lowering-tan.f64 91.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right), k\right)\right) \]

9. Applied egg-rr 91.9%

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

    1. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \left(\frac{\frac{\ell}{t}}{\sin k \cdot \tan k} \cdot \color{blue}{\frac{1}{k}}\right)\right) \]

    2. associate-/l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \left(\frac{\ell}{\left(\sin k \cdot \tan k\right) \cdot t} \cdot \frac{\color{blue}{1}}{k}\right)\right) \]

    3. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \left(\frac{\ell \cdot \frac{1}{k}}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot t}}\right)\right) \]

    4. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \left(\frac{\frac{\ell}{k}}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot t}\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{k}\right), \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot t\right)}\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot t\right)\right)\right) \]

    7. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\sin k \cdot \color{blue}{\left(\tan k \cdot t\right)}\right)\right)\right) \]

    8. /-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\sin k \cdot \left(\tan k \cdot \frac{t}{\color{blue}{1}}\right)\right)\right)\right) \]

    9. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\sin k \cdot \left(\tan k \cdot \frac{1}{\color{blue}{\frac{1}{t}}}\right)\right)\right)\right) \]

    10. div-inv N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \left(\sin k \cdot \frac{\tan k}{\color{blue}{\frac{1}{t}}}\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{*.f64}\left(\sin k, \color{blue}{\left(\frac{\tan k}{\frac{1}{t}}\right)}\right)\right)\right) \]

    12. sin-lowering-sin.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \left(\frac{\color{blue}{\tan k}}{\frac{1}{t}}\right)\right)\right)\right) \]

    13. div-inv N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \left(\tan k \cdot \color{blue}{\frac{1}{\frac{1}{t}}}\right)\right)\right)\right) \]

    14. clear-num N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \left(\tan k \cdot \frac{t}{\color{blue}{1}}\right)\right)\right)\right) \]

    15. /-rgt-identity N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \left(\tan k \cdot t\right)\right)\right)\right) \]

    16. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \left(t \cdot \color{blue}{\tan k}\right)\right)\right)\right) \]

    17. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(t, \color{blue}{\tan k}\right)\right)\right)\right) \]

    18. tan-lowering-tan.f64 97.8%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, k\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(t, \mathsf{tan.f64}\left(k\right)\right)\right)\right)\right) \]

11. Applied egg-rr 97.8%

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

Alternative 2: 84.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{k}{\ell}}}{k} \cdot \frac{1}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{k} \cdot \frac{\ell}{k \cdot \left(\sin k \cdot \left(t \cdot \tan k\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.5e-133)
   (* (/ (/ (/ 2.0 (/ k l)) (/ k l)) k) (/ 1.0 (* k t)))
   (* (/ (* 2.0 l) k) (/ l (* k (* (sin k) (* t (tan k))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.5e-133) {
		tmp = (((2.0 / (k / l)) / (k / l)) / k) * (1.0 / (k * t));
	} else {
		tmp = ((2.0 * l) / k) * (l / (k * (sin(k) * (t * tan(k)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.5d-133) then
        tmp = (((2.0d0 / (k / l)) / (k / l)) / k) * (1.0d0 / (k * t))
    else
        tmp = ((2.0d0 * l) / k) * (l / (k * (sin(k) * (t * tan(k)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.5e-133) {
		tmp = (((2.0 / (k / l)) / (k / l)) / k) * (1.0 / (k * t));
	} else {
		tmp = ((2.0 * l) / k) * (l / (k * (Math.sin(k) * (t * Math.tan(k)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3.5e-133:
		tmp = (((2.0 / (k / l)) / (k / l)) / k) * (1.0 / (k * t))
	else:
		tmp = ((2.0 * l) / k) * (l / (k * (math.sin(k) * (t * math.tan(k)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.5e-133)
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(k / l)) / Float64(k / l)) / k) * Float64(1.0 / Float64(k * t)));
	else
		tmp = Float64(Float64(Float64(2.0 * l) / k) * Float64(l / Float64(k * Float64(sin(k) * Float64(t * tan(k))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.5e-133)
		tmp = (((2.0 / (k / l)) / (k / l)) / k) * (1.0 / (k * t));
	else
		tmp = ((2.0 * l) / k) * (l / (k * (sin(k) * (t * tan(k)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.5e-133], N[(N[(N[(N[(2.0 / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(1.0 / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(k * N[(N[Sin[k], $MachinePrecision] * N[(t * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.50000000000000003e-133

   1. Initial program 35.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

   5. Simplified 67.6%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(2 \cdot \ell\right) \cdot \ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      5. /-lowering-/.f64 78.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   7. Applied egg-rr 78.1%

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

   8. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}, \mathsf{*.f64}\left(k, k\right)\right) \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2}}}{t}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{{k}^{2}}\right), t\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{{k}^{2}}\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{k \cdot k}\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot {\ell}^{2}}{k}}{k}\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{k}\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {\ell}^{2}\right), k\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), k\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), k\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      11. *-lowering-*.f64 65.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), k\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   10. Simplified 65.2%

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

       1. div-inv N/A

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

       2. times-frac N/A

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

       3. clear-num N/A

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

       4. div-inv N/A

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

       5. clear-num N/A

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

       6. /-rgt-identity N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{k}}{k}}{k}\right), \color{blue}{\left(\frac{1}{t \cdot k}\right)}\right) \]

   12. Applied egg-rr 76.1%

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

   if 3.50000000000000003e-133 < k

   1. Initial program 33.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

   5. Simplified 72.7%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(2 \cdot \ell\right) \cdot \ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      5. /-lowering-/.f64 78.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   7. Applied egg-rr 78.7%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \ell}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}}{k}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \ell\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}}}{k}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t}} \cdot \frac{\cos k}{{\sin k}^{2}}}{k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right), \color{blue}{k}\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{1}{\frac{{\sin k}^{2}}{\cos k}}\right), k\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\frac{{\sin k}^{2}}{\cos k}}\right), k\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\frac{\sin k \cdot \sin k}{\cos k}}\right), k\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\sin k \cdot \frac{\sin k}{\cos k}}\right), k\right)\right) \]

      11. tan-quot N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\sin k \cdot \tan k}\right), k\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(\sin k \cdot \tan k\right)\right), k\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\sin k \cdot \tan k\right)\right), k\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\sin k, \tan k\right)\right), k\right)\right) \]

      15. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \tan k\right)\right), k\right)\right) \]

      16. tan-lowering-tan.f64 93.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right), k\right)\right) \]

   9. Applied egg-rr 93.0%

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

       1. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \left(\frac{\frac{\ell}{\left(\sin k \cdot \tan k\right) \cdot t}}{k}\right)\right) \]

       2. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \left(\frac{\ell}{\color{blue}{k \cdot \left(\left(\sin k \cdot \tan k\right) \cdot t\right)}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(k \cdot \left(\left(\sin k \cdot \tan k\right) \cdot t\right)\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot t\right)}\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(\sin k \cdot \color{blue}{\left(\tan k \cdot t\right)}\right)\right)\right)\right) \]

       6. /-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(\sin k \cdot \left(\tan k \cdot \frac{t}{\color{blue}{1}}\right)\right)\right)\right)\right) \]

       7. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(\sin k \cdot \left(\tan k \cdot \frac{1}{\color{blue}{\frac{1}{t}}}\right)\right)\right)\right)\right) \]

       8. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(\sin k \cdot \frac{\tan k}{\color{blue}{\frac{1}{t}}}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\sin k, \color{blue}{\left(\frac{\tan k}{\frac{1}{t}}\right)}\right)\right)\right)\right) \]

       10. sin-lowering-sin.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \left(\frac{\color{blue}{\tan k}}{\frac{1}{t}}\right)\right)\right)\right)\right) \]

       11. div-inv N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \left(\tan k \cdot \color{blue}{\frac{1}{\frac{1}{t}}}\right)\right)\right)\right)\right) \]

       12. clear-num N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \left(\tan k \cdot \frac{t}{\color{blue}{1}}\right)\right)\right)\right)\right) \]

       13. /-rgt-identity N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \left(\tan k \cdot t\right)\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \left(t \cdot \color{blue}{\tan k}\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(t, \color{blue}{\tan k}\right)\right)\right)\right)\right) \]

       16. tan-lowering-tan.f64 91.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{*.f64}\left(t, \mathsf{tan.f64}\left(k\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 91.2%

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

Alternative 3: 81.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.65 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{k}{\ell}}}{k} \cdot \frac{1}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \ell\right) \cdot \frac{\frac{\frac{\ell}{t}}{\sin k \cdot \tan k}}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.65e-132)
   (* (/ (/ (/ 2.0 (/ k l)) (/ k l)) k) (/ 1.0 (* k t)))
   (* (* 2.0 l) (/ (/ (/ l t) (* (sin k) (tan k))) (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.65e-132) {
		tmp = (((2.0 / (k / l)) / (k / l)) / k) * (1.0 / (k * t));
	} else {
		tmp = (2.0 * l) * (((l / t) / (sin(k) * tan(k))) / (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.65d-132) then
        tmp = (((2.0d0 / (k / l)) / (k / l)) / k) * (1.0d0 / (k * t))
    else
        tmp = (2.0d0 * l) * (((l / t) / (sin(k) * tan(k))) / (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.65e-132) {
		tmp = (((2.0 / (k / l)) / (k / l)) / k) * (1.0 / (k * t));
	} else {
		tmp = (2.0 * l) * (((l / t) / (Math.sin(k) * Math.tan(k))) / (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.65e-132:
		tmp = (((2.0 / (k / l)) / (k / l)) / k) * (1.0 / (k * t))
	else:
		tmp = (2.0 * l) * (((l / t) / (math.sin(k) * math.tan(k))) / (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.65e-132)
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(k / l)) / Float64(k / l)) / k) * Float64(1.0 / Float64(k * t)));
	else
		tmp = Float64(Float64(2.0 * l) * Float64(Float64(Float64(l / t) / Float64(sin(k) * tan(k))) / Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.65e-132)
		tmp = (((2.0 / (k / l)) / (k / l)) / k) * (1.0 / (k * t));
	else
		tmp = (2.0 * l) * (((l / t) / (sin(k) * tan(k))) / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.65e-132], N[(N[(N[(N[(2.0 / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(1.0 / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * l), $MachinePrecision] * N[(N[(N[(l / t), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.65000000000000015e-132

   1. Initial program 35.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

   5. Simplified 67.6%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(2 \cdot \ell\right) \cdot \ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      5. /-lowering-/.f64 78.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   7. Applied egg-rr 78.1%

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

   8. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}, \mathsf{*.f64}\left(k, k\right)\right) \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2}}}{t}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{{k}^{2}}\right), t\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{{k}^{2}}\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{k \cdot k}\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot {\ell}^{2}}{k}}{k}\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{k}\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {\ell}^{2}\right), k\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), k\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), k\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      11. *-lowering-*.f64 65.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), k\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   10. Simplified 65.2%

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

       1. div-inv N/A

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

       2. times-frac N/A

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

       3. clear-num N/A

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

       4. div-inv N/A

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

       5. clear-num N/A

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

       6. /-rgt-identity N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{k}}{k}}{k}\right), \color{blue}{\left(\frac{1}{t \cdot k}\right)}\right) \]

   12. Applied egg-rr 76.1%

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

   if 2.65000000000000015e-132 < k

   1. Initial program 33.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

   5. Simplified 72.7%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(2 \cdot \ell\right) \cdot \ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      5. /-lowering-/.f64 78.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   7. Applied egg-rr 78.7%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(2 \cdot \ell\right), \color{blue}{\left(\frac{\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}}{k \cdot k}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\frac{\color{blue}{\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}}}{k \cdot k}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right), \color{blue}{\left(k \cdot k\right)}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{1}{\frac{{\sin k}^{2}}{\cos k}}\right), \left(k \cdot k\right)\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\frac{{\sin k}^{2}}{\cos k}}\right), \left(\color{blue}{k} \cdot k\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\frac{\sin k \cdot \sin k}{\cos k}}\right), \left(k \cdot k\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\sin k \cdot \frac{\sin k}{\cos k}}\right), \left(k \cdot k\right)\right)\right) \]

      10. tan-quot N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\sin k \cdot \tan k}\right), \left(k \cdot k\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(\sin k \cdot \tan k\right)\right), \left(\color{blue}{k} \cdot k\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\sin k \cdot \tan k\right)\right), \left(k \cdot k\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\sin k, \tan k\right)\right), \left(k \cdot k\right)\right)\right) \]

      14. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \tan k\right)\right), \left(k \cdot k\right)\right)\right) \]

      15. tan-lowering-tan.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right), \left(k \cdot k\right)\right)\right) \]

      16. *-lowering-*.f64 81.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   9. Applied egg-rr 81.0%

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

Alternative 4: 75.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-181}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{k}{\ell}}}{k} \cdot \frac{1}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t} \cdot \frac{\cos k}{k \cdot k}}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e-181)
   (* (/ (/ (/ 2.0 (/ k l)) (/ k l)) k) (/ 1.0 (* k t)))
   (/ (* (/ (* 2.0 (* l l)) t) (/ (cos k) (* k k))) (* k k))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-181) {
		tmp = (((2.0 / (k / l)) / (k / l)) / k) * (1.0 / (k * t));
	} else {
		tmp = (((2.0 * (l * l)) / t) * (cos(k) / (k * k))) / (k * k);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 5d-181) then
        tmp = (((2.0d0 / (k / l)) / (k / l)) / k) * (1.0d0 / (k * t))
    else
        tmp = (((2.0d0 * (l * l)) / t) * (cos(k) / (k * k))) / (k * k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-181) {
		tmp = (((2.0 / (k / l)) / (k / l)) / k) * (1.0 / (k * t));
	} else {
		tmp = (((2.0 * (l * l)) / t) * (Math.cos(k) / (k * k))) / (k * k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 5e-181:
		tmp = (((2.0 / (k / l)) / (k / l)) / k) * (1.0 / (k * t))
	else:
		tmp = (((2.0 * (l * l)) / t) * (math.cos(k) / (k * k))) / (k * k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e-181)
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(k / l)) / Float64(k / l)) / k) * Float64(1.0 / Float64(k * t)));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 * Float64(l * l)) / t) * Float64(cos(k) / Float64(k * k))) / Float64(k * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 5e-181)
		tmp = (((2.0 / (k / l)) / (k / l)) / k) * (1.0 / (k * t));
	else
		tmp = (((2.0 * (l * l)) / t) * (cos(k) / (k * k))) / (k * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e-181], N[(N[(N[(N[(2.0 / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(1.0 / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.0000000000000001e-181

   1. Initial program 30.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

   5. Simplified 67.0%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(2 \cdot \ell\right) \cdot \ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      5. /-lowering-/.f64 79.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   7. Applied egg-rr 79.8%

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

   8. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}, \mathsf{*.f64}\left(k, k\right)\right) \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2}}}{t}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{{k}^{2}}\right), t\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{{k}^{2}}\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{k \cdot k}\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot {\ell}^{2}}{k}}{k}\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{k}\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {\ell}^{2}\right), k\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), k\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), k\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      11. *-lowering-*.f64 69.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), k\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   10. Simplified 69.3%

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

       1. div-inv N/A

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

       2. times-frac N/A

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

       3. clear-num N/A

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

       4. div-inv N/A

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

       5. clear-num N/A

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

       6. /-rgt-identity N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{k}}{k}}{k}\right), \color{blue}{\left(\frac{1}{t \cdot k}\right)}\right) \]

   12. Applied egg-rr 90.7%

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

   if 5.0000000000000001e-181 < (*.f64 l l)

   1. Initial program 37.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

   5. Simplified 71.0%

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

   6. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), t\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \color{blue}{\left({k}^{2}\right)}\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), t\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \left(k \cdot k\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      2. *-lowering-*.f64 60.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), t\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{*.f64}\left(k, k\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   8. Simplified 60.9%

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

Alternative 5: 74.6% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{k}{\ell}}}{k} \cdot \frac{1}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{k} \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.45e-146)
   (* (/ (/ (/ 2.0 (/ k l)) (/ k l)) k) (/ 1.0 (* k t)))
   (* (/ (* 2.0 l) k) (/ (/ (/ l t) (* k k)) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.45e-146) {
		tmp = (((2.0 / (k / l)) / (k / l)) / k) * (1.0 / (k * t));
	} else {
		tmp = ((2.0 * l) / k) * (((l / t) / (k * k)) / k);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.45d-146) then
        tmp = (((2.0d0 / (k / l)) / (k / l)) / k) * (1.0d0 / (k * t))
    else
        tmp = ((2.0d0 * l) / k) * (((l / t) / (k * k)) / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.45e-146) {
		tmp = (((2.0 / (k / l)) / (k / l)) / k) * (1.0 / (k * t));
	} else {
		tmp = ((2.0 * l) / k) * (((l / t) / (k * k)) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.45e-146:
		tmp = (((2.0 / (k / l)) / (k / l)) / k) * (1.0 / (k * t))
	else:
		tmp = ((2.0 * l) / k) * (((l / t) / (k * k)) / k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.45e-146)
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(k / l)) / Float64(k / l)) / k) * Float64(1.0 / Float64(k * t)));
	else
		tmp = Float64(Float64(Float64(2.0 * l) / k) * Float64(Float64(Float64(l / t) / Float64(k * k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.45e-146)
		tmp = (((2.0 / (k / l)) / (k / l)) / k) * (1.0 / (k * t));
	else
		tmp = ((2.0 * l) / k) * (((l / t) / (k * k)) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.45e-146], N[(N[(N[(N[(2.0 / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * N[(1.0 / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] / k), $MachinePrecision] * N[(N[(N[(l / t), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.45000000000000005e-146

   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 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. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

   5. Simplified 68.2%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(2 \cdot \ell\right) \cdot \ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      5. /-lowering-/.f64 79.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   7. Applied egg-rr 79.0%

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

   8. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}, \mathsf{*.f64}\left(k, k\right)\right) \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2}}}{t}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{{k}^{2}}\right), t\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{{k}^{2}}\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{k \cdot k}\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot {\ell}^{2}}{k}}{k}\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{k}\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {\ell}^{2}\right), k\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), k\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), k\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      11. *-lowering-*.f64 65.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), k\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   10. Simplified 65.8%

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

       1. div-inv N/A

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

       2. times-frac N/A

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

       3. clear-num N/A

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

       4. div-inv N/A

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

       5. clear-num N/A

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

       6. /-rgt-identity N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{k}}{k}}{k}\right), \color{blue}{\left(\frac{1}{t \cdot k}\right)}\right) \]

   12. Applied egg-rr 75.7%

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

   if 1.45000000000000005e-146 < k

   1. Initial program 32.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

   5. Simplified 71.6%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(2 \cdot \ell\right) \cdot \ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      5. /-lowering-/.f64 77.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   7. Applied egg-rr 77.4%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \ell}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}}{k}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \ell\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}}}{k}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t}} \cdot \frac{\cos k}{{\sin k}^{2}}}{k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right), \color{blue}{k}\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{1}{\frac{{\sin k}^{2}}{\cos k}}\right), k\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\frac{{\sin k}^{2}}{\cos k}}\right), k\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\frac{\sin k \cdot \sin k}{\cos k}}\right), k\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\sin k \cdot \frac{\sin k}{\cos k}}\right), k\right)\right) \]

      11. tan-quot N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\sin k \cdot \tan k}\right), k\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(\sin k \cdot \tan k\right)\right), k\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\sin k \cdot \tan k\right)\right), k\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\sin k, \tan k\right)\right), k\right)\right) \]

      15. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \tan k\right)\right), k\right)\right) \]

      16. tan-lowering-tan.f64 93.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right), k\right)\right) \]

   9. Applied egg-rr 93.2%

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

   10. Taylor expanded in k around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \color{blue}{\left({k}^{2}\right)}\right), k\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot k\right)\right), k\right)\right) \]

       2. *-lowering-*.f64 60.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), k\right)\right) \]

   12. Simplified 60.9%

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

Alternative 6: 74.1% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{k}{\ell}}}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{k} \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3e-144)
   (/ (/ (/ 2.0 (/ k l)) (/ k l)) (* k (* k t)))
   (* (/ (* 2.0 l) k) (/ (/ (/ l t) (* k k)) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3e-144) {
		tmp = ((2.0 / (k / l)) / (k / l)) / (k * (k * t));
	} else {
		tmp = ((2.0 * l) / k) * (((l / t) / (k * k)) / k);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3d-144) then
        tmp = ((2.0d0 / (k / l)) / (k / l)) / (k * (k * t))
    else
        tmp = ((2.0d0 * l) / k) * (((l / t) / (k * k)) / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3e-144) {
		tmp = ((2.0 / (k / l)) / (k / l)) / (k * (k * t));
	} else {
		tmp = ((2.0 * l) / k) * (((l / t) / (k * k)) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3e-144:
		tmp = ((2.0 / (k / l)) / (k / l)) / (k * (k * t))
	else:
		tmp = ((2.0 * l) / k) * (((l / t) / (k * k)) / k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3e-144)
		tmp = Float64(Float64(Float64(2.0 / Float64(k / l)) / Float64(k / l)) / Float64(k * Float64(k * t)));
	else
		tmp = Float64(Float64(Float64(2.0 * l) / k) * Float64(Float64(Float64(l / t) / Float64(k * k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3e-144)
		tmp = ((2.0 / (k / l)) / (k / l)) / (k * (k * t));
	else
		tmp = ((2.0 * l) / k) * (((l / t) / (k * k)) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3e-144], N[(N[(N[(2.0 / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] / k), $MachinePrecision] * N[(N[(N[(l / t), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.9999999999999999e-144

   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 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. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

   5. Simplified 68.2%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(2 \cdot \ell\right) \cdot \ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      5. /-lowering-/.f64 79.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   7. Applied egg-rr 79.0%

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

   8. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}, \mathsf{*.f64}\left(k, k\right)\right) \]
   9. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2}}}{t}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{{k}^{2}}\right), t\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{{k}^{2}}\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{k \cdot k}\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot {\ell}^{2}}{k}}{k}\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot {\ell}^{2}}{k}\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot {\ell}^{2}\right), k\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2}\right)\right), k\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\ell \cdot \ell\right)\right), k\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      11. *-lowering-*.f64 65.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), k\right), k\right), t\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   10. Simplified 65.8%

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

       1. associate-/l/ N/A

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

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{k}}{k}\right), \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \left(\ell \cdot \ell\right)}{k \cdot k}\right), \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(2 \cdot \ell\right) \cdot \ell}{k \cdot k}\right), \left(\left(\color{blue}{k} \cdot k\right) \cdot t\right)\right) \]

       5. times-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \ell}{k} \cdot \frac{\ell}{k}\right), \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right) \]

       6. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{2 \cdot \ell}{k} \cdot \frac{1}{\frac{k}{\ell}}\right), \left(\left(k \cdot \color{blue}{k}\right) \cdot t\right)\right) \]

       7. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{2 \cdot \ell}{k}}{\frac{k}{\ell}}\right), \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \ell}{k}\right), \left(\frac{k}{\ell}\right)\right), \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right) \]

       9. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{\ell}{k}\right), \left(\frac{k}{\ell}\right)\right), \left(\left(\color{blue}{k} \cdot k\right) \cdot t\right)\right) \]

       10. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{1}{\frac{k}{\ell}}\right), \left(\frac{k}{\ell}\right)\right), \left(\left(k \cdot k\right) \cdot t\right)\right) \]

       11. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\frac{k}{\ell}}\right), \left(\frac{k}{\ell}\right)\right), \left(\left(\color{blue}{k} \cdot k\right) \cdot t\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{\ell}\right)\right), \left(\frac{k}{\ell}\right)\right), \left(\left(\color{blue}{k} \cdot k\right) \cdot t\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \ell\right)\right), \left(\frac{k}{\ell}\right)\right), \left(\left(k \cdot k\right) \cdot t\right)\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \ell\right)\right), \mathsf{/.f64}\left(k, \ell\right)\right), \left(\left(k \cdot \color{blue}{k}\right) \cdot t\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \ell\right)\right), \mathsf{/.f64}\left(k, \ell\right)\right), \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \ell\right)\right), \mathsf{/.f64}\left(k, \ell\right)\right), \left(k \cdot \left(t \cdot \color{blue}{k}\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \ell\right)\right), \mathsf{/.f64}\left(k, \ell\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot k\right)}\right)\right) \]

       18. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \ell\right)\right), \mathsf{/.f64}\left(k, \ell\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{t}\right)\right)\right) \]

       19. *-lowering-*.f64 74.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \ell\right)\right), \mathsf{/.f64}\left(k, \ell\right)\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{t}\right)\right)\right) \]

   12. Applied egg-rr 74.4%

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

   if 2.9999999999999999e-144 < k

   1. Initial program 32.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

   5. Simplified 71.6%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(2 \cdot \ell\right) \cdot \ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      5. /-lowering-/.f64 77.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   7. Applied egg-rr 77.4%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \ell}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}}{k}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \ell\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}}}{k}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t}} \cdot \frac{\cos k}{{\sin k}^{2}}}{k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right), \color{blue}{k}\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{1}{\frac{{\sin k}^{2}}{\cos k}}\right), k\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\frac{{\sin k}^{2}}{\cos k}}\right), k\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\frac{\sin k \cdot \sin k}{\cos k}}\right), k\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\sin k \cdot \frac{\sin k}{\cos k}}\right), k\right)\right) \]

      11. tan-quot N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\sin k \cdot \tan k}\right), k\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(\sin k \cdot \tan k\right)\right), k\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\sin k \cdot \tan k\right)\right), k\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\sin k, \tan k\right)\right), k\right)\right) \]

      15. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \tan k\right)\right), k\right)\right) \]

      16. tan-lowering-tan.f64 93.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right), k\right)\right) \]

   9. Applied egg-rr 93.2%

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

   10. Taylor expanded in k around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \color{blue}{\left({k}^{2}\right)}\right), k\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot k\right)\right), k\right)\right) \]

       2. *-lowering-*.f64 60.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), k\right)\right) \]

   12. Simplified 60.9%

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

Alternative 7: 72.8% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{+50}:\\ \;\;\;\;\frac{2 \cdot \ell}{k} \cdot \frac{\frac{\ell}{k \cdot \left(k \cdot k\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.3333333333333333}{\frac{k}{\frac{\ell}{\frac{t}{\ell}}}}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 9.5e+50)
   (* (/ (* 2.0 l) k) (/ (/ l (* k (* k k))) t))
   (/ (/ -0.3333333333333333 (/ k (/ l (/ t l)))) k)))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.5e+50) {
		tmp = ((2.0 * l) / k) * ((l / (k * (k * k))) / t);
	} else {
		tmp = (-0.3333333333333333 / (k / (l / (t / l)))) / k;
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9.5d+50) then
        tmp = ((2.0d0 * l) / k) * ((l / (k * (k * k))) / t)
    else
        tmp = ((-0.3333333333333333d0) / (k / (l / (t / l)))) / k
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.5e+50) {
		tmp = ((2.0 * l) / k) * ((l / (k * (k * k))) / t);
	} else {
		tmp = (-0.3333333333333333 / (k / (l / (t / l)))) / k;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 9.5e+50:
		tmp = ((2.0 * l) / k) * ((l / (k * (k * k))) / t)
	else:
		tmp = (-0.3333333333333333 / (k / (l / (t / l)))) / k
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 9.5e+50)
		tmp = Float64(Float64(Float64(2.0 * l) / k) * Float64(Float64(l / Float64(k * Float64(k * k))) / t));
	else
		tmp = Float64(Float64(-0.3333333333333333 / Float64(k / Float64(l / Float64(t / l)))) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 9.5e+50)
		tmp = ((2.0 * l) / k) * ((l / (k * (k * k))) / t);
	else
		tmp = (-0.3333333333333333 / (k / (l / (t / l)))) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 9.5e+50], N[(N[(N[(2.0 * l), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 / N[(k / N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.4999999999999993e50

   1. Initial program 35.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

   5. Simplified 69.8%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(2 \cdot \ell\right) \cdot \ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

      5. /-lowering-/.f64 80.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   7. Applied egg-rr 80.5%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \ell}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}}{k}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \ell\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}}}{k}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t}} \cdot \frac{\cos k}{{\sin k}^{2}}}{k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right), \color{blue}{k}\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{1}{\frac{{\sin k}^{2}}{\cos k}}\right), k\right)\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\frac{{\sin k}^{2}}{\cos k}}\right), k\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\frac{\sin k \cdot \sin k}{\cos k}}\right), k\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\sin k \cdot \frac{\sin k}{\cos k}}\right), k\right)\right) \]

      11. tan-quot N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\sin k \cdot \tan k}\right), k\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(\sin k \cdot \tan k\right)\right), k\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\sin k \cdot \tan k\right)\right), k\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\sin k, \tan k\right)\right), k\right)\right) \]

      15. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \tan k\right)\right), k\right)\right) \]

      16. tan-lowering-tan.f64 91.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right), k\right)\right) \]

   9. Applied egg-rr 91.3%

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

   10. Taylor expanded in k around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \color{blue}{\left(\frac{\ell}{{k}^{3} \cdot t}\right)}\right) \]
   11. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \left(\frac{\frac{\ell}{{k}^{3}}}{\color{blue}{t}}\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{{k}^{3}}\right), \color{blue}{t}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left({k}^{3}\right)\right), t\right)\right) \]

       4. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot \left(k \cdot k\right)\right)\right), t\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot {k}^{2}\right)\right), t\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left({k}^{2}\right)\right)\right), t\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(k \cdot k\right)\right)\right), t\right)\right) \]

       8. *-lowering-*.f64 72.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, k\right)\right)\right), t\right)\right) \]

   12. Simplified 72.8%

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

   if 9.4999999999999993e50 < k

   1. Initial program 33.9%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right) \]

   3. Simplified 37.1%

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

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   7. Simplified 21.7%

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

   8. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left(k \cdot \color{blue}{k}\right)\right) \]

      12. *-lowering-*.f64 53.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 53.6%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333}{k \cdot k}} \]
   11. Step-by-step derivation

       1. clear-num N/A

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

       2. inv-pow N/A

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

       3. pow-to-exp N/A

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

       4. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{k \cdot k}{\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}}\right) \cdot -1\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{k \cdot k}{\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}}\right), -1\right)\right) \]

       6. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{k \cdot k}{\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}}\right)\right), -1\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{k \cdot k}{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}}\right)\right), -1\right)\right) \]

       8. associate-/l* N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(k \cdot \frac{k}{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}}\right)\right), -1\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(k, \left(\frac{k}{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}}\right)\right)\right), -1\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \left(\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}\right)\right)\right)\right), -1\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right)\right)\right)\right), -1\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\left(\frac{\ell \cdot \ell}{t}\right), \frac{-1}{3}\right)\right)\right)\right), -1\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-1}{3}\right)\right)\right)\right), -1\right)\right) \]

       14. *-lowering-*.f64 24.6%

           \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right)\right)\right)\right), -1\right)\right) \]

   12. Applied egg-rr 24.6%

       \[\leadsto \color{blue}{e^{\log \left(k \cdot \frac{k}{\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333}\right) \cdot -1}} \]
   13. Step-by-step derivation

       1. exp-to-pow N/A

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

       2. *-commutative N/A

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

       3. unpow-prod-down N/A

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

       4. inv-pow N/A

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

       5. clear-num N/A

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

       6. inv-pow N/A

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

       7. un-div-inv N/A

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

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}}{k}\right), \color{blue}{k}\right) \]

       9. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{k}{\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}}}\right), k\right) \]

       10. associate-/r* N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{\frac{k}{\frac{\ell \cdot \ell}{t}}}{\frac{-1}{3}}}\right), k\right) \]

       11. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{-1}{3}}{\frac{k}{\frac{\ell \cdot \ell}{t}}}\right), k\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \left(\frac{k}{\frac{\ell \cdot \ell}{t}}\right)\right), k\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(k, \left(\frac{\ell \cdot \ell}{t}\right)\right)\right), k\right) \]

       14. associate-/l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(k, \left(\ell \cdot \frac{\ell}{t}\right)\right)\right), k\right) \]

       15. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(k, \left(\ell \cdot \frac{1}{\frac{t}{\ell}}\right)\right)\right), k\right) \]

       16. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(k, \left(\frac{\ell}{\frac{t}{\ell}}\right)\right)\right), k\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, \left(\frac{t}{\ell}\right)\right)\right)\right), k\right) \]

       18. /-lowering-/.f64 55.3%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right)\right)\right), k\right) \]

   14. Applied egg-rr 55.3%

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

Alternative 8: 24.9% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{\frac{t}{\ell}}\\ \mathbf{if}\;k \leq 3.1 \cdot 10^{+37}:\\ \;\;\;\;t\_1 \cdot -0.11666666666666667\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.3333333333333333}{\frac{k}{t\_1}}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (/ t l))))
   (if (<= k 3.1e+37)
     (* t_1 -0.11666666666666667)
     (/ (/ -0.3333333333333333 (/ k t_1)) k))))

C

double code(double t, double l, double k) {
	double t_1 = l / (t / l);
	double tmp;
	if (k <= 3.1e+37) {
		tmp = t_1 * -0.11666666666666667;
	} else {
		tmp = (-0.3333333333333333 / (k / t_1)) / k;
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l / (t / l)
    if (k <= 3.1d+37) then
        tmp = t_1 * (-0.11666666666666667d0)
    else
        tmp = ((-0.3333333333333333d0) / (k / t_1)) / k
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l / (t / l);
	double tmp;
	if (k <= 3.1e+37) {
		tmp = t_1 * -0.11666666666666667;
	} else {
		tmp = (-0.3333333333333333 / (k / t_1)) / k;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l / (t / l)
	tmp = 0
	if k <= 3.1e+37:
		tmp = t_1 * -0.11666666666666667
	else:
		tmp = (-0.3333333333333333 / (k / t_1)) / k
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / Float64(t / l))
	tmp = 0.0
	if (k <= 3.1e+37)
		tmp = Float64(t_1 * -0.11666666666666667);
	else
		tmp = Float64(Float64(-0.3333333333333333 / Float64(k / t_1)) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l / (t / l);
	tmp = 0.0;
	if (k <= 3.1e+37)
		tmp = t_1 * -0.11666666666666667;
	else
		tmp = (-0.3333333333333333 / (k / t_1)) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 3.1e+37], N[(t$95$1 * -0.11666666666666667), $MachinePrecision], N[(N[(-0.3333333333333333 / N[(k / t$95$1), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.1000000000000002e37

   1. Initial program 34.8%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right), \color{blue}{\left({k}^{4}\right)}\right) \]

   5. Simplified 29.0%

      \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t} + \left(k \cdot k\right) \cdot \left(-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t} + k \cdot \left(k \cdot \left(\frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667\right)\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{-7}{60} \cdot \frac{{\ell}^{2}}{t}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2}}{t} \cdot \color{blue}{\frac{-7}{60}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \color{blue}{\frac{-7}{60}}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-7}{60}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-7}{60}\right) \]

      5. *-lowering-*.f64 14.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-7}{60}\right) \]

   8. Simplified 14.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\ell \cdot \frac{\ell}{t}\right), \frac{-7}{60}\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\ell \cdot \frac{1}{\frac{t}{\ell}}\right), \frac{-7}{60}\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\frac{t}{\ell}}\right), \frac{-7}{60}\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(\ell\right)}{\mathsf{neg}\left(\frac{t}{\ell}\right)}\right), \frac{-7}{60}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\ell\right)\right), \left(\mathsf{neg}\left(\frac{t}{\ell}\right)\right)\right), \frac{-7}{60}\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(0 - \ell\right), \left(\mathsf{neg}\left(\frac{t}{\ell}\right)\right)\right), \frac{-7}{60}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\log 1 - \ell\right), \left(\mathsf{neg}\left(\frac{t}{\ell}\right)\right)\right), \frac{-7}{60}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\log 1, \ell\right), \left(\mathsf{neg}\left(\frac{t}{\ell}\right)\right)\right), \frac{-7}{60}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \ell\right), \left(\mathsf{neg}\left(\frac{t}{\ell}\right)\right)\right), \frac{-7}{60}\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \ell\right), \left(0 - \frac{t}{\ell}\right)\right), \frac{-7}{60}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \ell\right), \left(\log 1 - \frac{t}{\ell}\right)\right), \frac{-7}{60}\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \ell\right), \mathsf{\_.f64}\left(\log 1, \left(\frac{t}{\ell}\right)\right)\right), \frac{-7}{60}\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \ell\right), \mathsf{\_.f64}\left(0, \left(\frac{t}{\ell}\right)\right)\right), \frac{-7}{60}\right) \]

      14. /-lowering-/.f64 19.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \ell\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right)\right), \frac{-7}{60}\right) \]

   10. Applied egg-rr 19.4%

       \[\leadsto \color{blue}{\frac{0 - \ell}{0 - \frac{t}{\ell}}} \cdot -0.11666666666666667 \]

   if 3.1000000000000002e37 < k

   1. Initial program 35.3%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right) \]

   3. Simplified 38.4%

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

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   7. Simplified 22.3%

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

   8. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left(k \cdot \color{blue}{k}\right)\right) \]

      12. *-lowering-*.f64 52.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 52.8%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333}{k \cdot k}} \]
   11. Step-by-step derivation

       1. clear-num N/A

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

       2. inv-pow N/A

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

       3. pow-to-exp N/A

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

       4. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{k \cdot k}{\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}}\right) \cdot -1\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{k \cdot k}{\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}}\right), -1\right)\right) \]

       6. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{k \cdot k}{\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}}\right)\right), -1\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{k \cdot k}{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}}\right)\right), -1\right)\right) \]

       8. associate-/l* N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(k \cdot \frac{k}{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}}\right)\right), -1\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(k, \left(\frac{k}{\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}}\right)\right)\right), -1\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \left(\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}\right)\right)\right)\right), -1\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \left(\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}\right)\right)\right)\right), -1\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\left(\frac{\ell \cdot \ell}{t}\right), \frac{-1}{3}\right)\right)\right)\right), -1\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-1}{3}\right)\right)\right)\right), -1\right)\right) \]

       14. *-lowering-*.f64 25.2%

           \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right)\right)\right)\right), -1\right)\right) \]

   12. Applied egg-rr 25.2%

       \[\leadsto \color{blue}{e^{\log \left(k \cdot \frac{k}{\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333}\right) \cdot -1}} \]
   13. Step-by-step derivation

       1. exp-to-pow N/A

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

       2. *-commutative N/A

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

       3. unpow-prod-down N/A

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

       4. inv-pow N/A

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

       5. clear-num N/A

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

       6. inv-pow N/A

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

       7. un-div-inv N/A

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

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}}{k}\right), \color{blue}{k}\right) \]

       9. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{k}{\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}}}\right), k\right) \]

       10. associate-/r* N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{\frac{k}{\frac{\ell \cdot \ell}{t}}}{\frac{-1}{3}}}\right), k\right) \]

       11. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{-1}{3}}{\frac{k}{\frac{\ell \cdot \ell}{t}}}\right), k\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \left(\frac{k}{\frac{\ell \cdot \ell}{t}}\right)\right), k\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(k, \left(\frac{\ell \cdot \ell}{t}\right)\right)\right), k\right) \]

       14. associate-/l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(k, \left(\ell \cdot \frac{\ell}{t}\right)\right)\right), k\right) \]

       15. clear-num N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(k, \left(\ell \cdot \frac{1}{\frac{t}{\ell}}\right)\right)\right), k\right) \]

       16. un-div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(k, \left(\frac{\ell}{\frac{t}{\ell}}\right)\right)\right), k\right) \]

       17. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, \left(\frac{t}{\ell}\right)\right)\right)\right), k\right) \]

       18. /-lowering-/.f64 54.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right)\right)\right), k\right) \]

   14. Applied egg-rr 54.4%

       \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333}{\frac{k}{\frac{\ell}{\frac{t}{\ell}}}}}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{+37}:\\ \;\;\;\;\frac{\ell}{\frac{t}{\ell}} \cdot -0.11666666666666667\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.3333333333333333}{\frac{k}{\frac{\ell}{\frac{t}{\ell}}}}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 24.7% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{+37}:\\ \;\;\;\;\frac{\ell}{\frac{t}{\ell}} \cdot -0.11666666666666667\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{t} \cdot \frac{-0.3333333333333333}{k \cdot k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 3.1e+37)
   (* (/ l (/ t l)) -0.11666666666666667)
   (* l (* (/ l t) (/ -0.3333333333333333 (* k k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.1e+37) {
		tmp = (l / (t / l)) * -0.11666666666666667;
	} else {
		tmp = l * ((l / t) * (-0.3333333333333333 / (k * k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.1d+37) then
        tmp = (l / (t / l)) * (-0.11666666666666667d0)
    else
        tmp = l * ((l / t) * ((-0.3333333333333333d0) / (k * k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.1e+37) {
		tmp = (l / (t / l)) * -0.11666666666666667;
	} else {
		tmp = l * ((l / t) * (-0.3333333333333333 / (k * k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 3.1e+37:
		tmp = (l / (t / l)) * -0.11666666666666667
	else:
		tmp = l * ((l / t) * (-0.3333333333333333 / (k * k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 3.1e+37)
		tmp = Float64(Float64(l / Float64(t / l)) * -0.11666666666666667);
	else
		tmp = Float64(l * Float64(Float64(l / t) * Float64(-0.3333333333333333 / Float64(k * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.1e+37)
		tmp = (l / (t / l)) * -0.11666666666666667;
	else
		tmp = l * ((l / t) * (-0.3333333333333333 / (k * k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 3.1e+37], N[(N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision] * -0.11666666666666667), $MachinePrecision], N[(l * N[(N[(l / t), $MachinePrecision] * N[(-0.3333333333333333 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.1000000000000002e37

   1. Initial program 34.8%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right), \color{blue}{\left({k}^{4}\right)}\right) \]

   5. Simplified 29.0%

      \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t} + \left(k \cdot k\right) \cdot \left(-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t} + k \cdot \left(k \cdot \left(\frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667\right)\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{-7}{60} \cdot \frac{{\ell}^{2}}{t}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{\ell}^{2}}{t} \cdot \color{blue}{\frac{-7}{60}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \color{blue}{\frac{-7}{60}}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-7}{60}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-7}{60}\right) \]

      5. *-lowering-*.f64 14.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-7}{60}\right) \]

   8. Simplified 14.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\ell \cdot \frac{\ell}{t}\right), \frac{-7}{60}\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\ell \cdot \frac{1}{\frac{t}{\ell}}\right), \frac{-7}{60}\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{\frac{t}{\ell}}\right), \frac{-7}{60}\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(\ell\right)}{\mathsf{neg}\left(\frac{t}{\ell}\right)}\right), \frac{-7}{60}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\ell\right)\right), \left(\mathsf{neg}\left(\frac{t}{\ell}\right)\right)\right), \frac{-7}{60}\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(0 - \ell\right), \left(\mathsf{neg}\left(\frac{t}{\ell}\right)\right)\right), \frac{-7}{60}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\log 1 - \ell\right), \left(\mathsf{neg}\left(\frac{t}{\ell}\right)\right)\right), \frac{-7}{60}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\log 1, \ell\right), \left(\mathsf{neg}\left(\frac{t}{\ell}\right)\right)\right), \frac{-7}{60}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \ell\right), \left(\mathsf{neg}\left(\frac{t}{\ell}\right)\right)\right), \frac{-7}{60}\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \ell\right), \left(0 - \frac{t}{\ell}\right)\right), \frac{-7}{60}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \ell\right), \left(\log 1 - \frac{t}{\ell}\right)\right), \frac{-7}{60}\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \ell\right), \mathsf{\_.f64}\left(\log 1, \left(\frac{t}{\ell}\right)\right)\right), \frac{-7}{60}\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \ell\right), \mathsf{\_.f64}\left(0, \left(\frac{t}{\ell}\right)\right)\right), \frac{-7}{60}\right) \]

      14. /-lowering-/.f64 19.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \ell\right), \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \ell\right)\right)\right), \frac{-7}{60}\right) \]

   10. Applied egg-rr 19.4%

       \[\leadsto \color{blue}{\frac{0 - \ell}{0 - \frac{t}{\ell}}} \cdot -0.11666666666666667 \]

   if 3.1000000000000002e37 < k

   1. Initial program 35.3%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right) \]

   3. Simplified 38.4%

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

   5. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   7. Simplified 22.3%

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

   8. Taylor expanded in k around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left(k \cdot \color{blue}{k}\right)\right) \]

      12. *-lowering-*.f64 52.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   10. Simplified 52.8%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333}{k \cdot k}} \]
   11. Step-by-step derivation

       1. associate-/l* N/A

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

       2. associate-/l* N/A

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

       3. associate-*l* N/A

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

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\frac{-1}{3}}{k \cdot k}\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\frac{-1}{3}}{k \cdot k}\right)}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\frac{-1}{3}}}{k \cdot k}\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\frac{-1}{3}, \color{blue}{\left(k \cdot k\right)}\right)\right)\right) \]

       8. *-lowering-*.f64 54.6%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right) \]

   12. Applied egg-rr 54.6%

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

4. Final simplification 24.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{+37}:\\ \;\;\;\;\frac{\ell}{\frac{t}{\ell}} \cdot -0.11666666666666667\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{t} \cdot \frac{-0.3333333333333333}{k \cdot k}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.7% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \frac{2 \cdot \ell}{k} \cdot \frac{\frac{\frac{\ell}{t}}{k \cdot k}}{k} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ (* 2.0 l) k) (/ (/ (/ l t) (* k k)) k)))

C

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

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((2.0d0 * l) / k) * (((l / t) / (k * k)) / k)
end function

Java

public static double code(double t, double l, double k) {
	return ((2.0 * l) / k) * (((l / t) / (k * k)) / k);
}

Python

def code(t, l, k):
	return ((2.0 * l) / k) * (((l / t) / (k * k)) / k)

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(N[(2.0 * l), $MachinePrecision] / k), $MachinePrecision] * N[(N[(N[(l / t), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.0%

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

3. Taylor expanded in 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. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. /-lowering-/.f64 N/A

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

5. Simplified 69.6%

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

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(2 \cdot \ell\right) \cdot \ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{t}\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(\frac{\ell}{t}\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   5. /-lowering-/.f64 78.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(k\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(k\right), 2\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

7. Applied egg-rr 78.4%

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

   1. associate-*l* N/A

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

   2. times-frac N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \ell}{k}\right), \color{blue}{\left(\frac{\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}}{k}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \ell\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}}}{k}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \left(\frac{\color{blue}{\frac{\ell}{t}} \cdot \frac{\cos k}{{\sin k}^{2}}}{k}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right), \color{blue}{k}\right)\right) \]

   7. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t} \cdot \frac{1}{\frac{{\sin k}^{2}}{\cos k}}\right), k\right)\right) \]

   8. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\frac{{\sin k}^{2}}{\cos k}}\right), k\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\frac{\sin k \cdot \sin k}{\cos k}}\right), k\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\sin k \cdot \frac{\sin k}{\cos k}}\right), k\right)\right) \]

   11. tan-quot N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{t}}{\sin k \cdot \tan k}\right), k\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(\sin k \cdot \tan k\right)\right), k\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\sin k \cdot \tan k\right)\right), k\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\sin k, \tan k\right)\right), k\right)\right) \]

   15. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \tan k\right)\right), k\right)\right) \]

   16. tan-lowering-tan.f64 91.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(k\right), \mathsf{tan.f64}\left(k\right)\right)\right), k\right)\right) \]

9. Applied egg-rr 91.9%

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

10. Taylor expanded in k around 0

    \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \color{blue}{\left({k}^{2}\right)}\right), k\right)\right) \]
11. Step-by-step derivation

    1. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(k \cdot k\right)\right), k\right)\right) \]

    2. *-lowering-*.f64 69.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), k\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{*.f64}\left(k, k\right)\right), k\right)\right) \]

12. Simplified 69.0%

    \[\leadsto \frac{2 \cdot \ell}{k} \cdot \frac{\frac{\frac{\ell}{t}}{\color{blue}{k \cdot k}}}{k} \]
13. Add Preprocessing

Alternative 11: 56.5% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}}{k \cdot k} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (/ (* (* l l) 0.3333333333333333) t) (* k k)))

C

double code(double t, double l, double k) {
	return (((l * l) * 0.3333333333333333) / t) / (k * k);
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (((l * l) * 0.3333333333333333d0) / t) / (k * k)
end function

Java

public static double code(double t, double l, double k) {
	return (((l * l) * 0.3333333333333333) / t) / (k * k);
}

Python

def code(t, l, k):
	return (((l * l) * 0.3333333333333333) / t) / (k * k)

Julia

function code(t, l, k)
	return Float64(Float64(Float64(Float64(l * l) * 0.3333333333333333) / t) / Float64(k * k))
end

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / t), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.0%

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

   1. associate-/r* N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right) \]

3. Simplified 41.3%

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

5. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\frac{1}{3} \cdot \frac{{k}^{2} \cdot \ell}{{t}^{3}} + 2 \cdot \frac{\ell}{{t}^{3}}}{k}\right)}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\frac{{k}^{2} \cdot \ell}{{t}^{3}} \cdot \frac{1}{3} + 2 \cdot \frac{\ell}{{t}^{3}}}{k}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\left({k}^{2} \cdot \frac{\ell}{{t}^{3}}\right) \cdot \frac{1}{3} + 2 \cdot \frac{\ell}{{t}^{3}}}{k}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{{k}^{2} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \frac{1}{3}\right) + 2 \cdot \frac{\ell}{{t}^{3}}}{k}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{{k}^{2} \cdot \left(\frac{1}{3} \cdot \frac{\ell}{{t}^{3}}\right) + 2 \cdot \frac{\ell}{{t}^{3}}}{k}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left({k}^{2} \cdot \left(\frac{1}{3} \cdot \frac{\ell}{{t}^{3}}\right) + 2 \cdot \frac{\ell}{{t}^{3}}\right), k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

7. Simplified 17.2%

   \[\leadsto \frac{\frac{\ell \cdot \color{blue}{\frac{\frac{2 \cdot \ell}{t \cdot \left(t \cdot t\right)} + \frac{\left(0.3333333333333333 \cdot \left(k \cdot k\right)\right) \cdot \ell}{t \cdot \left(t \cdot t\right)}}{k}}}{\tan k}}{\frac{k \cdot k}{t \cdot t}} \]

8. Taylor expanded in k around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{3} \cdot \frac{k \cdot \ell}{{t}^{3}}\right)}\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{k \cdot \ell}{{t}^{3}} \cdot \frac{1}{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(k \cdot \frac{\ell}{{t}^{3}}\right) \cdot \frac{1}{3}\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(k \cdot \left(\frac{\ell}{{t}^{3}} \cdot \frac{1}{3}\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(k \cdot \left(\frac{1}{3} \cdot \frac{\ell}{{t}^{3}}\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(\frac{1}{3} \cdot \frac{\ell}{{t}^{3}}\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\ell}{{t}^{3}}\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\ell, \left({t}^{3}\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   8. cube-mult N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\ell, \left(t \cdot \left(t \cdot t\right)\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\ell, \left(t \cdot {t}^{2}\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left({t}^{2}\right)\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

   12. *-lowering-*.f64 20.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

10. Simplified 20.6%

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

11. Taylor expanded in k around 0

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

    1. associate-*r/ N/A

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

    2. *-commutative N/A

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

    3. associate-/r* N/A

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

    4. associate-*r/ N/A

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

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

    6. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{3} \cdot {\ell}^{2}}{t}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{3} \cdot {\ell}^{2}\right), t\right), \left({\color{blue}{k}}^{2}\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2} \cdot \frac{1}{3}\right), t\right), \left({k}^{2}\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{1}{3}\right), t\right), \left({k}^{2}\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{3}\right), t\right), \left({k}^{2}\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{3}\right), t\right), \left({k}^{2}\right)\right) \]

    12. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{3}\right), t\right), \left(k \cdot \color{blue}{k}\right)\right) \]

    13. *-lowering-*.f64 55.8%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{3}\right), t\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

13. Simplified 55.8%

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

Alternative 12: 30.4% accurate, 38.3× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* l (* (/ l t) (/ -0.3333333333333333 (* k k)))))

C

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

Fortran

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

Java

public static double code(double t, double l, double k) {
	return l * ((l / t) * (-0.3333333333333333 / (k * k)));
}

Python

def code(t, l, k):
	return l * ((l / t) * (-0.3333333333333333 / (k * k)))

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(l * N[(N[(l / t), $MachinePrecision] * N[(-0.3333333333333333 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.0%

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

   1. associate-/r* N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}\right), \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\right) \]

3. Simplified 41.3%

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

5. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{{t}^{3}} + 2 \cdot \frac{{\ell}^{2}}{{t}^{3}}}{{k}^{2}}\right)}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(t, t\right)\right)\right) \]

7. Simplified 19.6%

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

8. Taylor expanded in k around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. associate-*r/ N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \frac{-1}{3}\right), \left({\color{blue}{k}}^{2}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left({k}^{2}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \left(k \cdot \color{blue}{k}\right)\right) \]

   12. *-lowering-*.f64 29.4%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

10. Simplified 29.4%

    \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.3333333333333333}{k \cdot k}} \]
11. Step-by-step derivation

    1. associate-/l* N/A

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

    2. associate-/l* N/A

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

    3. associate-*l* N/A

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

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\frac{-1}{3}}{k \cdot k}\right)}\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \color{blue}{\left(\frac{\frac{-1}{3}}{k \cdot k}\right)}\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{\color{blue}{\frac{-1}{3}}}{k \cdot k}\right)\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\frac{-1}{3}, \color{blue}{\left(k \cdot k\right)}\right)\right)\right) \]

    8. *-lowering-*.f64 30.7%

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right)\right) \]

12. Applied egg-rr 30.7%

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

Alternative 13: 21.1% accurate, 60.1× speedup

Math

\[\begin{array}{l} \\ \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (* l l) (/ -0.11666666666666667 t)))

C

double code(double t, double l, double k) {
	return (l * l) * (-0.11666666666666667 / t);
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l * l) * ((-0.11666666666666667d0) / t)
end function

Java

public static double code(double t, double l, double k) {
	return (l * l) * (-0.11666666666666667 / t);
}

Python

def code(t, l, k):
	return (l * l) * (-0.11666666666666667 / t)

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(-0.11666666666666667 / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * l) * (-0.11666666666666667 / t);
end

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(-0.11666666666666667 / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.0%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right), \color{blue}{\left({k}^{4}\right)}\right) \]

5. Simplified 26.1%

   \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t} + \left(k \cdot k\right) \cdot \left(-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t} + k \cdot \left(k \cdot \left(\frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667\right)\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

6. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-7}{60} \cdot \frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2}}{t} \cdot \color{blue}{\frac{-7}{60}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \color{blue}{\frac{-7}{60}}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-7}{60}\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-7}{60}\right) \]

   5. *-lowering-*.f64 19.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-7}{60}\right) \]

8. Simplified 19.9%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667} \]
9. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{t}\right) \cdot \frac{-7}{60} \]

   2. associate-*l* N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{1}{t} \cdot \frac{-7}{60}\right)} \]

   3. *-commutative N/A

      \[\leadsto \left(\frac{1}{t} \cdot \frac{-7}{60}\right) \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{t} \cdot \frac{-7}{60}\right), \color{blue}{\left(\ell \cdot \ell\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{-7}{60} \cdot \frac{1}{t}\right), \left(\color{blue}{\ell} \cdot \ell\right)\right) \]

   6. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-7}{60}}{t}\right), \left(\color{blue}{\ell} \cdot \ell\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-7}{60}, t\right), \left(\color{blue}{\ell} \cdot \ell\right)\right) \]

   8. *-lowering-*.f64 19.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-7}{60}, t\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right) \]

10. Applied egg-rr 19.9%

    \[\leadsto \color{blue}{\frac{-0.11666666666666667}{t} \cdot \left(\ell \cdot \ell\right)} \]

11. Final simplification 19.9%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t} \]
12. Add Preprocessing

Alternative 14: 18.7% accurate, 60.1× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \left(\frac{\ell}{t} \cdot -0.11666666666666667\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* l (* (/ l t) -0.11666666666666667)))

C

double code(double t, double l, double k) {
	return l * ((l / t) * -0.11666666666666667);
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = l * ((l / t) * (-0.11666666666666667d0))
end function

Java

public static double code(double t, double l, double k) {
	return l * ((l / t) * -0.11666666666666667);
}

Python

def code(t, l, k):
	return l * ((l / t) * -0.11666666666666667)

Julia

function code(t, l, k)
	return Float64(l * Float64(Float64(l / t) * -0.11666666666666667))
end

MATLAB

function tmp = code(t, l, k)
	tmp = l * ((l / t) * -0.11666666666666667);
end

Wolfram

code[t_, l_, k_] := N[(l * N[(N[(l / t), $MachinePrecision] * -0.11666666666666667), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.0%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right), \color{blue}{\left({k}^{4}\right)}\right) \]

5. Simplified 26.1%

   \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t} + \left(k \cdot k\right) \cdot \left(-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t} + k \cdot \left(k \cdot \left(\frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667\right)\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

6. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-7}{60} \cdot \frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{{\ell}^{2}}{t} \cdot \color{blue}{\frac{-7}{60}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \color{blue}{\frac{-7}{60}}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-7}{60}\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-7}{60}\right) \]

   5. *-lowering-*.f64 19.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-7}{60}\right) \]

8. Simplified 19.9%

   \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{t} \cdot -0.11666666666666667} \]
9. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \left(\ell \cdot \frac{\ell}{t}\right) \cdot \frac{-7}{60} \]

   2. associate-*l* N/A

      \[\leadsto \ell \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{-7}{60}\right)} \]

   3. *-commutative N/A

      \[\leadsto \left(\frac{\ell}{t} \cdot \frac{-7}{60}\right) \cdot \color{blue}{\ell} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{t} \cdot \frac{-7}{60}\right), \color{blue}{\ell}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{t}\right), \frac{-7}{60}\right), \ell\right) \]

   6. /-lowering-/.f64 17.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \frac{-7}{60}\right), \ell\right) \]

10. Applied egg-rr 17.4%

    \[\leadsto \color{blue}{\left(\frac{\ell}{t} \cdot -0.11666666666666667\right) \cdot \ell} \]

11. Final simplification 17.4%

    \[\leadsto \ell \cdot \left(\frac{\ell}{t} \cdot -0.11666666666666667\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024141 
(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))))