Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.2% → 98.0%

Time: 19.4s

Alternatives: 17

Speedup: 28.1×

• 39.5% 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: 36.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 98.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.4%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-/r* N/A

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

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

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

3. Simplified 37.5%

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

   1. associate-/r* N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

   5. div-inv N/A

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

   6. associate-/r/ N/A

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

   7. associate-*l* N/A

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

   8. rgt-mult-inverse N/A

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

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

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

   10. associate-/l/ N/A

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

   11. pow2 N/A

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

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

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

   13. pow2 N/A

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

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

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

   15. /-lowering-/.f64 89.2%

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

6. Applied egg-rr 89.2%

   \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{2}{k \cdot k} \cdot 1}{\frac{t}{\ell}} \cdot \ell}}{\sin k}}{\tan k} \]
7. Step-by-step derivation

   1. *-rgt-identity N/A

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

   2. associate-/r* N/A

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

   3. associate-/l/ N/A

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

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

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

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

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

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

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

   7. /-lowering-/.f64 90.8%

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

8. Applied egg-rr 90.8%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

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

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

   4. /-lowering-/.f64 95.7%

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

10. Applied egg-rr 95.7%

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

    1. associate-*l/ N/A

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

    2. associate-/r* N/A

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

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

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

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

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

    5. associate-*l/ N/A

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

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

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

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

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

    8. /-lowering-/.f64 98.6%

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

12. Applied egg-rr 98.6%

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

Alternative 2: 82.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \frac{\frac{2 \cdot \ell}{k}}{k}}{t}}{k}}{\tan k}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+150}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{k \cdot k}}{\left(\sin k \cdot \tan k\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\ell}}}{\sin k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5.8e-63)
   (/ (/ (/ (* l (/ (/ (* 2.0 l) k) k)) t) k) (tan k))
   (if (<= k 3.8e+150)
     (/ (* l (/ 2.0 (* k k))) (* (* (sin k) (tan k)) (/ t l)))
     (* (/ l (tan k)) (/ (/ 2.0 (/ (* k (* k t)) l)) (sin k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.8e-63) {
		tmp = (((l * (((2.0 * l) / k) / k)) / t) / k) / tan(k);
	} else if (k <= 3.8e+150) {
		tmp = (l * (2.0 / (k * k))) / ((sin(k) * tan(k)) * (t / l));
	} else {
		tmp = (l / tan(k)) * ((2.0 / ((k * (k * t)) / l)) / sin(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 <= 5.8d-63) then
        tmp = (((l * (((2.0d0 * l) / k) / k)) / t) / k) / tan(k)
    else if (k <= 3.8d+150) then
        tmp = (l * (2.0d0 / (k * k))) / ((sin(k) * tan(k)) * (t / l))
    else
        tmp = (l / tan(k)) * ((2.0d0 / ((k * (k * t)) / l)) / sin(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.8e-63) {
		tmp = (((l * (((2.0 * l) / k) / k)) / t) / k) / Math.tan(k);
	} else if (k <= 3.8e+150) {
		tmp = (l * (2.0 / (k * k))) / ((Math.sin(k) * Math.tan(k)) * (t / l));
	} else {
		tmp = (l / Math.tan(k)) * ((2.0 / ((k * (k * t)) / l)) / Math.sin(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 5.8e-63:
		tmp = (((l * (((2.0 * l) / k) / k)) / t) / k) / math.tan(k)
	elif k <= 3.8e+150:
		tmp = (l * (2.0 / (k * k))) / ((math.sin(k) * math.tan(k)) * (t / l))
	else:
		tmp = (l / math.tan(k)) * ((2.0 / ((k * (k * t)) / l)) / math.sin(k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 5.8e-63)
		tmp = Float64(Float64(Float64(Float64(l * Float64(Float64(Float64(2.0 * l) / k) / k)) / t) / k) / tan(k));
	elseif (k <= 3.8e+150)
		tmp = Float64(Float64(l * Float64(2.0 / Float64(k * k))) / Float64(Float64(sin(k) * tan(k)) * Float64(t / l)));
	else
		tmp = Float64(Float64(l / tan(k)) * Float64(Float64(2.0 / Float64(Float64(k * Float64(k * t)) / l)) / sin(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5.8e-63)
		tmp = (((l * (((2.0 * l) / k) / k)) / t) / k) / tan(k);
	elseif (k <= 3.8e+150)
		tmp = (l * (2.0 / (k * k))) / ((sin(k) * tan(k)) * (t / l));
	else
		tmp = (l / tan(k)) * ((2.0 / ((k * (k * t)) / l)) / sin(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 5.8e-63], N[(N[(N[(N[(l * N[(N[(N[(2.0 * l), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / k), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.8e+150], N[(N[(l * N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.7999999999999995e-63

   1. Initial program 38.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

   3. Simplified 36.8%

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

      1. associate-/r* N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. div-inv N/A

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

      6. associate-/r/ N/A

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

      7. associate-*l* N/A

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

      8. rgt-mult-inverse N/A

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

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

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

      10. associate-/l/ N/A

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

      11. pow2 N/A

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

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

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

      13. pow2 N/A

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

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

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

      15. /-lowering-/.f64 89.4%

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

   6. Applied egg-rr 89.4%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{2}{k \cdot k} \cdot 1}{\frac{t}{\ell}} \cdot \ell}}{\sin k}}{\tan k} \]
   7. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. associate-/r* N/A

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

      3. associate-/l/ N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 90.6%

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

   8. Applied egg-rr 90.6%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 95.5%

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

   10. Applied egg-rr 95.5%

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

   11. Taylor expanded in k around 0

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. unpow3 N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-/r* N/A

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

       8. associate-*r/ N/A

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

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

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

   13. Simplified 81.1%

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

   if 5.7999999999999995e-63 < k < 3.79999999999999989e150

   1. Initial program 19.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

   3. Simplified 35.6%

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

      1. associate-/r* N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. div-inv N/A

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

      6. associate-/r/ N/A

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

      7. associate-*l* N/A

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

      8. rgt-mult-inverse N/A

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

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

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

      10. associate-/l/ N/A

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

      11. pow2 N/A

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

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

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

      13. pow2 N/A

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

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

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

      15. /-lowering-/.f64 96.0%

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

   6. Applied egg-rr 96.0%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{2}{k \cdot k} \cdot 1}{\frac{t}{\ell}} \cdot \ell}}{\sin k}}{\tan k} \]
   7. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\frac{\frac{2}{k \cdot k} \cdot 1}{\frac{t}{\ell}} \cdot \ell}{\sin k}}{\tan k} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\frac{\frac{\frac{2}{k \cdot k} \cdot 1}{\frac{\frac{t}{\ell}}{\ell}}}{\sin k}}{\tan k} \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\frac{2}{k \cdot k} \cdot 1}{\frac{t}{\ell \cdot \ell}}}{\sin k}}{\tan k} \]

      5. *-rgt-identity N/A

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

      6. associate-/l/ N/A

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

      7. *-rgt-identity N/A

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

      8. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{2}{k \cdot k} \cdot 1}{\frac{\frac{t}{\ell}}{\ell}}}{\tan k \cdot \sin k} \]

      9. associate-/r/ N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l/ N/A

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

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

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

   8. Applied egg-rr 98.0%

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

   if 3.79999999999999989e150 < k

   1. Initial program 41.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

   3. Simplified 46.3%

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

      1. associate-/r* N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. div-inv N/A

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

      6. associate-/r/ N/A

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

      7. associate-*l* N/A

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

      8. rgt-mult-inverse N/A

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

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

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

      10. associate-/l/ N/A

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

      11. pow2 N/A

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

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

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

      13. pow2 N/A

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

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

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

      15. /-lowering-/.f64 71.6%

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

   6. Applied egg-rr 71.6%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{2}{k \cdot k} \cdot 1}{\frac{t}{\ell}} \cdot \ell}}{\sin k}}{\tan k} \]
   7. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. associate-/r* N/A

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

      3. associate-/l/ N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 80.5%

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

   8. Applied egg-rr 80.5%

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

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

      8. associate-/l/ N/A

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

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

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

      10. associate-*r* N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

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

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. sin-lowering-sin.f64 84.1%

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

   10. Applied egg-rr 84.1%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \frac{\frac{2 \cdot \ell}{k}}{k}}{t}}{k}}{\tan k}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+150}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{k \cdot k}}{\left(\sin k \cdot \tan k\right) \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\ell}}}{\sin k}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 10^{-77}:\\ \;\;\;\;\frac{\frac{\frac{2 \cdot \ell}{k}}{\sin k} \cdot \frac{\ell}{k \cdot t}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{k}}{\sin k \cdot \frac{k}{\frac{\ell}{t}}}}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1e-77)
   (/ (* (/ (/ (* 2.0 l) k) (sin k)) (/ l (* k t))) (tan k))
   (/ (/ (* l (/ 2.0 k)) (* (sin k) (/ k (/ l t)))) (tan k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1e-77) {
		tmp = ((((2.0 * l) / k) / sin(k)) * (l / (k * t))) / tan(k);
	} else {
		tmp = ((l * (2.0 / k)) / (sin(k) * (k / (l / 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 <= 1d-77) then
        tmp = ((((2.0d0 * l) / k) / sin(k)) * (l / (k * t))) / tan(k)
    else
        tmp = ((l * (2.0d0 / k)) / (sin(k) * (k / (l / t)))) / tan(k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1e-77) {
		tmp = ((((2.0 * l) / k) / Math.sin(k)) * (l / (k * t))) / Math.tan(k);
	} else {
		tmp = ((l * (2.0 / k)) / (Math.sin(k) * (k / (l / t)))) / Math.tan(k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1e-77:
		tmp = ((((2.0 * l) / k) / math.sin(k)) * (l / (k * t))) / math.tan(k)
	else:
		tmp = ((l * (2.0 / k)) / (math.sin(k) * (k / (l / t)))) / math.tan(k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1e-77)
		tmp = Float64(Float64(Float64(Float64(Float64(2.0 * l) / k) / sin(k)) * Float64(l / Float64(k * t))) / tan(k));
	else
		tmp = Float64(Float64(Float64(l * Float64(2.0 / k)) / Float64(sin(k) * Float64(k / Float64(l / t)))) / tan(k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1e-77)
		tmp = ((((2.0 * l) / k) / sin(k)) * (l / (k * t))) / tan(k);
	else
		tmp = ((l * (2.0 / k)) / (sin(k) * (k / (l / t)))) / tan(k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1e-77], N[(N[(N[(N[(N[(2.0 * l), $MachinePrecision] / k), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * N[(2.0 / k), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.9999999999999993e-78

   1. Initial program 39.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

   3. Simplified 37.9%

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

      1. associate-/r* N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. div-inv N/A

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

      6. associate-/r/ N/A

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

      7. associate-*l* N/A

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

      8. rgt-mult-inverse N/A

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

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

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

      10. associate-/l/ N/A

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

      11. pow2 N/A

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

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

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

      13. pow2 N/A

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

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

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

      15. /-lowering-/.f64 89.1%

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

   6. Applied egg-rr 89.1%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{2}{k \cdot k} \cdot 1}{\frac{t}{\ell}} \cdot \ell}}{\sin k}}{\tan k} \]
   7. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. associate-/r* N/A

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

      3. associate-/l/ N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 90.3%

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

   8. Applied egg-rr 90.3%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 95.4%

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

   10. Applied egg-rr 95.4%

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

       1. div-inv N/A

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

       2. associate-*l/ N/A

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

       3. associate-*l/ N/A

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

       4. div-inv N/A

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

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

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

       6. un-div-inv N/A

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

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

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

       8. associate-*l/ N/A

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

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

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

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

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

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

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

       12. associate-*r/ N/A

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

       13. clear-num N/A

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

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

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

       15. *-commutative N/A

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

       16. *-lowering-*.f64 95.4%

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

   12. Applied egg-rr 95.4%

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

   if 9.9999999999999993e-78 < k

   1. Initial program 24.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

   3. Simplified 36.6%

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

      1. associate-/r* N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. div-inv N/A

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

      6. associate-/r/ N/A

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

      7. associate-*l* N/A

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

      8. rgt-mult-inverse N/A

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

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

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

      10. associate-/l/ N/A

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

      11. pow2 N/A

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

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

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

      13. pow2 N/A

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

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

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

      15. /-lowering-/.f64 89.4%

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

   6. Applied egg-rr 89.4%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{2}{k \cdot k} \cdot 1}{\frac{t}{\ell}} \cdot \ell}}{\sin k}}{\tan k} \]
   7. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. associate-/r* N/A

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

      3. associate-/l/ N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 91.9%

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

   8. Applied egg-rr 91.9%

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

      1. associate-/l* N/A

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

      2. frac-times N/A

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. sin-lowering-sin.f64 95.3%

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

   10. Applied egg-rr 95.3%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-77}:\\ \;\;\;\;\frac{\frac{\frac{2 \cdot \ell}{k}}{\sin k} \cdot \frac{\ell}{k \cdot t}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{k}}{\sin k \cdot \frac{k}{\frac{\ell}{t}}}}{\tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-49}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \frac{\frac{2 \cdot \ell}{k}}{k}}{t}}{k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot k} \cdot \frac{\frac{\ell \cdot \ell}{t}}{\sin k \cdot \tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.25e-49)
   (/ (/ (/ (* l (/ (/ (* 2.0 l) k) k)) t) k) (tan k))
   (* (/ 2.0 (* k k)) (/ (/ (* l l) t) (* (sin k) (tan k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.25e-49) {
		tmp = (((l * (((2.0 * l) / k) / k)) / t) / k) / tan(k);
	} else {
		tmp = (2.0 / (k * k)) * (((l * l) / t) / (sin(k) * 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 <= 1.25d-49) then
        tmp = (((l * (((2.0d0 * l) / k) / k)) / t) / k) / tan(k)
    else
        tmp = (2.0d0 / (k * k)) * (((l * l) / t) / (sin(k) * tan(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.25e-49) {
		tmp = (((l * (((2.0 * l) / k) / k)) / t) / k) / Math.tan(k);
	} else {
		tmp = (2.0 / (k * k)) * (((l * l) / t) / (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.25e-49:
		tmp = (((l * (((2.0 * l) / k) / k)) / t) / k) / math.tan(k)
	else:
		tmp = (2.0 / (k * k)) * (((l * l) / t) / (math.sin(k) * math.tan(k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.25e-49)
		tmp = Float64(Float64(Float64(Float64(l * Float64(Float64(Float64(2.0 * l) / k) / k)) / t) / k) / tan(k));
	else
		tmp = Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64(Float64(l * l) / t) / Float64(sin(k) * tan(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.25e-49)
		tmp = (((l * (((2.0 * l) / k) / k)) / t) / k) / tan(k);
	else
		tmp = (2.0 / (k * k)) * (((l * l) / t) / (sin(k) * tan(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.25e-49], N[(N[(N[(N[(l * N[(N[(N[(2.0 * l), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / k), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.25e-49

   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. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

   3. Simplified 37.1%

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

      1. associate-/r* N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. div-inv N/A

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

      6. associate-/r/ N/A

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

      7. associate-*l* N/A

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

      8. rgt-mult-inverse N/A

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

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

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

      10. associate-/l/ N/A

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

      11. pow2 N/A

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

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

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

      13. pow2 N/A

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

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

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

      15. /-lowering-/.f64 89.8%

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

   6. Applied egg-rr 89.8%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{2}{k \cdot k} \cdot 1}{\frac{t}{\ell}} \cdot \ell}}{\sin k}}{\tan k} \]
   7. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. associate-/r* N/A

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

      3. associate-/l/ N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 90.9%

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

   8. Applied egg-rr 90.9%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 95.7%

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

   10. Applied egg-rr 95.7%

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

   11. Taylor expanded in k around 0

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. unpow3 N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-/r* N/A

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

       8. associate-*r/ N/A

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

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

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

   13. Simplified 81.3%

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

   if 1.25e-49 < k

   1. Initial program 27.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/r* N/A

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

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

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

   3. Simplified 38.4%

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

      1. associate-/r* N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. div-inv N/A

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

      6. associate-/r/ N/A

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

      7. associate-*l* N/A

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

      8. rgt-mult-inverse N/A

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

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

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

      10. associate-/l/ N/A

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

      11. pow2 N/A

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

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

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

      13. pow2 N/A

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

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

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

      15. /-lowering-/.f64 87.7%

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

   6. Applied egg-rr 87.7%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{2}{k \cdot k} \cdot 1}{\frac{t}{\ell}} \cdot \ell}}{\sin k}}{\tan k} \]
   7. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\frac{\frac{2}{k \cdot k} \cdot 1}{\frac{t}{\ell}} \cdot \ell}{\sin k}}{\tan k} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\frac{\frac{\frac{2}{k \cdot k} \cdot 1}{\frac{\frac{t}{\ell}}{\ell}}}{\sin k}}{\tan k} \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{\frac{2}{k \cdot k} \cdot 1}{\frac{t}{\ell \cdot \ell}}}{\sin k}}{\tan k} \]

      5. *-rgt-identity N/A

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

      6. associate-/l/ N/A

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

      7. *-rgt-identity N/A

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

      8. div-inv N/A

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

      9. clear-num N/A

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

      10. associate-/l* N/A

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

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

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

   8. Applied egg-rr 86.3%

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

Alternative 5: 95.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.4%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-/r* N/A

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

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

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

3. Simplified 37.5%

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

   1. associate-/r* N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

   5. div-inv N/A

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

   6. associate-/r/ N/A

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

   7. associate-*l* N/A

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

   8. rgt-mult-inverse N/A

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

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

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

   10. associate-/l/ N/A

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

   11. pow2 N/A

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

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

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

   13. pow2 N/A

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

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

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

   15. /-lowering-/.f64 89.2%

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

6. Applied egg-rr 89.2%

   \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{2}{k \cdot k} \cdot 1}{\frac{t}{\ell}} \cdot \ell}}{\sin k}}{\tan k} \]
7. Step-by-step derivation

   1. *-rgt-identity N/A

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

   2. associate-/r* N/A

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

   3. associate-/l/ N/A

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

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

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

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

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

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

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

   7. /-lowering-/.f64 90.8%

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

8. Applied egg-rr 90.8%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

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

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

   4. /-lowering-/.f64 95.7%

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

10. Applied egg-rr 95.7%

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

    1. div-inv N/A

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

    2. associate-*l/ N/A

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

    3. associate-*l/ N/A

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

    4. div-inv N/A

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

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

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

    6. un-div-inv N/A

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

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

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

    8. associate-*l/ N/A

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

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

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

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

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

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

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

    12. associate-*r/ N/A

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

    13. clear-num N/A

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

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

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

    15. *-commutative N/A

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

    16. *-lowering-*.f64 95.8%

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

12. Applied egg-rr 95.8%

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

Alternative 6: 90.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return (l / tan(k)) * ((2.0 / ((k * (k * t)) / l)) / sin(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 / tan(k)) * ((2.0d0 / ((k * (k * t)) / l)) / sin(k))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.4%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-/r* N/A

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

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

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

3. Simplified 37.5%

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

   1. associate-/r* N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

   5. div-inv N/A

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

   6. associate-/r/ N/A

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

   7. associate-*l* N/A

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

   8. rgt-mult-inverse N/A

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

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

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

   10. associate-/l/ N/A

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

   11. pow2 N/A

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

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

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

   13. pow2 N/A

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

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

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

   15. /-lowering-/.f64 89.2%

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

6. Applied egg-rr 89.2%

   \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{2}{k \cdot k} \cdot 1}{\frac{t}{\ell}} \cdot \ell}}{\sin k}}{\tan k} \]
7. Step-by-step derivation

   1. *-rgt-identity N/A

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

   2. associate-/r* N/A

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

   3. associate-/l/ N/A

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

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

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

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

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

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

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

   7. /-lowering-/.f64 90.8%

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

8. Applied egg-rr 90.8%

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

   1. associate-/l/ N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

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

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

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

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

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

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

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

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

   8. associate-/l/ N/A

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

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

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

   10. associate-*r* N/A

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

   11. associate-*l/ N/A

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

   12. *-commutative N/A

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

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

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

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

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

   17. *-commutative N/A

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

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

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

   19. sin-lowering-sin.f64 90.1%

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

10. Applied egg-rr 90.1%

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

Alternative 7: 74.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\ell \cdot \frac{\frac{2 \cdot \ell}{k}}{k}}{t}}{k}}{\tan k} \end{array} \]

FPCore

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

C

double code(double t, double l, double k) {
	return (((l * (((2.0 * l) / k) / k)) / t) / k) / 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 = (((l * (((2.0d0 * l) / k) / k)) / t) / k) / tan(k)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.4%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-/r* N/A

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

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

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

3. Simplified 37.5%

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

   1. associate-/r* N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

   5. div-inv N/A

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

   6. associate-/r/ N/A

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

   7. associate-*l* N/A

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

   8. rgt-mult-inverse N/A

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

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

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

   10. associate-/l/ N/A

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

   11. pow2 N/A

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

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

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

   13. pow2 N/A

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

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

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

   15. /-lowering-/.f64 89.2%

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

6. Applied egg-rr 89.2%

   \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{2}{k \cdot k} \cdot 1}{\frac{t}{\ell}} \cdot \ell}}{\sin k}}{\tan k} \]
7. Step-by-step derivation

   1. *-rgt-identity N/A

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

   2. associate-/r* N/A

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

   3. associate-/l/ N/A

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

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

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

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

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

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

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

   7. /-lowering-/.f64 90.8%

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

8. Applied egg-rr 90.8%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

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

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

   4. /-lowering-/.f64 95.7%

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

10. Applied egg-rr 95.7%

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

11. Taylor expanded in k around 0

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

    1. associate-*r/ N/A

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

    2. *-commutative N/A

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

    3. unpow3 N/A

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

    4. unpow2 N/A

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

    5. associate-*r* N/A

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

    6. *-commutative N/A

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

    7. associate-/r* N/A

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

    8. associate-*r/ N/A

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

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

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

13. Simplified 79.6%

    \[\leadsto \frac{\color{blue}{\frac{\frac{\ell \cdot \frac{\frac{2 \cdot \ell}{k}}{k}}{t}}{k}}}{\tan k} \]
14. Add Preprocessing

Alternative 8: 72.5% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(k \cdot \frac{k \cdot k}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 2e-101)
   (/ 2.0 (* (/ t l) (* k (/ (* k k) (/ l k)))))
   (* (/ (* 2.0 l) (* k k)) (/ l (* k (* k t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 2e-101) {
		tmp = 2.0 / ((t / l) * (k * ((k * k) / (l / k))));
	} else {
		tmp = ((2.0 * l) / (k * k)) * (l / (k * (k * t)));
	}
	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 (t <= 2d-101) then
        tmp = 2.0d0 / ((t / l) * (k * ((k * k) / (l / k))))
    else
        tmp = ((2.0d0 * l) / (k * k)) * (l / (k * (k * t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 2e-101) {
		tmp = 2.0 / ((t / l) * (k * ((k * k) / (l / k))));
	} else {
		tmp = ((2.0 * l) / (k * k)) * (l / (k * (k * t)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 2e-101:
		tmp = 2.0 / ((t / l) * (k * ((k * k) / (l / k))))
	else:
		tmp = ((2.0 * l) / (k * k)) * (l / (k * (k * t)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 2e-101)
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64(k * Float64(Float64(k * k) / Float64(l / k)))));
	else
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(k * k)) * Float64(l / Float64(k * Float64(k * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 2e-101)
		tmp = 2.0 / ((t / l) * (k * ((k * k) / (l / k))));
	else
		tmp = ((2.0 * l) / (k * k)) * (l / (k * (k * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 2e-101], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(k * N[(N[(k * k), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.0000000000000001e-101

   1. Initial program 31.0%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 62.5%

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

   5. Simplified 62.5%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. associate-*r* N/A

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

      6. cube-unmult N/A

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

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

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

      8. cube-unmult N/A

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

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

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

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

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

      11. /-lowering-/.f64 71.1%

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

   7. Applied egg-rr 71.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

      10. /-lowering-/.f64 74.4%

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

   9. Applied egg-rr 74.4%

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

   if 2.0000000000000001e-101 < t

   1. Initial program 40.6%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 77.8%

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

   5. Simplified 77.8%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

      6. associate-/r* N/A

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

      7. *-rgt-identity N/A

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

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

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

      9. *-rgt-identity N/A

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

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

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

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

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

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

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

      13. /-lowering-/.f64 79.2%

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

   7. Applied egg-rr 79.2%

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

      1. associate-*l/ N/A

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

      2. div-inv N/A

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

      3. associate-/l/ N/A

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

      4. associate-*l/ N/A

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

      5. clear-num N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 81.5%

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

   9. Applied egg-rr 81.5%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \left(k \cdot \frac{k \cdot k}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.8% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-142}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \frac{k \cdot k}{\frac{\ell}{\frac{t}{\ell}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 5e-142)
   (/ 2.0 (* k (* k (/ (* k k) (/ l (/ t l))))))
   (* (/ (* 2.0 l) (* k k)) (/ l (* k (* k t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 5e-142) {
		tmp = 2.0 / (k * (k * ((k * k) / (l / (t / l)))));
	} else {
		tmp = ((2.0 * l) / (k * k)) * (l / (k * (k * t)));
	}
	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 (t <= 5d-142) then
        tmp = 2.0d0 / (k * (k * ((k * k) / (l / (t / l)))))
    else
        tmp = ((2.0d0 * l) / (k * k)) * (l / (k * (k * t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 5e-142) {
		tmp = 2.0 / (k * (k * ((k * k) / (l / (t / l)))));
	} else {
		tmp = ((2.0 * l) / (k * k)) * (l / (k * (k * t)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 5e-142:
		tmp = 2.0 / (k * (k * ((k * k) / (l / (t / l)))))
	else:
		tmp = ((2.0 * l) / (k * k)) * (l / (k * (k * t)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 5e-142)
		tmp = Float64(2.0 / Float64(k * Float64(k * Float64(Float64(k * k) / Float64(l / Float64(t / l))))));
	else
		tmp = Float64(Float64(Float64(2.0 * l) / Float64(k * k)) * Float64(l / Float64(k * Float64(k * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 5e-142)
		tmp = 2.0 / (k * (k * ((k * k) / (l / (t / l)))));
	else
		tmp = ((2.0 * l) / (k * k)) * (l / (k * (k * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 5e-142], N[(2.0 / N[(k * N[(k * N[(N[(k * k), $MachinePrecision] / N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.0000000000000002e-142

   1. Initial program 29.7%

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

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 62.6%

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

   5. Simplified 62.6%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. associate-*r* N/A

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

      6. cube-unmult N/A

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

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

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

      8. cube-unmult N/A

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

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

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

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

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

      11. /-lowering-/.f64 71.0%

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

   7. Applied egg-rr 71.0%

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

      1. frac-times N/A

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

      2. remove-double-div N/A

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

      3. unpow-1 N/A

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

      4. un-div-inv N/A

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

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{1}{t}\right)}^{-1} \cdot \frac{1}{\ell \cdot \ell}\right)}\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \left(\color{blue}{{\left(\frac{1}{t}\right)}^{-1}} \cdot \frac{1}{\ell \cdot \ell}\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot \left({\left(\frac{1}{t}\right)}^{-1} \cdot \frac{1}{\ell \cdot \ell}\right)\right)}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(k \cdot \color{blue}{\left(k \cdot \left(\left(k \cdot k\right) \cdot \left({\left(\frac{1}{t}\right)}^{-1} \cdot \frac{1}{\ell \cdot \ell}\right)\right)\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \left(\left(k \cdot k\right) \cdot \left({\left(\frac{1}{t}\right)}^{-1} \cdot \frac{1}{\ell \cdot \ell}\right)\right)\right)}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{\left(\left(k \cdot k\right) \cdot \left({\left(\frac{1}{t}\right)}^{-1} \cdot \frac{1}{\ell \cdot \ell}\right)\right)}\right)\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\left(k \cdot k\right) \cdot \frac{{\left(\frac{1}{t}\right)}^{-1}}{\color{blue}{\ell \cdot \ell}}\right)\right)\right)\right) \]

      12. unpow-1 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\left(k \cdot k\right) \cdot \frac{\frac{1}{\frac{1}{t}}}{\color{blue}{\ell} \cdot \ell}\right)\right)\right)\right) \]

      13. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\left(k \cdot k\right) \cdot \frac{1}{\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{t}}}\right)\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\left(k \cdot k\right) \cdot \frac{1}{\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{t}\right)}}\right)\right)\right)\right) \]

      15. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\left(k \cdot k\right) \cdot \frac{1}{\ell \cdot \frac{\ell}{\color{blue}{t}}}\right)\right)\right)\right) \]

      16. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(\frac{k \cdot k}{\color{blue}{\ell \cdot \frac{\ell}{t}}}\right)\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(k \cdot k\right), \color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)}\right)\right)\right)\right) \]

   9. Applied egg-rr 72.3%

      \[\leadsto \frac{2}{\color{blue}{k \cdot \left(k \cdot \frac{k \cdot k}{\frac{\ell}{\frac{t}{\ell}}}\right)}} \]

   if 5.0000000000000002e-142 < t

   1. Initial program 41.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 76.2%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   5. Simplified 76.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\color{blue}{\frac{t}{\ell \cdot \ell}}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{\frac{t}{\ell}}{\color{blue}{\ell}}} \]

      3. associate-/r/ N/A

         \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}} \cdot \color{blue}{\ell} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}}\right), \color{blue}{\ell}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot 1}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k} \cdot 1\right), \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

      13. /-lowering-/.f64 78.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \ell\right) \]

   7. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{k \cdot k}}{k \cdot k}}{\frac{t}{\ell}} \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\frac{\frac{2}{k \cdot k}}{k \cdot k} \cdot \ell}{\color{blue}{\frac{t}{\ell}}} \]

      2. div-inv N/A

         \[\leadsto \left(\frac{\frac{2}{k \cdot k}}{k \cdot k} \cdot \ell\right) \cdot \color{blue}{\frac{1}{\frac{t}{\ell}}} \]

      3. associate-/l/ N/A

         \[\leadsto \left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \ell\right) \cdot \frac{1}{\frac{t}{\ell}} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{2 \cdot \ell}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\color{blue}{1}}{\frac{t}{\ell}} \]

      5. clear-num N/A

         \[\leadsto \frac{2 \cdot \ell}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{\color{blue}{t}} \]

      6. times-frac N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]

      8. times-frac N/A

         \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot t}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right)}\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \ell\right), \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{\left(k \cdot k\right) \cdot t}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(k \cdot k\right)\right), \left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \left(k \cdot \left(t \cdot \color{blue}{k}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot k\right)}\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{t}\right)\right)\right)\right) \]

      18. *-lowering-*.f64 80.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{t}\right)\right)\right)\right) \]

   9. Applied egg-rr 80.6%

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 73.8% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ (* 2.0 l) (* k k)) (/ l (* k (* k t)))))

C

double code(double t, double l, double k) {
	return ((2.0 * l) / (k * k)) * (l / (k * (k * 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 = ((2.0d0 * l) / (k * k)) * (l / (k * (k * t)))
end function

Java

public static double code(double t, double l, double k) {
	return ((2.0 * l) / (k * k)) * (l / (k * (k * t)));
}

Python

def code(t, l, k):
	return ((2.0 * l) / (k * k)) * (l / (k * (k * t)))

Julia

function code(t, l, k)
	return Float64(Float64(Float64(2.0 * l) / Float64(k * k)) * Float64(l / Float64(k * Float64(k * t))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((2.0 * l) / (k * k)) * (l / (k * (k * t)));
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(2.0 * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   4. pow-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

   12. *-lowering-*.f64 67.9%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

5. Simplified 67.9%

   \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
6. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\color{blue}{\frac{t}{\ell \cdot \ell}}} \]

   2. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{\frac{t}{\ell}}{\color{blue}{\ell}}} \]

   3. associate-/r/ N/A

      \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}} \cdot \color{blue}{\ell} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}}\right), \color{blue}{\ell}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

   7. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot 1}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k} \cdot 1\right), \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

   9. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

   13. /-lowering-/.f64 72.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \ell\right) \]

7. Applied egg-rr 72.2%

   \[\leadsto \color{blue}{\frac{\frac{\frac{2}{k \cdot k}}{k \cdot k}}{\frac{t}{\ell}} \cdot \ell} \]
8. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \frac{\frac{\frac{2}{k \cdot k}}{k \cdot k} \cdot \ell}{\color{blue}{\frac{t}{\ell}}} \]

   2. div-inv N/A

      \[\leadsto \left(\frac{\frac{2}{k \cdot k}}{k \cdot k} \cdot \ell\right) \cdot \color{blue}{\frac{1}{\frac{t}{\ell}}} \]

   3. associate-/l/ N/A

      \[\leadsto \left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \ell\right) \cdot \frac{1}{\frac{t}{\ell}} \]

   4. associate-*l/ N/A

      \[\leadsto \frac{2 \cdot \ell}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\color{blue}{1}}{\frac{t}{\ell}} \]

   5. clear-num N/A

      \[\leadsto \frac{2 \cdot \ell}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{\color{blue}{t}} \]

   6. times-frac N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   7. associate-*l* N/A

      \[\leadsto \frac{\left(2 \cdot \ell\right) \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]

   8. times-frac N/A

      \[\leadsto \frac{2 \cdot \ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\left(k \cdot k\right) \cdot t}} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \ell}{k \cdot k}\right), \color{blue}{\left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right)}\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \ell\right), \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell}}{\left(k \cdot k\right) \cdot t}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(k \cdot k\right)\right), \left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell}{\left(k \cdot k\right) \cdot t}\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \left(k \cdot \left(t \cdot \color{blue}{k}\right)\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot k\right)}\right)\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{t}\right)\right)\right)\right) \]

   18. *-lowering-*.f64 76.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{t}\right)\right)\right)\right) \]

9. Applied egg-rr 76.0%

   \[\leadsto \color{blue}{\frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}} \]
10. Add Preprocessing

Alternative 11: 65.5% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ 2.0 (* k k)) (/ (* l l) (* k (* k t)))))

C

double code(double t, double l, double k) {
	return (2.0 / (k * k)) * ((l * l) / (k * (k * 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 = (2.0d0 / (k * k)) * ((l * l) / (k * (k * t)))
end function

Java

public static double code(double t, double l, double k) {
	return (2.0 / (k * k)) * ((l * l) / (k * (k * t)));
}

Python

def code(t, l, k):
	return (2.0 / (k * k)) * ((l * l) / (k * (k * t)))

Julia

function code(t, l, k)
	return Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64(l * l) / Float64(k * Float64(k * t))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (2.0 / (k * k)) * ((l * l) / (k * (k * t)));
end

Wolfram

code[t_, l_, k_] := N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(2, \left({k}^{4} \cdot \color{blue}{\frac{t}{{\ell}^{2}}}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{4}\right), \color{blue}{\left(\frac{t}{{\ell}^{2}}\right)}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{\left(2 \cdot 2\right)}\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   4. pow-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left({k}^{2} \cdot {k}^{2}\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({k}^{2}\right), \left({k}^{2}\right)\right), \left(\frac{\color{blue}{t}}{{\ell}^{2}}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(k \cdot k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left({k}^{2}\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{{\ell}^{2}}\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

   12. *-lowering-*.f64 67.9%

       \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

5. Simplified 67.9%

   \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
6. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\color{blue}{\frac{t}{\ell \cdot \ell}}} \]

   2. associate-/r* N/A

      \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{\frac{t}{\ell}}{\color{blue}{\ell}}} \]

   3. associate-/r/ N/A

      \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}} \cdot \color{blue}{\ell} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{\frac{t}{\ell}}\right), \color{blue}{\ell}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{k \cdot k}}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

   7. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2}{k \cdot k} \cdot 1}{k \cdot k}\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k} \cdot 1\right), \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

   9. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{k \cdot k}\right), \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(k \cdot k\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{t}{\ell}\right)\right), \ell\right) \]

   13. /-lowering-/.f64 72.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(t, \ell\right)\right), \ell\right) \]

7. Applied egg-rr 72.2%

   \[\leadsto \color{blue}{\frac{\frac{\frac{2}{k \cdot k}}{k \cdot k}}{\frac{t}{\ell}} \cdot \ell} \]
8. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \frac{\frac{\frac{2}{k \cdot k}}{k \cdot k} \cdot \ell}{\color{blue}{\frac{t}{\ell}}} \]

   2. div-inv N/A

      \[\leadsto \left(\frac{\frac{2}{k \cdot k}}{k \cdot k} \cdot \ell\right) \cdot \color{blue}{\frac{1}{\frac{t}{\ell}}} \]

   3. clear-num N/A

      \[\leadsto \left(\frac{\frac{2}{k \cdot k}}{k \cdot k} \cdot \ell\right) \cdot \frac{\ell}{\color{blue}{t}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\frac{2}{k \cdot k}}{k \cdot k} \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)} \]

   5. associate-/l/ N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \left(\color{blue}{\ell} \cdot \frac{\ell}{t}\right) \]

   6. associate-/l* N/A

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{\color{blue}{t}} \]

   7. times-frac N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}} \]

   8. associate-*l* N/A

      \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}} \]

   9. times-frac N/A

      \[\leadsto \frac{2}{k \cdot k} \cdot \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k \cdot k}\right), \color{blue}{\left(\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t}\right)}\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(k \cdot k\right)\right), \left(\frac{\color{blue}{\ell \cdot \ell}}{\left(k \cdot k\right) \cdot t}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{\ell \cdot \color{blue}{\ell}}{\left(k \cdot k\right) \cdot t}\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)\right) \]

   15. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \color{blue}{\left(k \cdot t\right)}\right)\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(k \cdot \left(t \cdot \color{blue}{k}\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \color{blue}{\left(t \cdot k\right)}\right)\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \left(k \cdot \color{blue}{t}\right)\right)\right)\right) \]

   19. *-lowering-*.f64 70.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \color{blue}{t}\right)\right)\right)\right) \]

9. Applied egg-rr 70.3%

   \[\leadsto \color{blue}{\frac{2}{k \cdot k} \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}} \]
10. Add Preprocessing

Alternative 12: 20.6% accurate, 46.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{t}{\ell \cdot \left(\ell \cdot -0.11666666666666667\right)}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 1.0 (/ t (* l (* l -0.11666666666666667)))))

C

double code(double t, double l, double k) {
	return 1.0 / (t / (l * (l * -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 = 1.0d0 / (t / (l * (l * (-0.11666666666666667d0))))
end function

Java

public static double code(double t, double l, double k) {
	return 1.0 / (t / (l * (l * -0.11666666666666667)));
}

Python

def code(t, l, k):
	return 1.0 / (t / (l * (l * -0.11666666666666667)))

Julia

function code(t, l, k)
	return Float64(1.0 / Float64(t / Float64(l * Float64(l * -0.11666666666666667))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 1.0 / (t / (l * (l * -0.11666666666666667)));
end

Wolfram

code[t_, l_, k_] := N[(1.0 / N[(t / N[(l * N[(l * -0.11666666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]

5. Simplified 37.0%

   \[\leadsto \color{blue}{\frac{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}\right)\right) + \frac{\ell \cdot \ell}{t} \cdot \left(2 + -0.3333333333333333 \cdot \left(k \cdot k\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

6. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-7}{60} \cdot \frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot {\ell}^{2}}{\color{blue}{t}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-7}{60} \cdot {\ell}^{2}\right), \color{blue}{t}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2} \cdot \frac{-7}{60}\right), t\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{-7}{60}\right), t\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{-7}{60}\right), t\right) \]

   6. *-lowering-*.f64 24.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{-7}{60}\right), t\right) \]

8. Simplified 24.9%

   \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}} \]
9. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \left(\left(\ell \cdot \ell\right) \cdot \frac{-7}{60}\right) \cdot \color{blue}{\frac{1}{t}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{\left(\left(\ell \cdot \ell\right) \cdot \frac{-7}{60}\right) \cdot 1}{\color{blue}{t}} \]

   3. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{t}{\left(\left(\ell \cdot \ell\right) \cdot \frac{-7}{60}\right) \cdot 1}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{t}{\left(\left(\ell \cdot \ell\right) \cdot \frac{-7}{60}\right) \cdot 1}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \color{blue}{\left(\left(\left(\ell \cdot \ell\right) \cdot \frac{-7}{60}\right) \cdot 1\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\left(\left(\ell \cdot \ell\right) \cdot \frac{-7}{60}\right), \color{blue}{1}\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\left(\ell \cdot \left(\ell \cdot \frac{-7}{60}\right)\right), 1\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{-7}{60}\right)\right), 1\right)\right)\right) \]

   9. *-lowering-*.f64 25.1%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{-7}{60}\right)\right), 1\right)\right)\right) \]

10. Applied egg-rr 25.1%

    \[\leadsto \color{blue}{\frac{1}{\frac{t}{\left(\ell \cdot \left(\ell \cdot -0.11666666666666667\right)\right) \cdot 1}}} \]

11. Final simplification 25.1%

    \[\leadsto \frac{1}{\frac{t}{\ell \cdot \left(\ell \cdot -0.11666666666666667\right)}} \]
12. Add Preprocessing

Alternative 13: 20.8% accurate, 46.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-0.11666666666666667}{t}}{\frac{1}{\ell \cdot \ell}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (/ -0.11666666666666667 t) (/ 1.0 (* l l))))

C

double code(double t, double l, double k) {
	return (-0.11666666666666667 / t) / (1.0 / (l * l));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((-0.11666666666666667d0) / t) / (1.0d0 / (l * l))
end function

Java

public static double code(double t, double l, double k) {
	return (-0.11666666666666667 / t) / (1.0 / (l * l));
}

Python

def code(t, l, k):
	return (-0.11666666666666667 / t) / (1.0 / (l * l))

Julia

function code(t, l, k)
	return Float64(Float64(-0.11666666666666667 / t) / Float64(1.0 / Float64(l * l)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (-0.11666666666666667 / t) / (1.0 / (l * l));
end

Wolfram

code[t_, l_, k_] := N[(N[(-0.11666666666666667 / t), $MachinePrecision] / N[(1.0 / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]

5. Simplified 37.0%

   \[\leadsto \color{blue}{\frac{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}\right)\right) + \frac{\ell \cdot \ell}{t} \cdot \left(2 + -0.3333333333333333 \cdot \left(k \cdot k\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

6. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-7}{60} \cdot \frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot {\ell}^{2}}{\color{blue}{t}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-7}{60} \cdot {\ell}^{2}\right), \color{blue}{t}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2} \cdot \frac{-7}{60}\right), t\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{-7}{60}\right), t\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{-7}{60}\right), t\right) \]

   6. *-lowering-*.f64 24.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{-7}{60}\right), t\right) \]

8. Simplified 24.9%

   \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}} \]
9. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\frac{-7}{60}}{t}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-7}{60}}{t} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

   3. /-rgt-identity N/A

      \[\leadsto \frac{\frac{-7}{60}}{t} \cdot \frac{\ell \cdot \ell}{\color{blue}{1}} \]

   4. clear-num N/A

      \[\leadsto \frac{\frac{-7}{60}}{t} \cdot \frac{1}{\color{blue}{\frac{1}{\ell \cdot \ell}}} \]

   5. un-div-inv N/A

      \[\leadsto \frac{\frac{\frac{-7}{60}}{t}}{\color{blue}{\frac{1}{\ell \cdot \ell}}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{-7}{60}}{t}\right), \color{blue}{\left(\frac{1}{\ell \cdot \ell}\right)}\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-7}{60}, t\right), \left(\frac{\color{blue}{1}}{\ell \cdot \ell}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-7}{60}, t\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   9. *-lowering-*.f64 24.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-7}{60}, t\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

10. Applied egg-rr 24.9%

    \[\leadsto \color{blue}{\frac{\frac{-0.11666666666666667}{t}}{\frac{1}{\ell \cdot \ell}}} \]
11. Add Preprocessing

Alternative 14: 20.5% accurate, 60.1× speedup

Math

\[\begin{array}{l} \\ \frac{\ell \cdot \left(\ell \cdot -0.11666666666666667\right)}{t} \end{array} \]

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 * Float64(l * -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 * N[(l * -0.11666666666666667), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]

Derivation

1. Initial program 34.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]

5. Simplified 37.0%

   \[\leadsto \color{blue}{\frac{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}\right)\right) + \frac{\ell \cdot \ell}{t} \cdot \left(2 + -0.3333333333333333 \cdot \left(k \cdot k\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

6. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-7}{60} \cdot \frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot {\ell}^{2}}{\color{blue}{t}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-7}{60} \cdot {\ell}^{2}\right), \color{blue}{t}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2} \cdot \frac{-7}{60}\right), t\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{-7}{60}\right), t\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{-7}{60}\right), t\right) \]

   6. *-lowering-*.f64 24.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{-7}{60}\right), t\right) \]

8. Simplified 24.9%

   \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}} \]
9. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\ell \cdot \left(\ell \cdot \frac{-7}{60}\right)\right), t\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\ell \cdot \frac{-7}{60}\right) \cdot \ell\right), t\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \frac{-7}{60}\right), \ell\right), t\right) \]

   4. *-lowering-*.f64 24.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \frac{-7}{60}\right), \ell\right), t\right) \]

10. Applied egg-rr 24.9%

    \[\leadsto \frac{\color{blue}{\left(\ell \cdot -0.11666666666666667\right) \cdot \ell}}{t} \]

11. Final simplification 24.9%

    \[\leadsto \frac{\ell \cdot \left(\ell \cdot -0.11666666666666667\right)}{t} \]
12. Add Preprocessing

Alternative 15: 20.5% 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 34.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]

5. Simplified 37.0%

   \[\leadsto \color{blue}{\frac{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}\right)\right) + \frac{\ell \cdot \ell}{t} \cdot \left(2 + -0.3333333333333333 \cdot \left(k \cdot k\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

6. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-7}{60} \cdot \frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot {\ell}^{2}}{\color{blue}{t}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-7}{60} \cdot {\ell}^{2}\right), \color{blue}{t}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2} \cdot \frac{-7}{60}\right), t\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{-7}{60}\right), t\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{-7}{60}\right), t\right) \]

   6. *-lowering-*.f64 24.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{-7}{60}\right), t\right) \]

8. Simplified 24.9%

   \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}} \]
9. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\frac{-7}{60}}{t}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{-7}{60}}{t} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-7}{60}}{t}\right), \color{blue}{\left(\ell \cdot \ell\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-7}{60}, t\right), \left(\color{blue}{\ell} \cdot \ell\right)\right) \]

   5. *-lowering-*.f64 24.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-7}{60}, t\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right) \]

10. Applied egg-rr 24.5%

    \[\leadsto \color{blue}{\frac{-0.11666666666666667}{t} \cdot \left(\ell \cdot \ell\right)} \]

11. Final simplification 24.5%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t} \]
12. Add Preprocessing

Alternative 16: 18.2% accurate, 60.1× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \frac{\ell}{\frac{t}{-0.11666666666666667}} \end{array} \]

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(l / Float64(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[(l / N[(t / -0.11666666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]

5. Simplified 37.0%

   \[\leadsto \color{blue}{\frac{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}\right)\right) + \frac{\ell \cdot \ell}{t} \cdot \left(2 + -0.3333333333333333 \cdot \left(k \cdot k\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

6. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-7}{60} \cdot \frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot {\ell}^{2}}{\color{blue}{t}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-7}{60} \cdot {\ell}^{2}\right), \color{blue}{t}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2} \cdot \frac{-7}{60}\right), t\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{-7}{60}\right), t\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{-7}{60}\right), t\right) \]

   6. *-lowering-*.f64 24.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{-7}{60}\right), t\right) \]

8. Simplified 24.9%

   \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}} \]
9. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\frac{-7}{60}}{t}} \]

   2. associate-*l* N/A

      \[\leadsto \ell \cdot \color{blue}{\left(\ell \cdot \frac{\frac{-7}{60}}{t}\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\ell \cdot \frac{\frac{-7}{60}}{t}\right)}\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{\color{blue}{\frac{t}{\frac{-7}{60}}}}\right)\right) \]

   5. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \left(\frac{\ell}{\color{blue}{\frac{t}{\frac{-7}{60}}}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \color{blue}{\left(\frac{t}{\frac{-7}{60}}\right)}\right)\right) \]

   7. /-lowering-/.f64 22.6%

      \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \color{blue}{\frac{-7}{60}}\right)\right)\right) \]

10. Applied egg-rr 22.6%

    \[\leadsto \color{blue}{\ell \cdot \frac{\ell}{\frac{t}{-0.11666666666666667}}} \]
11. Add Preprocessing

Alternative 17: 18.2% accurate, 60.1× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{\frac{t}{\ell}} \cdot -0.11666666666666667 \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ l (/ t l)) -0.11666666666666667))

C

double code(double t, double l, double k) {
	return (l / (t / l)) * -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 / (t / l)) * (-0.11666666666666667d0)
end function

Java

public static double code(double t, double l, double k) {
	return (l / (t / l)) * -0.11666666666666667;
}

Python

def code(t, l, k):
	return (l / (t / l)) * -0.11666666666666667

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(t / l)) * -0.11666666666666667)
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / (t / l)) * -0.11666666666666667;
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision] * -0.11666666666666667), $MachinePrecision]

Derivation

1. Initial program 34.4%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(-2 \cdot \left({k}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{{\ell}^{2}}{t} + \frac{31}{360} \cdot \frac{{\ell}^{2}}{t}\right)\right) + \frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}}} \]

5. Simplified 37.0%

   \[\leadsto \color{blue}{\frac{k \cdot \left(k \cdot \left(\left(k \cdot k\right) \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}\right)\right) + \frac{\ell \cdot \ell}{t} \cdot \left(2 + -0.3333333333333333 \cdot \left(k \cdot k\right)\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

6. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\frac{-7}{60} \cdot \frac{{\ell}^{2}}{t}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot {\ell}^{2}}{\color{blue}{t}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-7}{60} \cdot {\ell}^{2}\right), \color{blue}{t}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\ell}^{2} \cdot \frac{-7}{60}\right), t\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{-7}{60}\right), t\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{-7}{60}\right), t\right) \]

   6. *-lowering-*.f64 24.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{-7}{60}\right), t\right) \]

8. Simplified 24.9%

   \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot -0.11666666666666667}{t}} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\frac{-7}{60} \cdot \left(\ell \cdot \ell\right)}{t} \]

   2. associate-/l* N/A

      \[\leadsto \frac{-7}{60} \cdot \color{blue}{\frac{\ell \cdot \ell}{t}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{-7}{60} \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{t}}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-7}{60}, \color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)}\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-7}{60}, \left(\ell \cdot \frac{1}{\color{blue}{\frac{t}{\ell}}}\right)\right) \]

   6. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-7}{60}, \left(\frac{\ell}{\color{blue}{\frac{t}{\ell}}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-7}{60}, \mathsf{/.f64}\left(\ell, \color{blue}{\left(\frac{t}{\ell}\right)}\right)\right) \]

   8. /-lowering-/.f64 22.2%

      \[\leadsto \mathsf{*.f64}\left(\frac{-7}{60}, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \color{blue}{\ell}\right)\right)\right) \]

10. Applied egg-rr 22.2%

    \[\leadsto \color{blue}{-0.11666666666666667 \cdot \frac{\ell}{\frac{t}{\ell}}} \]

11. Final simplification 22.2%

    \[\leadsto \frac{\ell}{\frac{t}{\ell}} \cdot -0.11666666666666667 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024288 
(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))))