Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.3% → 98.9%

Time: 19.2s

Alternatives: 10

Speedup: 21.0×

• 40.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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: 36.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 98.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.5%

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

3. Taylor expanded in t around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. associate-*r/ N/A

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

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

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

5. Simplified 74.6%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

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

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

7. Applied egg-rr 82.9%

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

   1. *-commutative N/A

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

   2. times-frac N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

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

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

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

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

   8. *-commutative N/A

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

   9. un-div-inv N/A

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

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

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

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

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

   12. associate-*l* N/A

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

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

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

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

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

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

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

   16. tan-lowering-tan.f64 95.5%

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

9. Applied egg-rr 95.5%

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

    1. associate-/r/ N/A

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

    2. associate-*r* N/A

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

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

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

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

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

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

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

    6. *-commutative N/A

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

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

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

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

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

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

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

    10. tan-lowering-tan.f64 99.0%

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

11. Applied egg-rr 99.0%

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

Alternative 2: 82.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{k}{\frac{\ell \cdot 2}{k \cdot \left(k \cdot t\right)}}}\\ \mathbf{elif}\;k \leq 9.8 \cdot 10^{+153}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell \cdot 2}{\sin k \cdot \left(\tan k \cdot t\right)}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell \cdot \ell}{t \cdot \left(\sin k \cdot \tan k\right)}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4.2e-144)
   (/ (/ l k) (/ k (/ (* l 2.0) (* k (* k t)))))
   (if (<= k 9.8e+153)
     (* l (/ (/ (* l 2.0) (* (sin k) (* (tan k) t))) (* k k)))
     (* (/ 2.0 k) (/ (/ (* l l) (* t (* (sin k) (tan k)))) k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.2e-144) {
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))));
	} else if (k <= 9.8e+153) {
		tmp = l * (((l * 2.0) / (sin(k) * (tan(k) * t))) / (k * k));
	} else {
		tmp = (2.0 / k) * (((l * l) / (t * (sin(k) * tan(k)))) / k);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.2d-144) then
        tmp = (l / k) / (k / ((l * 2.0d0) / (k * (k * t))))
    else if (k <= 9.8d+153) then
        tmp = l * (((l * 2.0d0) / (sin(k) * (tan(k) * t))) / (k * k))
    else
        tmp = (2.0d0 / k) * (((l * l) / (t * (sin(k) * tan(k)))) / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.2e-144) {
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))));
	} else if (k <= 9.8e+153) {
		tmp = l * (((l * 2.0) / (Math.sin(k) * (Math.tan(k) * t))) / (k * k));
	} else {
		tmp = (2.0 / k) * (((l * l) / (t * (Math.sin(k) * Math.tan(k)))) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 4.2e-144:
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))))
	elif k <= 9.8e+153:
		tmp = l * (((l * 2.0) / (math.sin(k) * (math.tan(k) * t))) / (k * k))
	else:
		tmp = (2.0 / k) * (((l * l) / (t * (math.sin(k) * math.tan(k)))) / k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4.2e-144)
		tmp = Float64(Float64(l / k) / Float64(k / Float64(Float64(l * 2.0) / Float64(k * Float64(k * t)))));
	elseif (k <= 9.8e+153)
		tmp = Float64(l * Float64(Float64(Float64(l * 2.0) / Float64(sin(k) * Float64(tan(k) * t))) / Float64(k * k)));
	else
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(Float64(l * l) / Float64(t * Float64(sin(k) * tan(k)))) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4.2e-144)
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))));
	elseif (k <= 9.8e+153)
		tmp = l * (((l * 2.0) / (sin(k) * (tan(k) * t))) / (k * k));
	else
		tmp = (2.0 / k) * (((l * l) / (t * (sin(k) * tan(k)))) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4.2e-144], N[(N[(l / k), $MachinePrecision] / N[(k / N[(N[(l * 2.0), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 9.8e+153], N[(l * N[(N[(N[(l * 2.0), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] / N[(t * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.2000000000000002e-144

   1. Initial program 41.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

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

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

   5. Simplified 80.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

   7. Applied egg-rr 88.5%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. associate-*l* N/A

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

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

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

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

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

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

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

      16. tan-lowering-tan.f64 97.5%

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

   9. Applied egg-rr 97.5%

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

   10. Taylor expanded in k around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

       4. *-lowering-*.f64 82.3%

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

   12. Simplified 82.3%

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

   if 4.2000000000000002e-144 < k < 9.80000000000000003e153

   1. Initial program 19.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

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

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

   5. Simplified 77.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

   7. Applied egg-rr 89.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. un-div-inv N/A

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

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

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

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

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

      9. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 95.9%

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

   9. Applied egg-rr 95.9%

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

   if 9.80000000000000003e153 < k

   1. Initial program 31.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

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

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

   5. Simplified 50.5%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

   7. Applied egg-rr 60.7%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{k}{\frac{\ell \cdot 2}{k \cdot \left(k \cdot t\right)}}}\\ \mathbf{elif}\;k \leq 9.8 \cdot 10^{+153}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell \cdot 2}{\sin k \cdot \left(\tan k \cdot t\right)}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell \cdot \ell}{t \cdot \left(\sin k \cdot \tan k\right)}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{k}{\frac{\ell \cdot 2}{k \cdot \left(k \cdot t\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot 2}{\left(k \cdot t\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5.2e-17)
   (/ (/ l k) (/ k (/ (* l 2.0) (* k (* k t)))))
   (* (/ l k) (/ (* l 2.0) (* (* k t) (* (sin k) (tan k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.2e-17) {
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))));
	} else {
		tmp = (l / k) * ((l * 2.0) / ((k * 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 <= 5.2d-17) then
        tmp = (l / k) / (k / ((l * 2.0d0) / (k * (k * t))))
    else
        tmp = (l / k) * ((l * 2.0d0) / ((k * 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 <= 5.2e-17) {
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))));
	} else {
		tmp = (l / k) * ((l * 2.0) / ((k * t) * (Math.sin(k) * Math.tan(k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 5.2e-17:
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))))
	else:
		tmp = (l / k) * ((l * 2.0) / ((k * t) * (math.sin(k) * math.tan(k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 5.2e-17)
		tmp = Float64(Float64(l / k) / Float64(k / Float64(Float64(l * 2.0) / Float64(k * Float64(k * t)))));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l * 2.0) / Float64(Float64(k * t) * Float64(sin(k) * tan(k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5.2e-17)
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))));
	else
		tmp = (l / k) * ((l * 2.0) / ((k * t) * (sin(k) * tan(k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 5.2e-17], N[(N[(l / k), $MachinePrecision] / N[(k / N[(N[(l * 2.0), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l * 2.0), $MachinePrecision] / N[(N[(k * t), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.20000000000000006e-17

   1. Initial program 36.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

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

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

   5. Simplified 78.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

   7. Applied egg-rr 89.2%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. associate-*l* N/A

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

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

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

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

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

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

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

      16. tan-lowering-tan.f64 97.4%

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

   9. Applied egg-rr 97.4%

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

   10. Taylor expanded in k around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

       4. *-lowering-*.f64 85.3%

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

   12. Simplified 85.3%

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

   if 5.20000000000000006e-17 < k

   1. Initial program 26.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

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

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

   5. Simplified 66.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

   7. Applied egg-rr 68.4%

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

      1. times-frac N/A

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

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

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

      3. *-commutative N/A

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

      4. un-div-inv N/A

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

      5. associate-/l/ N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      15. /-lowering-/.f64 92.1%

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

   9. Applied egg-rr 92.1%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{k}{\frac{\ell \cdot 2}{k \cdot \left(k \cdot t\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot 2}{\left(k \cdot t\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{k}{\frac{\ell \cdot 2}{k \cdot \left(k \cdot t\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell \cdot 2}{\sin k \cdot \left(\tan k \cdot t\right)}}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4e-144)
   (/ (/ l k) (/ k (/ (* l 2.0) (* k (* k t)))))
   (* l (/ (/ (* l 2.0) (* (sin k) (* (tan k) t))) (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4e-144) {
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))));
	} else {
		tmp = l * (((l * 2.0) / (sin(k) * (tan(k) * t))) / (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4d-144) then
        tmp = (l / k) / (k / ((l * 2.0d0) / (k * (k * t))))
    else
        tmp = l * (((l * 2.0d0) / (sin(k) * (tan(k) * t))) / (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4e-144) {
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))));
	} else {
		tmp = l * (((l * 2.0) / (Math.sin(k) * (Math.tan(k) * t))) / (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 4e-144:
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))))
	else:
		tmp = l * (((l * 2.0) / (math.sin(k) * (math.tan(k) * t))) / (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4e-144)
		tmp = Float64(Float64(l / k) / Float64(k / Float64(Float64(l * 2.0) / Float64(k * Float64(k * t)))));
	else
		tmp = Float64(l * Float64(Float64(Float64(l * 2.0) / Float64(sin(k) * Float64(tan(k) * t))) / Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4e-144)
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))));
	else
		tmp = l * (((l * 2.0) / (sin(k) * (tan(k) * t))) / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4e-144], N[(N[(l / k), $MachinePrecision] / N[(k / N[(N[(l * 2.0), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(l * 2.0), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.9999999999999998e-144

   1. Initial program 41.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

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

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

   5. Simplified 80.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

   7. Applied egg-rr 88.5%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. associate-*l* N/A

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

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

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

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

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

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

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

      16. tan-lowering-tan.f64 97.5%

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

   9. Applied egg-rr 97.5%

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

   10. Taylor expanded in k around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

       4. *-lowering-*.f64 82.3%

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

   12. Simplified 82.3%

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

   if 3.9999999999999998e-144 < k

   1. Initial program 23.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

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

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

   5. Simplified 67.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

   7. Applied egg-rr 75.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. un-div-inv N/A

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

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

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

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

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

      9. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 80.4%

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

   9. Applied egg-rr 80.4%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{k}{\frac{\ell \cdot 2}{k \cdot \left(k \cdot t\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell \cdot 2}{\sin k \cdot \left(\tan k \cdot t\right)}}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.2:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{k}{\frac{\ell \cdot 2}{k \cdot \left(k \cdot t\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{\frac{\ell \cdot 2}{\frac{k}{\ell}}}{\frac{k}{\frac{\frac{\cos k}{t}}{k}}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.2)
   (/ (/ l k) (/ k (/ (* l 2.0) (* k (* k t)))))
   (/ 1.0 (/ k (/ (/ (* l 2.0) (/ k l)) (/ k (/ (/ (cos k) t) k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.2) {
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))));
	} else {
		tmp = 1.0 / (k / (((l * 2.0) / (k / l)) / (k / ((cos(k) / t) / 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 <= 1.2d0) then
        tmp = (l / k) / (k / ((l * 2.0d0) / (k * (k * t))))
    else
        tmp = 1.0d0 / (k / (((l * 2.0d0) / (k / l)) / (k / ((cos(k) / t) / k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.2) {
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))));
	} else {
		tmp = 1.0 / (k / (((l * 2.0) / (k / l)) / (k / ((Math.cos(k) / t) / k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.2:
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))))
	else:
		tmp = 1.0 / (k / (((l * 2.0) / (k / l)) / (k / ((math.cos(k) / t) / k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.2)
		tmp = Float64(Float64(l / k) / Float64(k / Float64(Float64(l * 2.0) / Float64(k * Float64(k * t)))));
	else
		tmp = Float64(1.0 / Float64(k / Float64(Float64(Float64(l * 2.0) / Float64(k / l)) / Float64(k / Float64(Float64(cos(k) / t) / k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.2)
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))));
	else
		tmp = 1.0 / (k / (((l * 2.0) / (k / l)) / (k / ((cos(k) / t) / k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.2], N[(N[(l / k), $MachinePrecision] / N[(k / N[(N[(l * 2.0), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(k / N[(N[(N[(l * 2.0), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(k / N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.19999999999999996

   1. Initial program 35.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

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

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

   5. Simplified 78.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

   7. Applied egg-rr 89.6%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. associate-*l* N/A

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

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

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

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

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

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

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

      16. tan-lowering-tan.f64 97.5%

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

   9. Applied egg-rr 97.5%

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

   10. Taylor expanded in k around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

       4. *-lowering-*.f64 85.2%

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

   12. Simplified 85.2%

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

   if 1.19999999999999996 < k

   1. Initial program 28.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

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

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

   5. Simplified 64.1%

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

   6. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 48.6%

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

   8. Simplified 48.6%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

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

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

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

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

      9. associate-*r* N/A

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

      10. associate-/l* N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

      14. *-commutative N/A

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

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

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

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

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

      17. clear-num N/A

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

      18. associate-/r* N/A

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

      19. clear-num N/A

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

   10. Applied egg-rr 51.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{k}{\frac{\frac{\ell \cdot 2}{\frac{k}{\ell}}}{\frac{k}{\frac{\frac{\cos k}{t}}{k}}}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{k}{\frac{\ell \cdot 2}{k \cdot \left(k \cdot t\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{\frac{\ell \cdot 2}{\frac{k}{\ell}}}{\frac{k}{\frac{\frac{\cos k}{t}}{k}}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.2:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{k}{\frac{\ell \cdot 2}{k \cdot \left(k \cdot t\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot 2}{\frac{k}{\ell}}}{\frac{k}{\frac{\frac{\cos k}{t}}{k}}}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.2)
   (/ (/ l k) (/ k (/ (* l 2.0) (* k (* k t)))))
   (/ (/ (/ (* l 2.0) (/ k l)) (/ k (/ (/ (cos k) t) k))) k)))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.2) {
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))));
	} else {
		tmp = (((l * 2.0) / (k / l)) / (k / ((cos(k) / t) / k))) / k;
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.2d0) then
        tmp = (l / k) / (k / ((l * 2.0d0) / (k * (k * t))))
    else
        tmp = (((l * 2.0d0) / (k / l)) / (k / ((cos(k) / t) / k))) / k
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.2) {
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))));
	} else {
		tmp = (((l * 2.0) / (k / l)) / (k / ((Math.cos(k) / t) / k))) / k;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.2:
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))))
	else:
		tmp = (((l * 2.0) / (k / l)) / (k / ((math.cos(k) / t) / k))) / k
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.2)
		tmp = Float64(Float64(l / k) / Float64(k / Float64(Float64(l * 2.0) / Float64(k * Float64(k * t)))));
	else
		tmp = Float64(Float64(Float64(Float64(l * 2.0) / Float64(k / l)) / Float64(k / Float64(Float64(cos(k) / t) / k))) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.2)
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))));
	else
		tmp = (((l * 2.0) / (k / l)) / (k / ((cos(k) / t) / k))) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.2], N[(N[(l / k), $MachinePrecision] / N[(k / N[(N[(l * 2.0), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * 2.0), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(k / N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.19999999999999996

   1. Initial program 35.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

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

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

   5. Simplified 78.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

   7. Applied egg-rr 89.6%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. associate-*l* N/A

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

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

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

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

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

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

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

      16. tan-lowering-tan.f64 97.5%

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

   9. Applied egg-rr 97.5%

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

   10. Taylor expanded in k around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

       4. *-lowering-*.f64 85.2%

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

   12. Simplified 85.2%

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

   if 1.19999999999999996 < k

   1. Initial program 28.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

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

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

   5. Simplified 64.1%

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

   6. Taylor expanded in k around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 48.6%

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

   8. Simplified 48.6%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

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

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

   10. Applied egg-rr 51.6%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{k}{\frac{\ell \cdot 2}{k \cdot \left(k \cdot t\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot 2}{\frac{k}{\ell}}}{\frac{k}{\frac{\frac{\cos k}{t}}{k}}}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.8% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 390:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{k}{\frac{\ell \cdot 2}{k \cdot \left(k \cdot t\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.6666666666666666}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 390.0)
   (/ (/ l k) (/ k (/ (* l 2.0) (* k (* k t)))))
   (/ (/ (* (/ (* l l) t) -0.6666666666666666) k) k)))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 390.0) {
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))));
	} else {
		tmp = ((((l * l) / t) * -0.6666666666666666) / k) / k;
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 390.0d0) then
        tmp = (l / k) / (k / ((l * 2.0d0) / (k * (k * t))))
    else
        tmp = ((((l * l) / t) * (-0.6666666666666666d0)) / k) / k
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 390.0) {
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))));
	} else {
		tmp = ((((l * l) / t) * -0.6666666666666666) / k) / k;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 390.0:
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))))
	else:
		tmp = ((((l * l) / t) * -0.6666666666666666) / k) / k
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 390.0)
		tmp = Float64(Float64(l / k) / Float64(k / Float64(Float64(l * 2.0) / Float64(k * Float64(k * t)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l * l) / t) * -0.6666666666666666) / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 390.0)
		tmp = (l / k) / (k / ((l * 2.0) / (k * (k * t))));
	else
		tmp = ((((l * l) / t) * -0.6666666666666666) / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 390.0], N[(N[(l / k), $MachinePrecision] / N[(k / N[(N[(l * 2.0), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * -0.6666666666666666), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 390

   1. Initial program 35.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

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

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

   5. Simplified 78.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

   7. Applied egg-rr 89.6%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. associate-*l* N/A

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

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

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

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

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

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

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

      16. tan-lowering-tan.f64 97.5%

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

   9. Applied egg-rr 97.5%

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

   10. Taylor expanded in k around 0

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

       4. *-lowering-*.f64 85.3%

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

   12. Simplified 85.3%

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

   if 390 < k

   1. Initial program 29.0%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 40.1%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 29.0%

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

   7. Simplified 29.0%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

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

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

   10. Simplified 14.2%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot \left(\left(k \cdot k\right) \cdot -0.6666666666666666 + 2\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   11. Taylor expanded in k around inf

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

       1. associate-*r/ N/A

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

       2. associate-/l/ N/A

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

       3. associate-*r/ N/A

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

       4. unpow2 N/A

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

       5. associate-/r* N/A

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

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

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

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

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

       8. *-commutative N/A

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

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

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 49.1%

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

   13. Simplified 49.1%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.6666666666666666}{k}}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 390:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{k}{\frac{\ell \cdot 2}{k \cdot \left(k \cdot t\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.6666666666666666}{k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.1% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 390:\\ \;\;\;\;\ell \cdot \left(\frac{2}{k} \cdot \frac{\frac{\frac{\ell}{k \cdot t}}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.6666666666666666}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 390.0)
   (* l (* (/ 2.0 k) (/ (/ (/ l (* k t)) k) k)))
   (/ (/ (* (/ (* l l) t) -0.6666666666666666) k) k)))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 390.0) {
		tmp = l * ((2.0 / k) * (((l / (k * t)) / k) / k));
	} else {
		tmp = ((((l * l) / t) * -0.6666666666666666) / k) / k;
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 390.0d0) then
        tmp = l * ((2.0d0 / k) * (((l / (k * t)) / k) / k))
    else
        tmp = ((((l * l) / t) * (-0.6666666666666666d0)) / k) / k
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 390.0) {
		tmp = l * ((2.0 / k) * (((l / (k * t)) / k) / k));
	} else {
		tmp = ((((l * l) / t) * -0.6666666666666666) / k) / k;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 390.0:
		tmp = l * ((2.0 / k) * (((l / (k * t)) / k) / k))
	else:
		tmp = ((((l * l) / t) * -0.6666666666666666) / k) / k
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 390.0)
		tmp = Float64(l * Float64(Float64(2.0 / k) * Float64(Float64(Float64(l / Float64(k * t)) / k) / k)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l * l) / t) * -0.6666666666666666) / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 390.0)
		tmp = l * ((2.0 / k) * (((l / (k * t)) / k) / k));
	else
		tmp = ((((l * l) / t) * -0.6666666666666666) / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 390.0], N[(l * N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * -0.6666666666666666), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 390

   1. Initial program 35.3%

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

   3. Taylor expanded in 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 64.4%

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

   5. Simplified 64.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   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. /-lowering-/.f64 N/A

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

      7. associate-*r* N/A

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

      8. cube-unmult N/A

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

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

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

      10. cube-unmult N/A

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

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

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

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

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

      13. /-lowering-/.f64 74.1%

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

   7. Applied egg-rr 74.1%

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

      1. div-inv N/A

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

      2. associate-/r* N/A

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

      3. frac-times N/A

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

      4. div-inv N/A

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

      5. *-rgt-identity N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 78.8%

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

   9. Applied egg-rr 78.8%

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

       1. associate-/r* N/A

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

       2. associate-/r* N/A

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

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

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

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

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

       5. associate-/l/ N/A

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

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

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

       7. *-lowering-*.f64 83.5%

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

   11. Applied egg-rr 83.5%

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

   if 390 < k

   1. Initial program 29.0%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 40.1%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 29.0%

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

   7. Simplified 29.0%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

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

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

   10. Simplified 14.2%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot \left(\left(k \cdot k\right) \cdot -0.6666666666666666 + 2\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   11. Taylor expanded in k around inf

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

       1. associate-*r/ N/A

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

       2. associate-/l/ N/A

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

       3. associate-*r/ N/A

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

       4. unpow2 N/A

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

       5. associate-/r* N/A

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

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

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

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

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

       8. *-commutative N/A

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

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

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 49.1%

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

   13. Simplified 49.1%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.6666666666666666}{k}}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 390:\\ \;\;\;\;\ell \cdot \left(\frac{2}{k} \cdot \frac{\frac{\frac{\ell}{k \cdot t}}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.6666666666666666}{k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.7% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 390:\\ \;\;\;\;\ell \cdot \left(\frac{2}{k} \cdot \frac{\frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.6666666666666666}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 390.0)
   (* l (* (/ 2.0 k) (/ (/ l t) (* k (* k k)))))
   (/ (/ (* (/ (* l l) t) -0.6666666666666666) k) k)))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 390.0) {
		tmp = l * ((2.0 / k) * ((l / t) / (k * (k * k))));
	} else {
		tmp = ((((l * l) / t) * -0.6666666666666666) / k) / k;
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 390.0d0) then
        tmp = l * ((2.0d0 / k) * ((l / t) / (k * (k * k))))
    else
        tmp = ((((l * l) / t) * (-0.6666666666666666d0)) / k) / k
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 390.0) {
		tmp = l * ((2.0 / k) * ((l / t) / (k * (k * k))));
	} else {
		tmp = ((((l * l) / t) * -0.6666666666666666) / k) / k;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 390.0:
		tmp = l * ((2.0 / k) * ((l / t) / (k * (k * k))))
	else:
		tmp = ((((l * l) / t) * -0.6666666666666666) / k) / k
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 390.0)
		tmp = Float64(l * Float64(Float64(2.0 / k) * Float64(Float64(l / t) / Float64(k * Float64(k * k)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(l * l) / t) * -0.6666666666666666) / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 390.0)
		tmp = l * ((2.0 / k) * ((l / t) / (k * (k * k))));
	else
		tmp = ((((l * l) / t) * -0.6666666666666666) / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 390.0], N[(l * N[(N[(2.0 / k), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] / N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * -0.6666666666666666), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 390

   1. Initial program 35.3%

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

   3. Taylor expanded in 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 64.4%

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

   5. Simplified 64.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot \frac{t}{\ell \cdot \ell}}} \]
   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. /-lowering-/.f64 N/A

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

      7. associate-*r* N/A

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

      8. cube-unmult N/A

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

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

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

      10. cube-unmult N/A

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

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

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

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

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

      13. /-lowering-/.f64 74.1%

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

   7. Applied egg-rr 74.1%

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

      1. div-inv N/A

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

      2. associate-/r* N/A

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

      3. frac-times N/A

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

      4. div-inv N/A

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

      5. *-rgt-identity N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 78.8%

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

   9. Applied egg-rr 78.8%

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

   if 390 < k

   1. Initial program 29.0%

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

      1. associate-/r* N/A

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

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

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

   3. Simplified 40.1%

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

   5. Taylor expanded in k around 0

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 29.0%

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

   7. Simplified 29.0%

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

   8. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

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

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

   10. Simplified 14.2%

       \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot \left(\left(k \cdot k\right) \cdot -0.6666666666666666 + 2\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

   11. Taylor expanded in k around inf

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

       1. associate-*r/ N/A

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

       2. associate-/l/ N/A

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

       3. associate-*r/ N/A

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

       4. unpow2 N/A

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

       5. associate-/r* N/A

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

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

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

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

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

       8. *-commutative N/A

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

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

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 49.1%

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

   13. Simplified 49.1%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.6666666666666666}{k}}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 390:\\ \;\;\;\;\ell \cdot \left(\frac{2}{k} \cdot \frac{\frac{\ell}{t}}{k \cdot \left(k \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.6666666666666666}{k}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 29.6% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.6666666666666666}{k}}{k} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (/ (* (/ (* l l) t) -0.6666666666666666) k) k))

C

double code(double t, double l, double k) {
	return ((((l * l) / t) * -0.6666666666666666) / k) / k;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((((l * l) / t) * (-0.6666666666666666d0)) / k) / k
end function

Java

public static double code(double t, double l, double k) {
	return ((((l * l) / t) * -0.6666666666666666) / k) / k;
}

Python

def code(t, l, k):
	return ((((l * l) / t) * -0.6666666666666666) / k) / k

Julia

function code(t, l, k)
	return Float64(Float64(Float64(Float64(Float64(l * l) / t) * -0.6666666666666666) / k) / k)
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((((l * l) / t) * -0.6666666666666666) / k) / k;
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * -0.6666666666666666), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]

Derivation

1. Initial program 33.5%

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

   1. associate-/r* N/A

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

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

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

3. Simplified 42.4%

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

5. Taylor expanded in k around 0

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

   1. associate-*r/ N/A

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. *-commutative N/A

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

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

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

   8. cube-mult N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left({t}^{2}\right)\right), k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \left(t \cdot t\right)\right), k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

   12. *-lowering-*.f64 32.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right), k\right)\right), \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{/.f64}\left(k, t\right), t\right)\right)\right) \]

7. Simplified 32.1%

   \[\leadsto \frac{\frac{\color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(t \cdot \left(t \cdot t\right)\right) \cdot k}}}{\tan k}}{k \cdot \frac{\frac{k}{t}}{t}} \]

8. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{\frac{-2}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-2}{3} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

   2. associate-/l* N/A

      \[\leadsto \frac{\left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right) \cdot \frac{-2}{3} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{{k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-2}{3}\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

   4. *-commutative N/A

      \[\leadsto \frac{{k}^{2} \cdot \left(\frac{-2}{3} \cdot \frac{{\ell}^{2}}{t}\right) + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}} \]

   5. +-commutative N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-2}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{\color{blue}{k}}^{4}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-2}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right), \color{blue}{\left({k}^{4}\right)}\right) \]

10. Simplified 44.4%

    \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{t} \cdot \left(\left(k \cdot k\right) \cdot -0.6666666666666666 + 2\right)}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]

11. Taylor expanded in k around inf

    \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
12. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{-2}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

    2. associate-/l/ N/A

       \[\leadsto \frac{\frac{\frac{-2}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

    3. associate-*r/ N/A

       \[\leadsto \frac{\frac{-2}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

    4. unpow2 N/A

       \[\leadsto \frac{\frac{-2}{3} \cdot \frac{{\ell}^{2}}{t}}{k \cdot \color{blue}{k}} \]

    5. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{-2}{3} \cdot \frac{{\ell}^{2}}{t}}{k}}{\color{blue}{k}} \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{-2}{3} \cdot \frac{{\ell}^{2}}{t}}{k}\right), \color{blue}{k}\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-2}{3} \cdot \frac{{\ell}^{2}}{t}\right), k\right), k\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{{\ell}^{2}}{t} \cdot \frac{-2}{3}\right), k\right), k\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{t}\right), \frac{-2}{3}\right), k\right), k\right) \]

    10. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right), \frac{-2}{3}\right), k\right), k\right) \]

    11. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right), \frac{-2}{3}\right), k\right), k\right) \]

    12. *-lowering-*.f64 27.9%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right), \frac{-2}{3}\right), k\right), k\right) \]

13. Simplified 27.9%

    \[\leadsto \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{t} \cdot -0.6666666666666666}{k}}{k}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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))))