Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.2% → 97.3%

Time: 21.5s

Alternatives: 12

Speedup: 60.1×

• 35.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

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: 34.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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 * ((l * (cos(k) / k)) / (((k / l) * t) * (sin(k) ** 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

   1. associate-/r* 37.3%

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

   2. *-commutative 37.3%

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

   3. associate-/r* 41.7%

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

   4. associate-*r/ 41.7%

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

   5. associate-/l* 41.7%

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

   6. +-commutative 41.7%

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

   7. unpow2 41.7%

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

   8. sqr-neg 41.7%

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

   9. distribute-frac-neg 41.7%

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

   10. distribute-frac-neg 41.7%

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

   11. unpow2 41.7%

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

   12. associate--l+ 50.1%

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

   13. metadata-eval 50.1%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   14. +-rgt-identity 50.1%

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

   15. unpow2 50.1%

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

   16. distribute-frac-neg 50.1%

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

3. Simplified 50.1%

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

4. Taylor expanded in k around inf 72.7%

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

   1. associate-*r/ 72.7%

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

   2. unpow2 72.7%

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

   3. associate-*r* 72.7%

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

   4. *-commutative 72.7%

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

   5. unpow2 72.7%

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

6. Simplified 72.7%

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

7. Taylor expanded in l around 0 72.7%

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

   1. associate-*r* 72.7%

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

   2. *-commutative 72.7%

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

   3. associate-*r* 72.6%

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

   4. unpow2 72.6%

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

   5. times-frac 71.0%

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

   6. unpow2 71.0%

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

   7. associate-*r/ 76.5%

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

   8. associate-*l* 76.5%

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

9. Simplified 76.5%

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

10. Taylor expanded in l around 0 72.7%

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

    1. associate-*r* 72.7%

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

    2. times-frac 73.1%

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

    3. *-commutative 73.1%

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

    4. associate-/r* 71.9%

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

    5. times-frac 71.4%

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

    6. unpow2 71.4%

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

    7. associate-*r* 71.4%

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

    8. unpow2 71.4%

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

    9. associate-*r/ 76.9%

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

    10. times-frac 82.4%

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

    11. associate-/l* 89.3%

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

    12. associate-/r* 89.4%

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

12. Simplified 89.4%

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

    1. frac-times 90.4%

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

    2. associate-/r/ 96.5%

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

14. Applied egg-rr 96.5%

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

15. Final simplification 96.5%

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

Alternative 2: 91.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 2.15 \cdot 10^{-61} \lor \neg \left(k \leq 6.5 \cdot 10^{+130}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{k}}{t_1} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (or (<= k 2.15e-61) (not (<= k 6.5e+130)))
     (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t t_1))))
     (* 2.0 (* (/ (/ (cos k) k) t_1) (* (/ l k) (/ l t)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if ((k <= 2.15e-61) || !(k <= 6.5e+130)) {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * t_1)));
	} else {
		tmp = 2.0 * (((cos(k) / k) / t_1) * ((l / k) * (l / t)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    if ((k <= 2.15d-61) .or. (.not. (k <= 6.5d+130))) then
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) / (t * t_1)))
    else
        tmp = 2.0d0 * (((cos(k) / k) / t_1) * ((l / k) * (l / t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if ((k <= 2.15e-61) || !(k <= 6.5e+130)) {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * t_1)));
	} else {
		tmp = 2.0 * (((Math.cos(k) / k) / t_1) * ((l / k) * (l / t)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if (k <= 2.15e-61) or not (k <= 6.5e+130):
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t * t_1)))
	else:
		tmp = 2.0 * (((math.cos(k) / k) / t_1) * ((l / k) * (l / t)))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if ((k <= 2.15e-61) || !(k <= 6.5e+130))
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * t_1))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) / k) / t_1) * Float64(Float64(l / k) * Float64(l / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if ((k <= 2.15e-61) || ~((k <= 6.5e+130)))
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * t_1)));
	else
		tmp = 2.0 * (((cos(k) / k) / t_1) * ((l / k) * (l / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[k, 2.15e-61], N[Not[LessEqual[k, 6.5e+130]], $MachinePrecision]], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.1500000000000002e-61 or 6.5e130 < k

   1. Initial program 38.2%

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

      1. associate-/r* 38.2%

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

      2. *-commutative 38.2%

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

      3. associate-/r* 42.9%

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

      4. associate-*r/ 42.9%

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

      5. associate-/l* 42.9%

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

      6. +-commutative 42.9%

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

      7. unpow2 42.9%

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

      8. sqr-neg 42.9%

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

      9. distribute-frac-neg 42.9%

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

      10. distribute-frac-neg 42.9%

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

      11. unpow2 42.9%

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

      12. associate--l+ 50.4%

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

      13. metadata-eval 50.4%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 50.4%

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

      15. unpow2 50.4%

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

      16. distribute-frac-neg 50.4%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in k around inf 70.3%

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

      1. associate-*r/ 70.3%

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

      2. unpow2 70.3%

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

      3. associate-*r* 70.3%

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

      4. *-commutative 70.3%

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

      5. unpow2 70.3%

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

   6. Simplified 70.3%

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

   7. Taylor expanded in l around 0 70.3%

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

      1. associate-*r* 70.3%

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

      2. *-commutative 70.3%

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

      3. associate-*r* 70.2%

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

      4. unpow2 70.2%

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

      5. times-frac 67.9%

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

      6. unpow2 67.9%

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

      7. associate-*r/ 73.6%

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

      8. associate-*l* 73.6%

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

   9. Simplified 73.6%

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

   10. Taylor expanded in l around 0 70.3%

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

       1. associate-*r* 70.3%

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

       2. times-frac 70.8%

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

       3. *-commutative 70.8%

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

       4. associate-/r* 68.9%

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

       5. times-frac 68.4%

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

       6. unpow2 68.4%

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

       7. associate-*r* 68.4%

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

       8. unpow2 68.4%

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

       9. associate-*r/ 74.0%

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

       10. times-frac 80.6%

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

       11. associate-/l* 87.9%

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

       12. associate-/r* 87.9%

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

   12. Simplified 87.9%

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

   13. Taylor expanded in l around 0 70.3%

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

       1. times-frac 70.8%

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

       2. unpow2 70.8%

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

       3. unpow2 70.8%

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

       4. times-frac 92.4%

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

   15. Simplified 92.4%

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

   if 2.1500000000000002e-61 < k < 6.5e130

   1. Initial program 32.8%

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

      1. associate-/r* 32.8%

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

      2. *-commutative 32.8%

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

      3. associate-/r* 35.1%

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

      4. associate-*r/ 35.1%

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

      5. associate-/l* 35.1%

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

      6. +-commutative 35.1%

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

      7. unpow2 35.1%

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

      8. sqr-neg 35.1%

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

      9. distribute-frac-neg 35.1%

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

      10. distribute-frac-neg 35.1%

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

      11. unpow2 35.1%

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

      12. associate--l+ 48.0%

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

      13. metadata-eval 48.0%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 48.0%

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

      15. unpow2 48.0%

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

      16. distribute-frac-neg 48.0%

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

   3. Simplified 48.0%

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

   4. Taylor expanded in k around inf 85.5%

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

      1. associate-*r/ 85.5%

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

      2. unpow2 85.5%

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

      3. associate-*r* 85.5%

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

      4. *-commutative 85.5%

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

      5. unpow2 85.5%

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

   6. Simplified 85.5%

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

   7. Taylor expanded in l around 0 85.5%

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

      1. associate-*r* 85.5%

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

      2. *-commutative 85.5%

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

      3. associate-*r* 85.5%

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

      4. unpow2 85.5%

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

      5. times-frac 87.9%

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

      6. unpow2 87.9%

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

      7. associate-*r/ 92.6%

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

      8. associate-*l* 92.4%

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

   9. Simplified 92.4%

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

   10. Taylor expanded in l around 0 85.5%

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

       1. associate-*r* 85.5%

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

       2. times-frac 85.5%

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

       3. *-commutative 85.5%

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

       4. associate-/r* 88.0%

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

       5. times-frac 87.9%

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

       6. unpow2 87.9%

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

       7. associate-*r* 87.9%

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

       8. unpow2 87.9%

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

       9. associate-*r/ 92.4%

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

       10. times-frac 92.5%

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

       11. associate-/l* 97.3%

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

       12. associate-/r* 97.4%

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

   12. Simplified 97.4%

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

       1. associate-/r/ 97.4%

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

   14. Applied egg-rr 97.4%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{-61} \lor \neg \left(k \leq 6.5 \cdot 10^{+130}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{k}}{{\sin k}^{2}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t}\right)\right)\\ \end{array} \]

Alternative 3: 91.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 2.3 \cdot 10^{-61} \lor \neg \left(k \leq 3.7 \cdot 10^{+142}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\frac{k}{\frac{\ell}{t}}} \cdot \frac{\frac{\cos k}{k}}{t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (or (<= k 2.3e-61) (not (<= k 3.7e+142)))
     (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t t_1))))
     (* 2.0 (* (/ l (/ k (/ l t))) (/ (/ (cos k) k) t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if ((k <= 2.3e-61) || !(k <= 3.7e+142)) {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * t_1)));
	} else {
		tmp = 2.0 * ((l / (k / (l / t))) * ((cos(k) / k) / t_1));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    if ((k <= 2.3d-61) .or. (.not. (k <= 3.7d+142))) then
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) / (t * t_1)))
    else
        tmp = 2.0d0 * ((l / (k / (l / t))) * ((cos(k) / k) / t_1))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if ((k <= 2.3e-61) || !(k <= 3.7e+142)) {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * t_1)));
	} else {
		tmp = 2.0 * ((l / (k / (l / t))) * ((Math.cos(k) / k) / t_1));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if (k <= 2.3e-61) or not (k <= 3.7e+142):
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t * t_1)))
	else:
		tmp = 2.0 * ((l / (k / (l / t))) * ((math.cos(k) / k) / t_1))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if ((k <= 2.3e-61) || !(k <= 3.7e+142))
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * t_1))));
	else
		tmp = Float64(2.0 * Float64(Float64(l / Float64(k / Float64(l / t))) * Float64(Float64(cos(k) / k) / t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if ((k <= 2.3e-61) || ~((k <= 3.7e+142)))
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * t_1)));
	else
		tmp = 2.0 * ((l / (k / (l / t))) * ((cos(k) / k) / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[k, 2.3e-61], N[Not[LessEqual[k, 3.7e+142]], $MachinePrecision]], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.29999999999999992e-61 or 3.6999999999999997e142 < k

   1. Initial program 38.0%

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

      1. associate-/r* 38.1%

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

      2. *-commutative 38.1%

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

      3. associate-/r* 42.8%

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

      4. associate-*r/ 42.8%

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

      5. associate-/l* 42.8%

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

      6. +-commutative 42.8%

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

      7. unpow2 42.8%

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

      8. sqr-neg 42.8%

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

      9. distribute-frac-neg 42.8%

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

      10. distribute-frac-neg 42.8%

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

      11. unpow2 42.8%

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

      12. associate--l+ 50.4%

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

      13. metadata-eval 50.4%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 50.4%

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

      15. unpow2 50.4%

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

      16. distribute-frac-neg 50.4%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in k around inf 70.5%

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

      1. associate-*r/ 70.5%

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

      2. unpow2 70.5%

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

      3. associate-*r* 70.5%

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

      4. *-commutative 70.5%

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

      5. unpow2 70.5%

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

   6. Simplified 70.5%

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

   7. Taylor expanded in l around 0 70.5%

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

      1. associate-*r* 70.5%

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

      2. *-commutative 70.5%

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

      3. associate-*r* 70.4%

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

      4. unpow2 70.4%

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

      5. times-frac 68.0%

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

      6. unpow2 68.0%

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

      7. associate-*r/ 73.3%

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

      8. associate-*l* 73.3%

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

   9. Simplified 73.3%

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

   10. Taylor expanded in l around 0 70.5%

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

       1. associate-*r* 70.5%

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

       2. times-frac 71.0%

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

       3. *-commutative 71.0%

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

       4. associate-/r* 69.1%

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

       5. times-frac 68.5%

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

       6. unpow2 68.5%

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

       7. associate-*r* 68.5%

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

       8. unpow2 68.5%

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

       9. associate-*r/ 73.8%

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

       10. times-frac 80.4%

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

       11. associate-/l* 87.8%

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

       12. associate-/r* 87.8%

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

   12. Simplified 87.8%

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

   13. Taylor expanded in l around 0 70.5%

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

       1. times-frac 71.0%

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

       2. unpow2 71.0%

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

       3. unpow2 71.0%

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

       4. times-frac 92.3%

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

   15. Simplified 92.3%

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

   if 2.29999999999999992e-61 < k < 3.6999999999999997e142

   1. Initial program 33.6%

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

      1. associate-/r* 33.6%

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

      2. *-commutative 33.6%

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

      3. associate-/r* 35.9%

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

      4. associate-*r/ 35.8%

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

      5. associate-/l* 35.9%

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

      6. +-commutative 35.9%

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

      7. unpow2 35.9%

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

      8. sqr-neg 35.9%

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

      9. distribute-frac-neg 35.9%

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

      10. distribute-frac-neg 35.9%

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

      11. unpow2 35.9%

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

      12. associate--l+ 48.2%

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

      13. metadata-eval 48.2%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 48.2%

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

      15. unpow2 48.2%

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

      16. distribute-frac-neg 48.2%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in k around inf 83.8%

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

      1. associate-*r/ 83.8%

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

      2. unpow2 83.8%

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

      3. associate-*r* 83.8%

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

      4. *-commutative 83.8%

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

      5. unpow2 83.8%

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

   6. Simplified 83.8%

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

   7. Taylor expanded in l around 0 83.8%

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

      1. associate-*r* 83.8%

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

      2. *-commutative 83.8%

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

      3. associate-*r* 83.8%

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

      4. unpow2 83.8%

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

      5. times-frac 86.2%

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

      6. unpow2 86.2%

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

      7. associate-*r/ 92.9%

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

      8. associate-*l* 92.8%

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

   9. Simplified 92.8%

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

   10. Taylor expanded in l around 0 83.8%

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

       1. associate-*r* 83.8%

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

       2. times-frac 83.8%

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

       3. *-commutative 83.8%

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

       4. associate-/r* 86.2%

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

       5. times-frac 86.2%

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

       6. unpow2 86.2%

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

       7. associate-*r* 86.2%

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

       8. unpow2 86.2%

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

       9. associate-*r/ 92.8%

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

       10. times-frac 92.8%

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

       11. associate-/l* 97.4%

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

       12. associate-/r* 97.5%

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

   12. Simplified 97.5%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-61} \lor \neg \left(k \leq 3.7 \cdot 10^{+142}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\frac{k}{\frac{\ell}{t}}} \cdot \frac{\frac{\cos k}{k}}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 4: 74.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-38}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\frac{k}{\frac{\ell}{t}}} \cdot \frac{\frac{\cos k}{k}}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{k \cdot \left(k \cdot {\sin k}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2e-38)
   (* 2.0 (* (/ l (/ k (/ l t))) (/ (/ (cos k) k) (* k k))))
   (* 2.0 (* (* l (/ l t)) (/ (cos k) (* k (* k (pow (sin k) 2.0))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2e-38) {
		tmp = 2.0 * ((l / (k / (l / t))) * ((cos(k) / k) / (k * k)));
	} else {
		tmp = 2.0 * ((l * (l / t)) * (cos(k) / (k * (k * pow(sin(k), 2.0)))));
	}
	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 <= 2d-38) then
        tmp = 2.0d0 * ((l / (k / (l / t))) * ((cos(k) / k) / (k * k)))
    else
        tmp = 2.0d0 * ((l * (l / t)) * (cos(k) / (k * (k * (sin(k) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2e-38) {
		tmp = 2.0 * ((l / (k / (l / t))) * ((Math.cos(k) / k) / (k * k)));
	} else {
		tmp = 2.0 * ((l * (l / t)) * (Math.cos(k) / (k * (k * Math.pow(Math.sin(k), 2.0)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2e-38:
		tmp = 2.0 * ((l / (k / (l / t))) * ((math.cos(k) / k) / (k * k)))
	else:
		tmp = 2.0 * ((l * (l / t)) * (math.cos(k) / (k * (k * math.pow(math.sin(k), 2.0)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2e-38)
		tmp = Float64(2.0 * Float64(Float64(l / Float64(k / Float64(l / t))) * Float64(Float64(cos(k) / k) / Float64(k * k))));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / t)) * Float64(cos(k) / Float64(k * Float64(k * (sin(k) ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2e-38)
		tmp = 2.0 * ((l / (k / (l / t))) * ((cos(k) / k) / (k * k)));
	else
		tmp = 2.0 * ((l * (l / t)) * (cos(k) / (k * (k * (sin(k) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2e-38], N[(2.0 * N[(N[(l / N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(k * N[(k * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.9999999999999999e-38

   1. Initial program 36.5%

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

      1. associate-/r* 36.5%

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

      2. *-commutative 36.5%

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

      3. associate-/r* 41.7%

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

      4. associate-*r/ 41.7%

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

      5. associate-/l* 41.7%

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

      6. +-commutative 41.7%

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

      7. unpow2 41.7%

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

      8. sqr-neg 41.7%

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

      9. distribute-frac-neg 41.7%

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

      10. distribute-frac-neg 41.7%

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

      11. unpow2 41.7%

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

      12. associate--l+ 48.1%

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

      13. metadata-eval 48.1%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 48.1%

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

      15. unpow2 48.1%

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

      16. distribute-frac-neg 48.1%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in k around inf 69.4%

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

      1. associate-*r/ 69.4%

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

      2. unpow2 69.4%

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

      3. associate-*r* 69.5%

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

      4. *-commutative 69.5%

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

      5. unpow2 69.5%

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

   6. Simplified 69.5%

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

   7. Taylor expanded in l around 0 69.4%

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

      1. associate-*r* 69.5%

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

      2. *-commutative 69.5%

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

      3. associate-*r* 69.3%

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

      4. unpow2 69.3%

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

      5. times-frac 66.5%

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

      6. unpow2 66.5%

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

      7. associate-*r/ 72.8%

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

      8. associate-*l* 72.8%

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

   9. Simplified 72.8%

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

   10. Taylor expanded in l around 0 69.4%

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

       1. associate-*r* 69.5%

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

       2. times-frac 70.0%

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

       3. *-commutative 70.0%

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

       4. associate-/r* 67.8%

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

       5. times-frac 67.1%

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

       6. unpow2 67.1%

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

       7. associate-*r* 67.1%

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

       8. unpow2 67.1%

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

       9. associate-*r/ 73.4%

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

       10. times-frac 78.0%

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

       11. associate-/l* 87.7%

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

       12. associate-/r* 87.7%

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

   12. Simplified 87.7%

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

   13. Taylor expanded in k around 0 72.9%

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

       1. unpow2 66.2%

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

   15. Simplified 72.9%

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

   if 1.9999999999999999e-38 < k

   1. Initial program 39.2%

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

      1. associate-/r* 39.2%

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

      2. *-commutative 39.2%

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

      3. associate-/r* 41.6%

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

      4. associate-*r/ 41.6%

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

      5. associate-/l* 41.6%

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

      6. +-commutative 41.6%

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

      7. unpow2 41.6%

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

      8. sqr-neg 41.6%

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

      9. distribute-frac-neg 41.6%

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

      10. distribute-frac-neg 41.6%

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

      11. unpow2 41.6%

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

      12. associate--l+ 54.1%

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

      13. metadata-eval 54.1%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 54.1%

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

      15. unpow2 54.1%

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

      16. distribute-frac-neg 54.1%

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

   3. Simplified 54.1%

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

   4. Taylor expanded in k around inf 79.5%

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

      1. associate-*r/ 79.5%

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

      2. unpow2 79.5%

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

      3. associate-*r* 79.5%

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

      4. *-commutative 79.5%

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

      5. unpow2 79.5%

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

   6. Simplified 79.5%

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

   7. Taylor expanded in l around 0 79.5%

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

      1. associate-*r* 79.5%

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

      2. *-commutative 79.5%

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

      3. associate-*r* 79.5%

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

      4. unpow2 79.5%

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

      5. times-frac 80.6%

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

      6. unpow2 80.6%

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

      7. associate-*r/ 84.4%

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

      8. associate-*l* 84.3%

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

   9. Simplified 84.3%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-38}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\frac{k}{\frac{\ell}{t}}} \cdot \frac{\frac{\cos k}{k}}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{k \cdot \left(k \cdot {\sin k}^{2}\right)}\right)\\ \end{array} \]

Alternative 5: 90.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ** 2.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

   1. associate-/r* 37.3%

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

   2. *-commutative 37.3%

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

   3. associate-/r* 41.7%

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

   4. associate-*r/ 41.7%

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

   5. associate-/l* 41.7%

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

   6. +-commutative 41.7%

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

   7. unpow2 41.7%

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

   8. sqr-neg 41.7%

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

   9. distribute-frac-neg 41.7%

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

   10. distribute-frac-neg 41.7%

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

   11. unpow2 41.7%

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

   12. associate--l+ 50.1%

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

   13. metadata-eval 50.1%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   14. +-rgt-identity 50.1%

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

   15. unpow2 50.1%

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

   16. distribute-frac-neg 50.1%

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

3. Simplified 50.1%

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

4. Taylor expanded in k around inf 72.7%

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

   1. associate-*r/ 72.7%

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

   2. unpow2 72.7%

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

   3. associate-*r* 72.7%

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

   4. *-commutative 72.7%

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

   5. unpow2 72.7%

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

6. Simplified 72.7%

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

7. Taylor expanded in l around 0 72.7%

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

   1. associate-*r* 72.7%

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

   2. *-commutative 72.7%

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

   3. associate-*r* 72.6%

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

   4. unpow2 72.6%

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

   5. times-frac 71.0%

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

   6. unpow2 71.0%

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

   7. associate-*r/ 76.5%

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

   8. associate-*l* 76.5%

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

9. Simplified 76.5%

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

10. Taylor expanded in l around 0 72.7%

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

    1. associate-*r* 72.7%

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

    2. times-frac 73.1%

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

    3. *-commutative 73.1%

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

    4. associate-/r* 71.9%

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

    5. times-frac 71.4%

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

    6. unpow2 71.4%

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

    7. associate-*r* 71.4%

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

    8. unpow2 71.4%

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

    9. associate-*r/ 76.9%

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

    10. times-frac 82.4%

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

    11. associate-/l* 89.3%

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

    12. associate-/r* 89.4%

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

12. Simplified 89.4%

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

13. Taylor expanded in l around 0 72.7%

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

    1. times-frac 72.8%

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

    2. unpow2 72.8%

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

    3. unpow2 72.8%

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

    4. times-frac 91.3%

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

15. Simplified 91.3%

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

16. Final simplification 91.3%

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

Alternative 6: 68.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.3%

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

   1. associate-/r* 37.3%

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

   2. *-commutative 37.3%

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

   3. associate-/r* 41.7%

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

   4. associate-*r/ 41.7%

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

   5. associate-/l* 41.7%

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

   6. +-commutative 41.7%

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

   7. unpow2 41.7%

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

   8. sqr-neg 41.7%

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

   9. distribute-frac-neg 41.7%

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

   10. distribute-frac-neg 41.7%

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

   11. unpow2 41.7%

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

   12. associate--l+ 50.1%

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

   13. metadata-eval 50.1%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   14. +-rgt-identity 50.1%

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

   15. unpow2 50.1%

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

   16. distribute-frac-neg 50.1%

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

3. Simplified 50.1%

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

4. Taylor expanded in k around inf 72.7%

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

   1. associate-*r/ 72.7%

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

   2. unpow2 72.7%

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

   3. associate-*r* 72.7%

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

   4. *-commutative 72.7%

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

   5. unpow2 72.7%

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

6. Simplified 72.7%

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

7. Taylor expanded in l around 0 72.7%

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

   1. associate-*r* 72.7%

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

   2. *-commutative 72.7%

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

   3. associate-*r* 72.6%

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

   4. unpow2 72.6%

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

   5. times-frac 71.0%

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

   6. unpow2 71.0%

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

   7. associate-*r/ 76.5%

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

   8. associate-*l* 76.5%

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

9. Simplified 76.5%

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

10. Taylor expanded in l around 0 72.7%

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

    1. associate-*r* 72.7%

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

    2. times-frac 73.1%

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

    3. *-commutative 73.1%

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

    4. associate-/r* 71.9%

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

    5. times-frac 71.4%

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

    6. unpow2 71.4%

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

    7. associate-*r* 71.4%

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

    8. unpow2 71.4%

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

    9. associate-*r/ 76.9%

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

    10. times-frac 82.4%

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

    11. associate-/l* 89.3%

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

    12. associate-/r* 89.4%

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

12. Simplified 89.4%

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

13. Taylor expanded in k around 0 72.3%

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

    1. unpow2 67.7%

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

15. Simplified 72.3%

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

16. Final simplification 72.3%

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

Alternative 7: 66.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 11600:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\frac{t}{\frac{\ell}{k \cdot k}}} \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 11600.0)
   (/ 2.0 (* (/ (pow k 4.0) l) (/ t l)))
   (* 2.0 (* (/ l (/ t (/ l (* k k)))) -0.16666666666666666))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 11600.0) {
		tmp = 2.0 / ((pow(k, 4.0) / l) * (t / l));
	} else {
		tmp = 2.0 * ((l / (t / (l / (k * k)))) * -0.16666666666666666);
	}
	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 <= 11600.0d0) then
        tmp = 2.0d0 / (((k ** 4.0d0) / l) * (t / l))
    else
        tmp = 2.0d0 * ((l / (t / (l / (k * k)))) * (-0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 11600.0) {
		tmp = 2.0 / ((Math.pow(k, 4.0) / l) * (t / l));
	} else {
		tmp = 2.0 * ((l / (t / (l / (k * k)))) * -0.16666666666666666);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 11600.0:
		tmp = 2.0 / ((math.pow(k, 4.0) / l) * (t / l))
	else:
		tmp = 2.0 * ((l / (t / (l / (k * k)))) * -0.16666666666666666)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 11600.0)
		tmp = Float64(2.0 / Float64(Float64((k ^ 4.0) / l) * Float64(t / l)));
	else
		tmp = Float64(2.0 * Float64(Float64(l / Float64(t / Float64(l / Float64(k * k)))) * -0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 11600.0)
		tmp = 2.0 / (((k ^ 4.0) / l) * (t / l));
	else
		tmp = 2.0 * ((l / (t / (l / (k * k)))) * -0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 11600.0], N[(2.0 / N[(N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / N[(t / N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 11600

   1. Initial program 36.2%

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

      1. associate-/r* 36.2%

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

      2. *-commutative 36.2%

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

      3. associate-/r* 41.2%

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

      4. associate-*r/ 41.2%

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

      5. associate-/l* 41.2%

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

      6. +-commutative 41.2%

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

      7. unpow2 41.2%

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

      8. sqr-neg 41.2%

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

      9. distribute-frac-neg 41.2%

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

      10. distribute-frac-neg 41.2%

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

      11. unpow2 41.2%

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

      12. associate--l+ 49.0%

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

      13. metadata-eval 49.0%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 49.0%

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

      15. unpow2 49.0%

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

      16. distribute-frac-neg 49.0%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in k around 0 65.4%

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

      1. associate-*r/ 65.4%

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

      2. associate-/l* 65.4%

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

      3. unpow2 65.4%

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

   6. Simplified 65.4%

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

      1. times-frac 71.5%

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

   8. Applied egg-rr 71.5%

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

   if 11600 < k

   1. Initial program 40.1%

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

      1. associate-/r* 40.1%

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

      2. *-commutative 40.1%

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

      3. associate-/r* 42.8%

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

      4. associate-*r/ 42.8%

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

      5. associate-/l* 42.8%

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

      6. +-commutative 42.8%

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

      7. unpow2 42.8%

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

      8. sqr-neg 42.8%

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

      9. distribute-frac-neg 42.8%

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

      10. distribute-frac-neg 42.8%

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

      11. unpow2 42.8%

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

      12. associate--l+ 52.6%

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

      13. metadata-eval 52.6%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 52.6%

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

      15. unpow2 52.6%

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

      16. distribute-frac-neg 52.6%

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

   3. Simplified 52.6%

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

   4. Taylor expanded in k around inf 77.6%

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

      1. associate-*r/ 77.6%

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

      2. unpow2 77.6%

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

      3. associate-*r* 77.6%

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

      4. *-commutative 77.6%

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

      5. unpow2 77.6%

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

   6. Simplified 77.6%

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

   7. Taylor expanded in l around 0 77.6%

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

      1. associate-*r* 77.6%

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

      2. *-commutative 77.6%

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

      3. associate-*r* 77.6%

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

      4. unpow2 77.6%

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

      5. times-frac 78.9%

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

      6. unpow2 78.9%

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

      7. associate-*r/ 82.9%

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

      8. associate-*l* 82.9%

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

   9. Simplified 82.9%

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

   10. Taylor expanded in k around 0 70.6%

       \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \color{blue}{\left(\frac{1}{{k}^{4}} - 0.16666666666666666 \cdot \frac{1}{{k}^{2}}\right)}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ 70.6%

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(\frac{1}{{k}^{4}} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{{k}^{2}}}\right)\right) \]

       2. metadata-eval 70.6%

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(\frac{1}{{k}^{4}} - \frac{\color{blue}{0.16666666666666666}}{{k}^{2}}\right)\right) \]

       3. unpow2 70.6%

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(\frac{1}{{k}^{4}} - \frac{0.16666666666666666}{\color{blue}{k \cdot k}}\right)\right) \]

   12. Simplified 70.6%

       \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \color{blue}{\left(\frac{1}{{k}^{4}} - \frac{0.16666666666666666}{k \cdot k}\right)}\right) \]

   13. Taylor expanded in k around inf 69.7%

       \[\leadsto 2 \cdot \color{blue}{\left(-0.16666666666666666 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
   14. Step-by-step derivation

       1. *-commutative 69.7%

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

       2. unpow2 69.7%

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

       3. associate-/l* 70.6%

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

       4. *-commutative 70.6%

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

       5. associate-/l* 70.6%

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

       6. unpow2 70.6%

          \[\leadsto 2 \cdot \left(\frac{\ell}{\frac{t}{\frac{\ell}{\color{blue}{k \cdot k}}}} \cdot -0.16666666666666666\right) \]

   15. Simplified 70.6%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{\frac{t}{\frac{\ell}{k \cdot k}}} \cdot -0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 11600:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\frac{t}{\frac{\ell}{k \cdot k}}} \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 8: 63.5% accurate, 24.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{+54}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\frac{t}{\frac{\ell}{k \cdot k}}} \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.15e+54)
   (* (/ 2.0 (* t (* k k))) (/ (* l l) (* k k)))
   (* 2.0 (* (/ l (/ t (/ l (* k k)))) -0.16666666666666666))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.15e+54) {
		tmp = (2.0 / (t * (k * k))) * ((l * l) / (k * k));
	} else {
		tmp = 2.0 * ((l / (t / (l / (k * k)))) * -0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.15d+54) then
        tmp = (2.0d0 / (t * (k * k))) * ((l * l) / (k * k))
    else
        tmp = 2.0d0 * ((l / (t / (l / (k * k)))) * (-0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.15e+54) {
		tmp = (2.0 / (t * (k * k))) * ((l * l) / (k * k));
	} else {
		tmp = 2.0 * ((l / (t / (l / (k * k)))) * -0.16666666666666666);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.15e+54:
		tmp = (2.0 / (t * (k * k))) * ((l * l) / (k * k))
	else:
		tmp = 2.0 * ((l / (t / (l / (k * k)))) * -0.16666666666666666)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.15e+54)
		tmp = Float64(Float64(2.0 / Float64(t * Float64(k * k))) * Float64(Float64(l * l) / Float64(k * k)));
	else
		tmp = Float64(2.0 * Float64(Float64(l / Float64(t / Float64(l / Float64(k * k)))) * -0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.15e+54)
		tmp = (2.0 / (t * (k * k))) * ((l * l) / (k * k));
	else
		tmp = 2.0 * ((l / (t / (l / (k * k)))) * -0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.15e+54], N[(N[(2.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / N[(t / N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.14999999999999988e54

   1. Initial program 35.8%

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

      1. associate-/r* 35.8%

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

      2. *-commutative 35.8%

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

      3. associate-/r* 40.6%

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

      4. associate-*r/ 40.7%

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

      5. associate-/l* 40.6%

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

      6. +-commutative 40.6%

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

      7. unpow2 40.6%

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

      8. sqr-neg 40.6%

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

      9. distribute-frac-neg 40.6%

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

      10. distribute-frac-neg 40.6%

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

      11. unpow2 40.6%

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

      12. associate--l+ 48.1%

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

      13. metadata-eval 48.1%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 48.1%

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

      15. unpow2 48.1%

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

      16. distribute-frac-neg 48.1%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in k around inf 71.2%

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

      1. associate-*r/ 71.2%

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

      2. unpow2 71.2%

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

      3. associate-*r* 71.2%

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

      4. *-commutative 71.2%

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

      5. unpow2 71.2%

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

   6. Simplified 71.2%

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

   7. Taylor expanded in k around 0 66.9%

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

      1. unpow2 66.9%

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

   9. Simplified 66.9%

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

   10. Taylor expanded in k around 0 64.7%

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

       1. unpow2 64.7%

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

   12. Simplified 64.7%

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

       1. times-frac 65.2%

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

   14. Applied egg-rr 65.2%

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

   if 2.14999999999999988e54 < k

   1. Initial program 41.7%

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

      1. associate-/r* 41.7%

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

      2. *-commutative 41.7%

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

      3. associate-/r* 44.7%

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

      4. associate-*r/ 44.7%

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

      5. associate-/l* 44.7%

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

      6. +-commutative 44.7%

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

      7. unpow2 44.7%

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

      8. sqr-neg 44.7%

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

      9. distribute-frac-neg 44.7%

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

      10. distribute-frac-neg 44.7%

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

      11. unpow2 44.7%

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

      12. associate--l+ 56.0%

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

      13. metadata-eval 56.0%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 56.0%

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

      15. unpow2 56.0%

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

      16. distribute-frac-neg 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in k around inf 77.1%

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

      1. associate-*r/ 77.1%

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

      2. unpow2 77.1%

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

      3. associate-*r* 77.1%

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

      4. *-commutative 77.1%

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

      5. unpow2 77.1%

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

   6. Simplified 77.1%

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

   7. Taylor expanded in l around 0 77.1%

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

      1. associate-*r* 77.1%

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

      2. *-commutative 77.1%

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

      3. associate-*r* 77.1%

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

      4. unpow2 77.1%

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

      5. times-frac 78.5%

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

      6. unpow2 78.5%

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

      7. associate-*r/ 80.4%

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

      8. associate-*l* 80.3%

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

   9. Simplified 80.3%

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

   10. Taylor expanded in k around 0 71.0%

       \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \color{blue}{\left(\frac{1}{{k}^{4}} - 0.16666666666666666 \cdot \frac{1}{{k}^{2}}\right)}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ 71.0%

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(\frac{1}{{k}^{4}} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{{k}^{2}}}\right)\right) \]

       2. metadata-eval 71.0%

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(\frac{1}{{k}^{4}} - \frac{\color{blue}{0.16666666666666666}}{{k}^{2}}\right)\right) \]

       3. unpow2 71.0%

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(\frac{1}{{k}^{4}} - \frac{0.16666666666666666}{\color{blue}{k \cdot k}}\right)\right) \]

   12. Simplified 71.0%

       \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \color{blue}{\left(\frac{1}{{k}^{4}} - \frac{0.16666666666666666}{k \cdot k}\right)}\right) \]

   13. Taylor expanded in k around inf 70.4%

       \[\leadsto 2 \cdot \color{blue}{\left(-0.16666666666666666 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
   14. Step-by-step derivation

       1. *-commutative 70.4%

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

       2. unpow2 70.4%

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

       3. associate-/l* 71.0%

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

       4. *-commutative 71.0%

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

       5. associate-/l* 71.0%

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

       6. unpow2 71.0%

          \[\leadsto 2 \cdot \left(\frac{\ell}{\frac{t}{\frac{\ell}{\color{blue}{k \cdot k}}}} \cdot -0.16666666666666666\right) \]

   15. Simplified 71.0%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{\frac{t}{\frac{\ell}{k \cdot k}}} \cdot -0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{+54}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\frac{t}{\frac{\ell}{k \cdot k}}} \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 9: 41.2% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{+54}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t} \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \frac{-0.16666666666666666}{k \cdot k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.15e+54)
   (* (/ (* l l) t) 0.08333333333333333)
   (* 2.0 (* (* l (/ l t)) (/ -0.16666666666666666 (* k k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.15e+54) {
		tmp = ((l * l) / t) * 0.08333333333333333;
	} else {
		tmp = 2.0 * ((l * (l / t)) * (-0.16666666666666666 / (k * k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.15d+54) then
        tmp = ((l * l) / t) * 0.08333333333333333d0
    else
        tmp = 2.0d0 * ((l * (l / t)) * ((-0.16666666666666666d0) / (k * k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.15e+54) {
		tmp = ((l * l) / t) * 0.08333333333333333;
	} else {
		tmp = 2.0 * ((l * (l / t)) * (-0.16666666666666666 / (k * k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.15e+54:
		tmp = ((l * l) / t) * 0.08333333333333333
	else:
		tmp = 2.0 * ((l * (l / t)) * (-0.16666666666666666 / (k * k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.15e+54)
		tmp = Float64(Float64(Float64(l * l) / t) * 0.08333333333333333);
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / t)) * Float64(-0.16666666666666666 / Float64(k * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.15e+54)
		tmp = ((l * l) / t) * 0.08333333333333333;
	else
		tmp = 2.0 * ((l * (l / t)) * (-0.16666666666666666 / (k * k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.15e+54], N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * 0.08333333333333333), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.14999999999999988e54

   1. Initial program 35.8%

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

      1. associate-/r* 35.8%

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

      2. *-commutative 35.8%

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

      3. associate-/r* 40.6%

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

      4. associate-*r/ 40.7%

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

      5. associate-/l* 40.6%

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

      6. +-commutative 40.6%

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

      7. unpow2 40.6%

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

      8. sqr-neg 40.6%

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

      9. distribute-frac-neg 40.6%

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

      10. distribute-frac-neg 40.6%

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

      11. unpow2 40.6%

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

      12. associate--l+ 48.1%

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

      13. metadata-eval 48.1%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 48.1%

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

      15. unpow2 48.1%

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

      16. distribute-frac-neg 48.1%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in k around inf 71.2%

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

      1. associate-*r/ 71.2%

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

      2. unpow2 71.2%

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

      3. associate-*r* 71.2%

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

      4. *-commutative 71.2%

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

      5. unpow2 71.2%

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

   6. Simplified 71.2%

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

   7. Taylor expanded in k around 0 66.9%

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

      1. unpow2 66.9%

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

   9. Simplified 66.9%

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

   10. Taylor expanded in k around 0 38.6%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(-0.5 \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + \left(0.041666666666666664 \cdot \left({k}^{4} \cdot {\ell}^{2}\right) + {\ell}^{2}\right)\right)}}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]
   11. Step-by-step derivation

       1. associate-+r+ 38.6%

          \[\leadsto \frac{2 \cdot \color{blue}{\left(\left(-0.5 \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 0.041666666666666664 \cdot \left({k}^{4} \cdot {\ell}^{2}\right)\right) + {\ell}^{2}\right)}}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

       2. +-commutative 38.6%

          \[\leadsto \frac{2 \cdot \color{blue}{\left({\ell}^{2} + \left(-0.5 \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 0.041666666666666664 \cdot \left({k}^{4} \cdot {\ell}^{2}\right)\right)\right)}}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

       3. unpow2 38.6%

          \[\leadsto \frac{2 \cdot \left(\color{blue}{\ell \cdot \ell} + \left(-0.5 \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 0.041666666666666664 \cdot \left({k}^{4} \cdot {\ell}^{2}\right)\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

       4. associate-*r* 38.6%

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \left(\color{blue}{\left(-0.5 \cdot {k}^{2}\right) \cdot {\ell}^{2}} + 0.041666666666666664 \cdot \left({k}^{4} \cdot {\ell}^{2}\right)\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

       5. associate-*r* 38.6%

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \left(\left(-0.5 \cdot {k}^{2}\right) \cdot {\ell}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {k}^{4}\right) \cdot {\ell}^{2}}\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

       6. distribute-rgt-out 40.2%

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \color{blue}{{\ell}^{2} \cdot \left(-0.5 \cdot {k}^{2} + 0.041666666666666664 \cdot {k}^{4}\right)}\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

       7. unpow2 40.2%

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(-0.5 \cdot {k}^{2} + 0.041666666666666664 \cdot {k}^{4}\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

       8. *-commutative 40.2%

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{{k}^{2} \cdot -0.5} + 0.041666666666666664 \cdot {k}^{4}\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

       9. unpow2 40.2%

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot -0.5 + 0.041666666666666664 \cdot {k}^{4}\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

   12. Simplified 40.2%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell + \left(\ell \cdot \ell\right) \cdot \left(\left(k \cdot k\right) \cdot -0.5 + 0.041666666666666664 \cdot {k}^{4}\right)\right)}}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

   13. Taylor expanded in k around inf 34.9%

       \[\leadsto \color{blue}{0.08333333333333333 \cdot \frac{{\ell}^{2}}{t}} \]
   14. Step-by-step derivation

       1. *-commutative 34.9%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{t} \cdot 0.08333333333333333} \]

       2. unpow2 34.9%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{t} \cdot 0.08333333333333333 \]

   15. Simplified 34.9%

       \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{t} \cdot 0.08333333333333333} \]

   if 2.14999999999999988e54 < k

   1. Initial program 41.7%

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

      1. associate-/r* 41.7%

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

      2. *-commutative 41.7%

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

      3. associate-/r* 44.7%

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

      4. associate-*r/ 44.7%

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

      5. associate-/l* 44.7%

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

      6. +-commutative 44.7%

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

      7. unpow2 44.7%

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

      8. sqr-neg 44.7%

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

      9. distribute-frac-neg 44.7%

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

      10. distribute-frac-neg 44.7%

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

      11. unpow2 44.7%

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

      12. associate--l+ 56.0%

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

      13. metadata-eval 56.0%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 56.0%

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

      15. unpow2 56.0%

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

      16. distribute-frac-neg 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in k around inf 77.1%

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

      1. associate-*r/ 77.1%

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

      2. unpow2 77.1%

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

      3. associate-*r* 77.1%

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

      4. *-commutative 77.1%

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

      5. unpow2 77.1%

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

   6. Simplified 77.1%

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

   7. Taylor expanded in l around 0 77.1%

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

      1. associate-*r* 77.1%

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

      2. *-commutative 77.1%

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

      3. associate-*r* 77.1%

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

      4. unpow2 77.1%

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

      5. times-frac 78.5%

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

      6. unpow2 78.5%

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

      7. associate-*r/ 80.4%

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

      8. associate-*l* 80.3%

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

   9. Simplified 80.3%

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

   10. Taylor expanded in k around 0 71.0%

       \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \color{blue}{\left(\frac{1}{{k}^{4}} - 0.16666666666666666 \cdot \frac{1}{{k}^{2}}\right)}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ 71.0%

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(\frac{1}{{k}^{4}} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{{k}^{2}}}\right)\right) \]

       2. metadata-eval 71.0%

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(\frac{1}{{k}^{4}} - \frac{\color{blue}{0.16666666666666666}}{{k}^{2}}\right)\right) \]

       3. unpow2 71.0%

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(\frac{1}{{k}^{4}} - \frac{0.16666666666666666}{\color{blue}{k \cdot k}}\right)\right) \]

   12. Simplified 71.0%

       \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \color{blue}{\left(\frac{1}{{k}^{4}} - \frac{0.16666666666666666}{k \cdot k}\right)}\right) \]

   13. Taylor expanded in k around inf 71.0%

       \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \color{blue}{\frac{-0.16666666666666666}{{k}^{2}}}\right) \]
   14. Step-by-step derivation

       1. unpow2 71.0%

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \frac{-0.16666666666666666}{\color{blue}{k \cdot k}}\right) \]

   15. Simplified 71.0%

       \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \color{blue}{\frac{-0.16666666666666666}{k \cdot k}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{+54}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t} \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \frac{-0.16666666666666666}{k \cdot k}\right)\\ \end{array} \]

Alternative 10: 41.4% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+54}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t} \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\frac{t}{\frac{\ell}{k \cdot k}}} \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2e+54)
   (* (/ (* l l) t) 0.08333333333333333)
   (* 2.0 (* (/ l (/ t (/ l (* k k)))) -0.16666666666666666))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2e+54) {
		tmp = ((l * l) / t) * 0.08333333333333333;
	} else {
		tmp = 2.0 * ((l / (t / (l / (k * k)))) * -0.16666666666666666);
	}
	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 <= 2d+54) then
        tmp = ((l * l) / t) * 0.08333333333333333d0
    else
        tmp = 2.0d0 * ((l / (t / (l / (k * k)))) * (-0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2e+54) {
		tmp = ((l * l) / t) * 0.08333333333333333;
	} else {
		tmp = 2.0 * ((l / (t / (l / (k * k)))) * -0.16666666666666666);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2e+54:
		tmp = ((l * l) / t) * 0.08333333333333333
	else:
		tmp = 2.0 * ((l / (t / (l / (k * k)))) * -0.16666666666666666)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2e+54)
		tmp = Float64(Float64(Float64(l * l) / t) * 0.08333333333333333);
	else
		tmp = Float64(2.0 * Float64(Float64(l / Float64(t / Float64(l / Float64(k * k)))) * -0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2e+54)
		tmp = ((l * l) / t) * 0.08333333333333333;
	else
		tmp = 2.0 * ((l / (t / (l / (k * k)))) * -0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2e+54], N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * 0.08333333333333333), $MachinePrecision], N[(2.0 * N[(N[(l / N[(t / N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.0000000000000002e54

   1. Initial program 35.8%

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

      1. associate-/r* 35.8%

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

      2. *-commutative 35.8%

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

      3. associate-/r* 40.6%

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

      4. associate-*r/ 40.7%

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

      5. associate-/l* 40.6%

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

      6. +-commutative 40.6%

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

      7. unpow2 40.6%

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

      8. sqr-neg 40.6%

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

      9. distribute-frac-neg 40.6%

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

      10. distribute-frac-neg 40.6%

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

      11. unpow2 40.6%

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

      12. associate--l+ 48.1%

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

      13. metadata-eval 48.1%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 48.1%

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

      15. unpow2 48.1%

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

      16. distribute-frac-neg 48.1%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in k around inf 71.2%

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

      1. associate-*r/ 71.2%

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

      2. unpow2 71.2%

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

      3. associate-*r* 71.2%

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

      4. *-commutative 71.2%

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

      5. unpow2 71.2%

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

   6. Simplified 71.2%

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

   7. Taylor expanded in k around 0 66.9%

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

      1. unpow2 66.9%

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

   9. Simplified 66.9%

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

   10. Taylor expanded in k around 0 38.6%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(-0.5 \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + \left(0.041666666666666664 \cdot \left({k}^{4} \cdot {\ell}^{2}\right) + {\ell}^{2}\right)\right)}}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]
   11. Step-by-step derivation

       1. associate-+r+ 38.6%

          \[\leadsto \frac{2 \cdot \color{blue}{\left(\left(-0.5 \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 0.041666666666666664 \cdot \left({k}^{4} \cdot {\ell}^{2}\right)\right) + {\ell}^{2}\right)}}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

       2. +-commutative 38.6%

          \[\leadsto \frac{2 \cdot \color{blue}{\left({\ell}^{2} + \left(-0.5 \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 0.041666666666666664 \cdot \left({k}^{4} \cdot {\ell}^{2}\right)\right)\right)}}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

       3. unpow2 38.6%

          \[\leadsto \frac{2 \cdot \left(\color{blue}{\ell \cdot \ell} + \left(-0.5 \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 0.041666666666666664 \cdot \left({k}^{4} \cdot {\ell}^{2}\right)\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

       4. associate-*r* 38.6%

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \left(\color{blue}{\left(-0.5 \cdot {k}^{2}\right) \cdot {\ell}^{2}} + 0.041666666666666664 \cdot \left({k}^{4} \cdot {\ell}^{2}\right)\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

       5. associate-*r* 38.6%

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \left(\left(-0.5 \cdot {k}^{2}\right) \cdot {\ell}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {k}^{4}\right) \cdot {\ell}^{2}}\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

       6. distribute-rgt-out 40.2%

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \color{blue}{{\ell}^{2} \cdot \left(-0.5 \cdot {k}^{2} + 0.041666666666666664 \cdot {k}^{4}\right)}\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

       7. unpow2 40.2%

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(-0.5 \cdot {k}^{2} + 0.041666666666666664 \cdot {k}^{4}\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

       8. *-commutative 40.2%

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{{k}^{2} \cdot -0.5} + 0.041666666666666664 \cdot {k}^{4}\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

       9. unpow2 40.2%

          \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot -0.5 + 0.041666666666666664 \cdot {k}^{4}\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

   12. Simplified 40.2%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell + \left(\ell \cdot \ell\right) \cdot \left(\left(k \cdot k\right) \cdot -0.5 + 0.041666666666666664 \cdot {k}^{4}\right)\right)}}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

   13. Taylor expanded in k around inf 34.9%

       \[\leadsto \color{blue}{0.08333333333333333 \cdot \frac{{\ell}^{2}}{t}} \]
   14. Step-by-step derivation

       1. *-commutative 34.9%

          \[\leadsto \color{blue}{\frac{{\ell}^{2}}{t} \cdot 0.08333333333333333} \]

       2. unpow2 34.9%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{t} \cdot 0.08333333333333333 \]

   15. Simplified 34.9%

       \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{t} \cdot 0.08333333333333333} \]

   if 2.0000000000000002e54 < k

   1. Initial program 41.7%

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

      1. associate-/r* 41.7%

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

      2. *-commutative 41.7%

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

      3. associate-/r* 44.7%

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

      4. associate-*r/ 44.7%

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

      5. associate-/l* 44.7%

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

      6. +-commutative 44.7%

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

      7. unpow2 44.7%

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

      8. sqr-neg 44.7%

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

      9. distribute-frac-neg 44.7%

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

      10. distribute-frac-neg 44.7%

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

      11. unpow2 44.7%

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

      12. associate--l+ 56.0%

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

      13. metadata-eval 56.0%

          \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      14. +-rgt-identity 56.0%

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

      15. unpow2 56.0%

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

      16. distribute-frac-neg 56.0%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in k around inf 77.1%

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

      1. associate-*r/ 77.1%

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

      2. unpow2 77.1%

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

      3. associate-*r* 77.1%

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

      4. *-commutative 77.1%

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

      5. unpow2 77.1%

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

   6. Simplified 77.1%

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

   7. Taylor expanded in l around 0 77.1%

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

      1. associate-*r* 77.1%

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

      2. *-commutative 77.1%

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

      3. associate-*r* 77.1%

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

      4. unpow2 77.1%

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

      5. times-frac 78.5%

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

      6. unpow2 78.5%

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

      7. associate-*r/ 80.4%

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

      8. associate-*l* 80.3%

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

   9. Simplified 80.3%

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

   10. Taylor expanded in k around 0 71.0%

       \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \color{blue}{\left(\frac{1}{{k}^{4}} - 0.16666666666666666 \cdot \frac{1}{{k}^{2}}\right)}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ 71.0%

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(\frac{1}{{k}^{4}} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{{k}^{2}}}\right)\right) \]

       2. metadata-eval 71.0%

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(\frac{1}{{k}^{4}} - \frac{\color{blue}{0.16666666666666666}}{{k}^{2}}\right)\right) \]

       3. unpow2 71.0%

          \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \left(\frac{1}{{k}^{4}} - \frac{0.16666666666666666}{\color{blue}{k \cdot k}}\right)\right) \]

   12. Simplified 71.0%

       \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{t}\right) \cdot \color{blue}{\left(\frac{1}{{k}^{4}} - \frac{0.16666666666666666}{k \cdot k}\right)}\right) \]

   13. Taylor expanded in k around inf 70.4%

       \[\leadsto 2 \cdot \color{blue}{\left(-0.16666666666666666 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
   14. Step-by-step derivation

       1. *-commutative 70.4%

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

       2. unpow2 70.4%

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

       3. associate-/l* 71.0%

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

       4. *-commutative 71.0%

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

       5. associate-/l* 71.0%

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

       6. unpow2 71.0%

          \[\leadsto 2 \cdot \left(\frac{\ell}{\frac{t}{\frac{\ell}{\color{blue}{k \cdot k}}}} \cdot -0.16666666666666666\right) \]

   15. Simplified 71.0%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{\frac{t}{\frac{\ell}{k \cdot k}}} \cdot -0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+54}:\\ \;\;\;\;\frac{\ell \cdot \ell}{t} \cdot 0.08333333333333333\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\frac{t}{\frac{\ell}{k \cdot k}}} \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 11: 31.1% accurate, 60.1× speedup

Math

\[\begin{array}{l} \\ 0.08333333333333333 \cdot \frac{\ell}{\frac{t}{\ell}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 0.08333333333333333 (/ l (/ t l))))

C

double code(double t, double l, double k) {
	return 0.08333333333333333 * (l / (t / l));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 0.08333333333333333d0 * (l / (t / l))
end function

Java

public static double code(double t, double l, double k) {
	return 0.08333333333333333 * (l / (t / l));
}

Python

def code(t, l, k):
	return 0.08333333333333333 * (l / (t / l))

Julia

function code(t, l, k)
	return Float64(0.08333333333333333 * Float64(l / Float64(t / l)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 0.08333333333333333 * (l / (t / l));
end

Wolfram

code[t_, l_, k_] := N[(0.08333333333333333 * N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.3%

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

   1. associate-/r* 37.3%

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

   2. *-commutative 37.3%

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

   3. associate-/r* 41.7%

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

   4. associate-*r/ 41.7%

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

   5. associate-/l* 41.7%

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

   6. +-commutative 41.7%

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

   7. unpow2 41.7%

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

   8. sqr-neg 41.7%

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

   9. distribute-frac-neg 41.7%

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

   10. distribute-frac-neg 41.7%

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

   11. unpow2 41.7%

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

   12. associate--l+ 50.1%

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

   13. metadata-eval 50.1%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   14. +-rgt-identity 50.1%

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

   15. unpow2 50.1%

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

   16. distribute-frac-neg 50.1%

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

3. Simplified 50.1%

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

4. Taylor expanded in k around inf 72.7%

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

   1. associate-*r/ 72.7%

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

   2. unpow2 72.7%

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

   3. associate-*r* 72.7%

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

   4. *-commutative 72.7%

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

   5. unpow2 72.7%

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

6. Simplified 72.7%

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

7. Taylor expanded in k around 0 67.7%

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

   1. unpow2 67.7%

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

9. Simplified 67.7%

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

10. Taylor expanded in k around 0 28.9%

    \[\leadsto \frac{2 \cdot \color{blue}{\left(-0.5 \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + \left(0.041666666666666664 \cdot \left({k}^{4} \cdot {\ell}^{2}\right) + {\ell}^{2}\right)\right)}}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]
11. Step-by-step derivation

    1. associate-+r+ 28.9%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(\left(-0.5 \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 0.041666666666666664 \cdot \left({k}^{4} \cdot {\ell}^{2}\right)\right) + {\ell}^{2}\right)}}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

    2. +-commutative 28.9%

       \[\leadsto \frac{2 \cdot \color{blue}{\left({\ell}^{2} + \left(-0.5 \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 0.041666666666666664 \cdot \left({k}^{4} \cdot {\ell}^{2}\right)\right)\right)}}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

    3. unpow2 28.9%

       \[\leadsto \frac{2 \cdot \left(\color{blue}{\ell \cdot \ell} + \left(-0.5 \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 0.041666666666666664 \cdot \left({k}^{4} \cdot {\ell}^{2}\right)\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

    4. associate-*r* 28.9%

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \left(\color{blue}{\left(-0.5 \cdot {k}^{2}\right) \cdot {\ell}^{2}} + 0.041666666666666664 \cdot \left({k}^{4} \cdot {\ell}^{2}\right)\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

    5. associate-*r* 28.9%

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \left(\left(-0.5 \cdot {k}^{2}\right) \cdot {\ell}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {k}^{4}\right) \cdot {\ell}^{2}}\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

    6. distribute-rgt-out 30.5%

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \color{blue}{{\ell}^{2} \cdot \left(-0.5 \cdot {k}^{2} + 0.041666666666666664 \cdot {k}^{4}\right)}\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

    7. unpow2 30.5%

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(-0.5 \cdot {k}^{2} + 0.041666666666666664 \cdot {k}^{4}\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

    8. *-commutative 30.5%

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{{k}^{2} \cdot -0.5} + 0.041666666666666664 \cdot {k}^{4}\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

    9. unpow2 30.5%

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot -0.5 + 0.041666666666666664 \cdot {k}^{4}\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

12. Simplified 30.5%

    \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell + \left(\ell \cdot \ell\right) \cdot \left(\left(k \cdot k\right) \cdot -0.5 + 0.041666666666666664 \cdot {k}^{4}\right)\right)}}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

13. Taylor expanded in k around inf 38.2%

    \[\leadsto \color{blue}{0.08333333333333333 \cdot \frac{{\ell}^{2}}{t}} \]
14. Step-by-step derivation

    1. *-commutative 38.2%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{t} \cdot 0.08333333333333333} \]

    2. unpow2 38.2%

       \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{t} \cdot 0.08333333333333333 \]

    3. associate-/l* 33.7%

       \[\leadsto \color{blue}{\frac{\ell}{\frac{t}{\ell}}} \cdot 0.08333333333333333 \]

15. Simplified 33.7%

    \[\leadsto \color{blue}{\frac{\ell}{\frac{t}{\ell}} \cdot 0.08333333333333333} \]

16. Final simplification 33.7%

    \[\leadsto 0.08333333333333333 \cdot \frac{\ell}{\frac{t}{\ell}} \]

Alternative 12: 36.1% accurate, 60.1× speedup

Math

\[\begin{array}{l} \\ \frac{\ell \cdot \ell}{t} \cdot 0.08333333333333333 \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ (* l l) t) 0.08333333333333333))

C

double code(double t, double l, double k) {
	return ((l * l) / t) * 0.08333333333333333;
}

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) * 0.08333333333333333d0
end function

Java

public static double code(double t, double l, double k) {
	return ((l * l) / t) * 0.08333333333333333;
}

Python

def code(t, l, k):
	return ((l * l) / t) * 0.08333333333333333

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l * l) / t) * 0.08333333333333333)
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l * l) / t) * 0.08333333333333333;
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]

Derivation

1. Initial program 37.3%

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

   1. associate-/r* 37.3%

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

   2. *-commutative 37.3%

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

   3. associate-/r* 41.7%

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

   4. associate-*r/ 41.7%

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

   5. associate-/l* 41.7%

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

   6. +-commutative 41.7%

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

   7. unpow2 41.7%

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

   8. sqr-neg 41.7%

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

   9. distribute-frac-neg 41.7%

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

   10. distribute-frac-neg 41.7%

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

   11. unpow2 41.7%

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

   12. associate--l+ 50.1%

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

   13. metadata-eval 50.1%

       \[\leadsto \frac{\frac{2}{\frac{\sin k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}} \cdot \tan k}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   14. +-rgt-identity 50.1%

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

   15. unpow2 50.1%

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

   16. distribute-frac-neg 50.1%

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

3. Simplified 50.1%

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

4. Taylor expanded in k around inf 72.7%

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

   1. associate-*r/ 72.7%

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

   2. unpow2 72.7%

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

   3. associate-*r* 72.7%

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

   4. *-commutative 72.7%

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

   5. unpow2 72.7%

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

6. Simplified 72.7%

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

7. Taylor expanded in k around 0 67.7%

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

   1. unpow2 67.7%

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

9. Simplified 67.7%

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

10. Taylor expanded in k around 0 28.9%

    \[\leadsto \frac{2 \cdot \color{blue}{\left(-0.5 \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + \left(0.041666666666666664 \cdot \left({k}^{4} \cdot {\ell}^{2}\right) + {\ell}^{2}\right)\right)}}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]
11. Step-by-step derivation

    1. associate-+r+ 28.9%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(\left(-0.5 \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 0.041666666666666664 \cdot \left({k}^{4} \cdot {\ell}^{2}\right)\right) + {\ell}^{2}\right)}}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

    2. +-commutative 28.9%

       \[\leadsto \frac{2 \cdot \color{blue}{\left({\ell}^{2} + \left(-0.5 \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 0.041666666666666664 \cdot \left({k}^{4} \cdot {\ell}^{2}\right)\right)\right)}}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

    3. unpow2 28.9%

       \[\leadsto \frac{2 \cdot \left(\color{blue}{\ell \cdot \ell} + \left(-0.5 \cdot \left({k}^{2} \cdot {\ell}^{2}\right) + 0.041666666666666664 \cdot \left({k}^{4} \cdot {\ell}^{2}\right)\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

    4. associate-*r* 28.9%

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \left(\color{blue}{\left(-0.5 \cdot {k}^{2}\right) \cdot {\ell}^{2}} + 0.041666666666666664 \cdot \left({k}^{4} \cdot {\ell}^{2}\right)\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

    5. associate-*r* 28.9%

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \left(\left(-0.5 \cdot {k}^{2}\right) \cdot {\ell}^{2} + \color{blue}{\left(0.041666666666666664 \cdot {k}^{4}\right) \cdot {\ell}^{2}}\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

    6. distribute-rgt-out 30.5%

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \color{blue}{{\ell}^{2} \cdot \left(-0.5 \cdot {k}^{2} + 0.041666666666666664 \cdot {k}^{4}\right)}\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

    7. unpow2 30.5%

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(-0.5 \cdot {k}^{2} + 0.041666666666666664 \cdot {k}^{4}\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

    8. *-commutative 30.5%

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{{k}^{2} \cdot -0.5} + 0.041666666666666664 \cdot {k}^{4}\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

    9. unpow2 30.5%

       \[\leadsto \frac{2 \cdot \left(\ell \cdot \ell + \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot -0.5 + 0.041666666666666664 \cdot {k}^{4}\right)\right)}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

12. Simplified 30.5%

    \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell + \left(\ell \cdot \ell\right) \cdot \left(\left(k \cdot k\right) \cdot -0.5 + 0.041666666666666664 \cdot {k}^{4}\right)\right)}}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)} \]

13. Taylor expanded in k around inf 38.2%

    \[\leadsto \color{blue}{0.08333333333333333 \cdot \frac{{\ell}^{2}}{t}} \]
14. Step-by-step derivation

    1. *-commutative 38.2%

       \[\leadsto \color{blue}{\frac{{\ell}^{2}}{t} \cdot 0.08333333333333333} \]

    2. unpow2 38.2%

       \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{t} \cdot 0.08333333333333333 \]

15. Simplified 38.2%

    \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{t} \cdot 0.08333333333333333} \]

16. Final simplification 38.2%

    \[\leadsto \frac{\ell \cdot \ell}{t} \cdot 0.08333333333333333 \]

Reproduce

herbie shell --seed 2023293 
(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))))