Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.9% → 96.0%

Time: 19.1s

Alternatives: 14

Speedup: 28.1×

• 35.0% 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: 34.9% 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: 96.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+254}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot 2}{k \cdot t}}{k}}{\sin k} \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot {\left(\frac{\ell}{k}\right)}^{2}}{t}}{\sin k \cdot \tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e+254)
   (* (/ (/ (/ (* l 2.0) (* k t)) k) (sin k)) (/ l (tan k)))
   (/ (/ (* 2.0 (pow (/ l k) 2.0)) t) (* (sin k) (tan k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+254) {
		tmp = ((((l * 2.0) / (k * t)) / k) / sin(k)) * (l / tan(k));
	} else {
		tmp = ((2.0 * pow((l / k), 2.0)) / t) / (sin(k) * tan(k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d+254) then
        tmp = ((((l * 2.0d0) / (k * t)) / k) / sin(k)) * (l / tan(k))
    else
        tmp = ((2.0d0 * ((l / k) ** 2.0d0)) / t) / (sin(k) * tan(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+254) {
		tmp = ((((l * 2.0) / (k * t)) / k) / Math.sin(k)) * (l / Math.tan(k));
	} else {
		tmp = ((2.0 * Math.pow((l / k), 2.0)) / t) / (Math.sin(k) * Math.tan(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e+254:
		tmp = ((((l * 2.0) / (k * t)) / k) / math.sin(k)) * (l / math.tan(k))
	else:
		tmp = ((2.0 * math.pow((l / k), 2.0)) / t) / (math.sin(k) * math.tan(k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e+254)
		tmp = Float64(Float64(Float64(Float64(Float64(l * 2.0) / Float64(k * t)) / k) / sin(k)) * Float64(l / tan(k)));
	else
		tmp = Float64(Float64(Float64(2.0 * (Float64(l / k) ^ 2.0)) / t) / Float64(sin(k) * tan(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e+254)
		tmp = ((((l * 2.0) / (k * t)) / k) / sin(k)) * (l / tan(k));
	else
		tmp = ((2.0 * ((l / k) ^ 2.0)) / t) / (sin(k) * tan(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e+254], N[(N[(N[(N[(N[(l * 2.0), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.9999999999999994e253

   1. Initial program 38.1%

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

      1. associate-*l* 38.1%

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

      2. associate-*l* 38.1%

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

      3. associate-/r* 38.1%

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

      4. associate-/r/ 38.1%

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

      5. *-commutative 38.1%

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

      6. times-frac 39.3%

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

      7. +-commutative 39.3%

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

      8. associate--l+ 49.2%

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

      9. metadata-eval 49.2%

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

      10. +-rgt-identity 49.2%

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

      11. times-frac 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in t around 0 85.6%

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

      1. unpow2 85.6%

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

   6. Simplified 85.6%

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

      1. associate-*l/ 85.6%

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

      2. associate-*l* 90.4%

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

   8. Applied egg-rr 90.4%

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

      1. associate-*l/ 89.9%

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

      2. associate-*r* 92.7%

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

   10. Simplified 92.7%

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

   11. Taylor expanded in k around inf 88.4%

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

       1. *-commutative 88.4%

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

       2. associate-*r* 88.4%

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

       3. unpow2 88.4%

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

       4. associate-*r* 91.1%

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

       5. associate-*r/ 91.1%

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

       6. associate-/r* 93.2%

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

       7. *-commutative 93.2%

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

       8. associate-/r* 97.6%

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

   13. Simplified 97.6%

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

   if 9.9999999999999994e253 < (*.f64 l l)

   1. Initial program 29.0%

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

      1. associate-*l* 29.0%

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

      2. associate-*l* 29.0%

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

      3. associate-/r* 29.0%

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

      4. associate-/r/ 29.0%

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

      5. *-commutative 29.0%

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

      6. times-frac 28.9%

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

      7. +-commutative 28.9%

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

      8. associate--l+ 29.1%

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

      9. metadata-eval 29.1%

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

      10. +-rgt-identity 29.1%

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

      11. times-frac 29.1%

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

   3. Simplified 29.1%

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

   4. Taylor expanded in t around 0 52.0%

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

      1. unpow2 52.0%

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

   6. Simplified 52.0%

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

      1. associate-*l/ 52.0%

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

      2. associate-*l* 57.0%

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

   8. Applied egg-rr 57.0%

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

      1. associate-*l/ 57.0%

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

      2. associate-*r* 71.3%

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

   10. Simplified 71.3%

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

   11. Taylor expanded in k around inf 63.1%

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

       1. *-commutative 63.1%

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

       2. associate-*r* 63.1%

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

       3. unpow2 63.1%

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

       4. associate-*r* 71.2%

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

       5. associate-*r/ 71.2%

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

       6. associate-/r* 71.3%

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

       7. *-commutative 71.3%

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

       8. associate-/r* 87.3%

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

   13. Simplified 87.3%

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

       1. frac-times 87.3%

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

       2. associate-/l/ 71.3%

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

   15. Applied egg-rr 71.3%

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

       1. associate-*l/ 57.0%

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

       2. associate-*r* 52.0%

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

       3. unpow2 52.0%

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

       4. associate-/r* 54.4%

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

       5. associate-*r* 54.4%

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

       6. unpow2 54.4%

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

       7. associate-*r/ 54.4%

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

       8. unpow2 54.4%

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

       9. unpow2 54.4%

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

       10. times-frac 96.2%

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

       11. unpow2 96.2%

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

   17. Simplified 96.2%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+254}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot 2}{k \cdot t}}{k}}{\sin k} \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot {\left(\frac{\ell}{k}\right)}^{2}}{t}}{\sin k \cdot \tan k}\\ \end{array} \]

Alternative 2: 89.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{\sin k}\\ t_2 := \frac{\ell}{\tan k}\\ \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+281}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{t_2 \cdot t_1}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(t_1 \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (sin k))) (t_2 (/ l (tan k))))
   (if (<= (* l l) 2e+281)
     (* (/ 2.0 k) (/ (* t_2 t_1) (* k t)))
     (* t_2 (* t_1 (/ 2.0 (* k (* k t))))))))

C

double code(double t, double l, double k) {
	double t_1 = l / sin(k);
	double t_2 = l / tan(k);
	double tmp;
	if ((l * l) <= 2e+281) {
		tmp = (2.0 / k) * ((t_2 * t_1) / (k * t));
	} else {
		tmp = t_2 * (t_1 * (2.0 / (k * (k * t))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = l / sin(k)
    t_2 = l / tan(k)
    if ((l * l) <= 2d+281) then
        tmp = (2.0d0 / k) * ((t_2 * t_1) / (k * t))
    else
        tmp = t_2 * (t_1 * (2.0d0 / (k * (k * t))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l / Math.sin(k);
	double t_2 = l / Math.tan(k);
	double tmp;
	if ((l * l) <= 2e+281) {
		tmp = (2.0 / k) * ((t_2 * t_1) / (k * t));
	} else {
		tmp = t_2 * (t_1 * (2.0 / (k * (k * t))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l / math.sin(k)
	t_2 = l / math.tan(k)
	tmp = 0
	if (l * l) <= 2e+281:
		tmp = (2.0 / k) * ((t_2 * t_1) / (k * t))
	else:
		tmp = t_2 * (t_1 * (2.0 / (k * (k * t))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / sin(k))
	t_2 = Float64(l / tan(k))
	tmp = 0.0
	if (Float64(l * l) <= 2e+281)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(t_2 * t_1) / Float64(k * t)));
	else
		tmp = Float64(t_2 * Float64(t_1 * Float64(2.0 / Float64(k * Float64(k * t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l / sin(k);
	t_2 = l / tan(k);
	tmp = 0.0;
	if ((l * l) <= 2e+281)
		tmp = (2.0 / k) * ((t_2 * t_1) / (k * t));
	else
		tmp = t_2 * (t_1 * (2.0 / (k * (k * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 2e+281], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(t$95$2 * t$95$1), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(t$95$1 * N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2.0000000000000001e281

   1. Initial program 37.2%

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

      1. associate-*l* 37.2%

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

      2. associate-*l* 37.2%

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

      3. associate-/r* 37.2%

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

      4. associate-/r/ 37.2%

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

      5. *-commutative 37.2%

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

      6. times-frac 38.3%

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

      7. +-commutative 38.3%

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

      8. associate--l+ 47.9%

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

      9. metadata-eval 47.9%

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

      10. +-rgt-identity 47.9%

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

      11. times-frac 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in t around 0 84.0%

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

      1. unpow2 84.0%

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

   6. Simplified 84.0%

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

      1. associate-*l/ 84.1%

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

      2. associate-*l* 88.6%

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

   8. Applied egg-rr 88.6%

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

      1. times-frac 93.4%

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

   10. Applied egg-rr 93.4%

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

   if 2.0000000000000001e281 < (*.f64 l l)

   1. Initial program 30.2%

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

      1. associate-*l* 30.2%

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

      2. associate-*l* 30.2%

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

      3. associate-/r* 30.2%

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

      4. associate-/r/ 30.2%

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

      5. *-commutative 30.2%

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

      6. times-frac 30.1%

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

      7. +-commutative 30.1%

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

      8. associate--l+ 30.2%

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

      9. metadata-eval 30.2%

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

      10. +-rgt-identity 30.2%

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

      11. times-frac 30.2%

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

   3. Simplified 30.2%

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

   4. Taylor expanded in t around 0 52.5%

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

      1. unpow2 52.5%

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

   6. Simplified 52.5%

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

      1. associate-*l/ 52.5%

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

      2. associate-*l* 58.0%

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

   8. Applied egg-rr 58.0%

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

      1. associate-*l/ 58.0%

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

      2. associate-*r* 73.6%

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

   10. Simplified 73.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+281}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell}{\tan k} \cdot \frac{\ell}{\sin k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\\ \end{array} \]

Alternative 3: 88.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot 2}{k \cdot t}}{k}}{\sin k} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{\sin k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e-69)
   (* (/ (/ (/ (* l 2.0) (* k t)) k) (sin k)) (/ l k))
   (* (/ l (tan k)) (/ (/ (* (/ 2.0 k) (/ l t)) k) (sin k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-69) {
		tmp = ((((l * 2.0) / (k * t)) / k) / sin(k)) * (l / k);
	} else {
		tmp = (l / tan(k)) * ((((2.0 / k) * (l / t)) / k) / sin(k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 5d-69) then
        tmp = ((((l * 2.0d0) / (k * t)) / k) / sin(k)) * (l / k)
    else
        tmp = (l / tan(k)) * ((((2.0d0 / k) * (l / t)) / k) / sin(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-69) {
		tmp = ((((l * 2.0) / (k * t)) / k) / Math.sin(k)) * (l / k);
	} else {
		tmp = (l / Math.tan(k)) * ((((2.0 / k) * (l / t)) / k) / Math.sin(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 5e-69:
		tmp = ((((l * 2.0) / (k * t)) / k) / math.sin(k)) * (l / k)
	else:
		tmp = (l / math.tan(k)) * ((((2.0 / k) * (l / t)) / k) / math.sin(k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e-69)
		tmp = Float64(Float64(Float64(Float64(Float64(l * 2.0) / Float64(k * t)) / k) / sin(k)) * Float64(l / k));
	else
		tmp = Float64(Float64(l / tan(k)) * Float64(Float64(Float64(Float64(2.0 / k) * Float64(l / t)) / k) / sin(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 5e-69)
		tmp = ((((l * 2.0) / (k * t)) / k) / sin(k)) * (l / k);
	else
		tmp = (l / tan(k)) * ((((2.0 / k) * (l / t)) / k) / sin(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e-69], N[(N[(N[(N[(N[(l * 2.0), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.00000000000000033e-69

   1. Initial program 31.0%

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

      1. associate-*l* 31.0%

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

      2. associate-*l* 31.0%

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

      3. associate-/r* 31.0%

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

      4. associate-/r/ 31.0%

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

      5. *-commutative 31.0%

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

      6. times-frac 33.0%

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

      7. +-commutative 33.0%

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

      8. associate--l+ 47.1%

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

      9. metadata-eval 47.1%

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

      10. +-rgt-identity 47.1%

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

      11. times-frac 60.2%

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

   3. Simplified 60.2%

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

   4. Taylor expanded in t around 0 84.8%

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

      1. unpow2 84.8%

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

   6. Simplified 84.8%

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

      1. associate-*l/ 84.8%

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

      2. associate-*l* 89.2%

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

   8. Applied egg-rr 89.2%

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

      1. associate-*l/ 88.5%

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

      2. associate-*r* 92.3%

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

   10. Simplified 92.3%

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

   11. Taylor expanded in k around inf 88.6%

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

       1. *-commutative 88.6%

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

       2. associate-*r* 88.6%

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

       3. unpow2 88.6%

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

       4. associate-*r* 89.6%

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

       5. associate-*r/ 89.6%

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

       6. associate-/r* 93.1%

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

       7. *-commutative 93.1%

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

       8. associate-/r* 97.0%

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

   13. Simplified 97.0%

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

   14. Taylor expanded in k around 0 94.6%

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

   if 5.00000000000000033e-69 < (*.f64 l l)

   1. Initial program 37.7%

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

      1. associate-*l* 37.7%

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

      2. associate-*l* 37.7%

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

      3. associate-/r* 37.7%

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

      4. associate-/r/ 37.7%

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

      5. *-commutative 37.7%

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

      6. times-frac 37.7%

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

      7. +-commutative 37.7%

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

      8. associate--l+ 39.7%

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

      9. metadata-eval 39.7%

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

      10. +-rgt-identity 39.7%

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

      11. times-frac 39.7%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in t around 0 68.0%

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

      1. unpow2 68.0%

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

   6. Simplified 68.0%

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

      1. associate-*l/ 68.0%

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

      2. associate-*l* 73.1%

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

   8. Applied egg-rr 73.1%

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

      1. associate-*l/ 73.1%

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

      2. associate-*r* 81.4%

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

   10. Simplified 81.4%

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

   11. Taylor expanded in k around inf 74.6%

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

       1. *-commutative 74.6%

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

       2. associate-*r* 74.6%

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

       3. unpow2 74.6%

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

       4. associate-*r* 81.4%

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

       5. associate-*r/ 81.4%

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

       6. associate-/r* 81.5%

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

       7. *-commutative 81.5%

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

       8. associate-/r* 92.4%

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

   13. Simplified 92.4%

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

       1. times-frac 89.4%

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

   15. Applied egg-rr 89.4%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot 2}{k \cdot t}}{k}}{\sin k} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k}}{\sin k}\\ \end{array} \]

Alternative 4: 85.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-*l* 35.1%

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

   3. associate-/r* 35.1%

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

   4. associate-/r/ 35.1%

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

   5. *-commutative 35.1%

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

   6. times-frac 35.9%

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

   7. +-commutative 35.9%

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

   8. associate--l+ 42.6%

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

   9. metadata-eval 42.6%

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

   10. +-rgt-identity 42.6%

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

   11. times-frac 47.7%

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

3. Simplified 47.7%

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

4. Taylor expanded in t around 0 74.6%

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

   1. unpow2 74.6%

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

6. Simplified 74.6%

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

   1. associate-*l/ 74.6%

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

   2. associate-*l* 79.4%

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

8. Applied egg-rr 79.4%

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

   1. associate-*l/ 79.1%

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

   2. associate-*r* 85.7%

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

10. Simplified 85.7%

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

11. Taylor expanded in k around inf 80.1%

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

    1. *-commutative 80.1%

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

    2. associate-*r* 80.1%

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

    3. unpow2 80.1%

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

    4. associate-*r* 84.6%

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

    5. associate-*r/ 84.6%

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

    6. associate-/r* 86.0%

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

    7. *-commutative 86.0%

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

    8. associate-/r* 94.2%

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

13. Simplified 94.2%

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

14. Taylor expanded in l around 0 80.1%

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

    1. associate-/r* 83.2%

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

    2. unpow2 83.2%

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

    3. *-commutative 83.2%

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

16. Simplified 83.2%

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

17. Final simplification 83.2%

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

Alternative 5: 88.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-*l* 35.1%

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

   3. associate-/r* 35.1%

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

   4. associate-/r/ 35.1%

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

   5. *-commutative 35.1%

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

   6. times-frac 35.9%

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

   7. +-commutative 35.9%

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

   8. associate--l+ 42.6%

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

   9. metadata-eval 42.6%

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

   10. +-rgt-identity 42.6%

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

   11. times-frac 47.7%

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

3. Simplified 47.7%

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

4. Taylor expanded in t around 0 74.6%

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

   1. unpow2 74.6%

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

6. Simplified 74.6%

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

   1. associate-*l/ 74.6%

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

   2. associate-*l* 79.4%

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

8. Applied egg-rr 79.4%

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

   1. associate-*l/ 79.1%

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

   2. associate-*r* 85.7%

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

10. Simplified 85.7%

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

11. Final simplification 85.7%

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

Alternative 6: 88.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-*l* 35.1%

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

   3. associate-/r* 35.1%

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

   4. associate-/r/ 35.1%

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

   5. *-commutative 35.1%

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

   6. times-frac 35.9%

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

   7. +-commutative 35.9%

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

   8. associate--l+ 42.6%

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

   9. metadata-eval 42.6%

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

   10. +-rgt-identity 42.6%

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

   11. times-frac 47.7%

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

3. Simplified 47.7%

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

4. Taylor expanded in t around 0 74.6%

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

   1. unpow2 74.6%

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

6. Simplified 74.6%

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

   1. associate-*l/ 74.6%

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

   2. associate-*l* 79.4%

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

8. Applied egg-rr 79.4%

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

   1. associate-*l/ 79.1%

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

   2. associate-*r* 85.7%

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

10. Simplified 85.7%

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

    1. associate-*l/ 86.0%

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

12. Applied egg-rr 86.0%

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

13. Final simplification 86.0%

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

Alternative 7: 88.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-*l* 35.1%

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

   3. associate-/r* 35.1%

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

   4. associate-/r/ 35.1%

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

   5. *-commutative 35.1%

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

   6. times-frac 35.9%

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

   7. +-commutative 35.9%

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

   8. associate--l+ 42.6%

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

   9. metadata-eval 42.6%

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

   10. +-rgt-identity 42.6%

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

   11. times-frac 47.7%

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

3. Simplified 47.7%

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

4. Taylor expanded in t around 0 74.6%

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

   1. unpow2 74.6%

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

6. Simplified 74.6%

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

   1. associate-*l/ 74.6%

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

   2. associate-*l* 79.4%

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

8. Applied egg-rr 79.4%

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

   1. associate-*l/ 79.1%

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

   2. associate-*r* 85.7%

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

10. Simplified 85.7%

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

11. Taylor expanded in k around inf 80.1%

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

    1. *-commutative 80.1%

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

    2. associate-*r* 80.1%

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

    3. unpow2 80.1%

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

    4. associate-*r* 84.6%

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

    5. associate-*r/ 84.6%

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

    6. associate-/r* 86.0%

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

    7. *-commutative 86.0%

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

    8. associate-/r* 94.2%

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

13. Simplified 94.2%

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

14. Taylor expanded in l around 0 80.1%

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

    1. unpow2 80.1%

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

    2. associate-*r* 86.0%

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

16. Simplified 86.0%

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

17. Final simplification 86.0%

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

Alternative 8: 93.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-*l* 35.1%

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

   3. associate-/r* 35.1%

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

   4. associate-/r/ 35.1%

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

   5. *-commutative 35.1%

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

   6. times-frac 35.9%

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

   7. +-commutative 35.9%

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

   8. associate--l+ 42.6%

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

   9. metadata-eval 42.6%

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

   10. +-rgt-identity 42.6%

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

   11. times-frac 47.7%

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

3. Simplified 47.7%

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

4. Taylor expanded in t around 0 74.6%

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

   1. unpow2 74.6%

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

6. Simplified 74.6%

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

   1. associate-*l/ 74.6%

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

   2. associate-*l* 79.4%

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

8. Applied egg-rr 79.4%

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

   1. associate-*l/ 79.1%

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

   2. associate-*r* 85.7%

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

10. Simplified 85.7%

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

11. Taylor expanded in k around inf 80.1%

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

    1. *-commutative 80.1%

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

    2. associate-*r* 80.1%

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

    3. unpow2 80.1%

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

    4. associate-*r* 84.6%

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

    5. associate-*r/ 84.6%

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

    6. associate-/r* 86.0%

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

    7. *-commutative 86.0%

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

    8. associate-/r* 94.2%

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

13. Simplified 94.2%

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

14. Final simplification 94.2%

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

Alternative 9: 71.2% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-*l* 35.1%

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

   3. associate-/r* 35.1%

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

   4. associate-/r/ 35.1%

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

   5. *-commutative 35.1%

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

   6. times-frac 35.9%

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

   7. +-commutative 35.9%

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

   8. associate--l+ 42.6%

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

   9. metadata-eval 42.6%

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

   10. +-rgt-identity 42.6%

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

   11. times-frac 47.7%

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

3. Simplified 47.7%

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

4. Taylor expanded in t around 0 74.6%

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

   1. unpow2 74.6%

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

6. Simplified 74.6%

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

   1. associate-*l/ 74.6%

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

   2. associate-*l* 79.4%

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

8. Applied egg-rr 79.4%

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

   1. associate-*l/ 79.1%

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

   2. associate-*r* 85.7%

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

10. Simplified 85.7%

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

11. Taylor expanded in k around 0 73.5%

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

12. Final simplification 73.5%

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

Alternative 10: 72.0% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-*l* 35.1%

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

   3. associate-/r* 35.1%

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

   4. associate-/r/ 35.1%

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

   5. *-commutative 35.1%

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

   6. times-frac 35.9%

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

   7. +-commutative 35.9%

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

   8. associate--l+ 42.6%

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

   9. metadata-eval 42.6%

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

   10. +-rgt-identity 42.6%

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

   11. times-frac 47.7%

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

3. Simplified 47.7%

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

4. Taylor expanded in t around 0 74.6%

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

   1. unpow2 74.6%

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

6. Simplified 74.6%

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

   1. associate-*l/ 74.6%

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

   2. associate-*l* 79.4%

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

8. Applied egg-rr 79.4%

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

   1. associate-*l/ 79.1%

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

   2. associate-*r* 85.7%

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

10. Simplified 85.7%

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

11. Taylor expanded in k around inf 80.1%

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

    1. *-commutative 80.1%

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

    2. associate-*r* 80.1%

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

    3. unpow2 80.1%

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

    4. associate-*r* 84.6%

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

    5. associate-*r/ 84.6%

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

    6. associate-/r* 86.0%

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

    7. *-commutative 86.0%

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

    8. associate-/r* 94.2%

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

13. Simplified 94.2%

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

14. Taylor expanded in k around 0 75.7%

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

15. Final simplification 75.7%

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

Alternative 11: 68.6% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-*l* 35.1%

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

   3. associate-/r* 35.1%

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

   4. associate-/r/ 35.1%

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

   5. *-commutative 35.1%

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

   6. times-frac 35.9%

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

   7. +-commutative 35.9%

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

   8. associate--l+ 42.6%

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

   9. metadata-eval 42.6%

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

   10. +-rgt-identity 42.6%

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

   11. times-frac 47.7%

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

3. Simplified 47.7%

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

4. Taylor expanded in t around 0 74.6%

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

   1. unpow2 74.6%

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

6. Simplified 74.6%

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

7. Taylor expanded in k around 0 64.1%

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

   1. unpow2 64.1%

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

   2. unpow2 64.1%

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

9. Simplified 64.1%

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

    1. times-frac 70.5%

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

11. Applied egg-rr 70.5%

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

12. Final simplification 70.5%

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

Alternative 12: 68.6% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-*l* 35.1%

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

   3. associate-/r* 35.1%

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

   4. associate-/r/ 35.1%

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

   5. *-commutative 35.1%

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

   6. times-frac 35.9%

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

   7. +-commutative 35.9%

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

   8. associate--l+ 42.6%

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

   9. metadata-eval 42.6%

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

   10. +-rgt-identity 42.6%

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

   11. times-frac 47.7%

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

3. Simplified 47.7%

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

4. Taylor expanded in t around 0 74.6%

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

   1. unpow2 74.6%

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

6. Simplified 74.6%

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

7. Taylor expanded in k around 0 64.1%

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

   1. unpow2 64.1%

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

   2. unpow2 64.1%

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

9. Simplified 64.1%

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

10. Taylor expanded in l around 0 64.1%

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

    1. unpow2 64.1%

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

    2. associate-/l* 70.5%

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

    3. unpow2 70.5%

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

12. Simplified 70.5%

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

13. Final simplification 70.5%

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

Alternative 13: 68.4% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-*l* 35.1%

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

   3. associate-/r* 35.1%

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

   4. associate-/r/ 35.1%

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

   5. *-commutative 35.1%

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

   6. times-frac 35.9%

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

   7. +-commutative 35.9%

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

   8. associate--l+ 42.6%

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

   9. metadata-eval 42.6%

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

   10. +-rgt-identity 42.6%

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

   11. times-frac 47.7%

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

3. Simplified 47.7%

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

4. Taylor expanded in t around 0 74.6%

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

   1. unpow2 74.6%

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

6. Simplified 74.6%

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

7. Taylor expanded in k around 0 64.1%

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

   1. unpow2 64.1%

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

   2. unpow2 64.1%

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

9. Simplified 64.1%

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

    1. times-frac 70.5%

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

11. Applied egg-rr 70.5%

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

12. Taylor expanded in k around 0 70.5%

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

    1. associate-/r* 70.5%

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

    2. unpow2 70.5%

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

    3. associate-/r* 70.6%

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

14. Simplified 70.6%

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

15. Final simplification 70.6%

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

Alternative 14: 69.8% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-*l* 35.1%

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

   3. associate-/r* 35.1%

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

   4. associate-/r/ 35.1%

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

   5. *-commutative 35.1%

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

   6. times-frac 35.9%

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

   7. +-commutative 35.9%

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

   8. associate--l+ 42.6%

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

   9. metadata-eval 42.6%

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

   10. +-rgt-identity 42.6%

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

   11. times-frac 47.7%

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

3. Simplified 47.7%

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

4. Taylor expanded in t around 0 74.6%

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

   1. unpow2 74.6%

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

6. Simplified 74.6%

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

7. Taylor expanded in k around 0 64.1%

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

   1. unpow2 64.1%

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

   2. unpow2 64.1%

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

9. Simplified 64.1%

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

    1. associate-*r* 64.1%

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

    2. associate-*l/ 64.1%

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

    3. times-frac 72.3%

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

11. Applied egg-rr 72.3%

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

12. Final simplification 72.3%

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

Reproduce

herbie shell --seed 2023189 
(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))))