Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.1% → 95.1%

Time: 20.3s

Alternatives: 9

Speedup: 28.1×

• 36.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{+201}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell}{\sin k \cdot \frac{\tan k}{\ell}}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t \cdot {\sin k}^{2}}{{\left(\frac{\ell}{k}\right)}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 4e+201)
   (* (/ 2.0 k) (/ (/ l (* (sin k) (/ (tan k) l))) (* k t)))
   (* 2.0 (/ (cos k) (/ (* t (pow (sin k) 2.0)) (pow (/ l k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 4e+201) {
		tmp = (2.0 / k) * ((l / (sin(k) * (tan(k) / l))) / (k * t));
	} else {
		tmp = 2.0 * (cos(k) / ((t * pow(sin(k), 2.0)) / pow((l / k), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 4d+201) then
        tmp = (2.0d0 / k) * ((l / (sin(k) * (tan(k) / l))) / (k * t))
    else
        tmp = 2.0d0 * (cos(k) / ((t * (sin(k) ** 2.0d0)) / ((l / k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 4e+201) {
		tmp = (2.0 / k) * ((l / (Math.sin(k) * (Math.tan(k) / l))) / (k * t));
	} else {
		tmp = 2.0 * (Math.cos(k) / ((t * Math.pow(Math.sin(k), 2.0)) / Math.pow((l / k), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 4e+201:
		tmp = (2.0 / k) * ((l / (math.sin(k) * (math.tan(k) / l))) / (k * t))
	else:
		tmp = 2.0 * (math.cos(k) / ((t * math.pow(math.sin(k), 2.0)) / math.pow((l / k), 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 4e+201)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(l / Float64(sin(k) * Float64(tan(k) / l))) / Float64(k * t)));
	else
		tmp = Float64(2.0 * Float64(cos(k) / Float64(Float64(t * (sin(k) ^ 2.0)) / (Float64(l / k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 4e+201)
		tmp = (2.0 / k) * ((l / (sin(k) * (tan(k) / l))) / (k * t));
	else
		tmp = 2.0 * (cos(k) / ((t * (sin(k) ^ 2.0)) / ((l / k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 4e+201], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(l / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.00000000000000015e201

   1. Initial program 35.5%

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

      1. associate-*l* 35.5%

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

      2. associate-*l* 35.6%

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

      3. associate-/r* 35.0%

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

      4. associate-/r/ 35.0%

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

      5. *-commutative 35.0%

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

      6. times-frac 36.6%

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

      7. +-commutative 36.6%

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

      8. associate--l+ 48.3%

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

      9. metadata-eval 48.3%

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

      10. +-rgt-identity 48.3%

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

      11. times-frac 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in t around 0 90.0%

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

      1. unpow2 90.0%

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

   6. Simplified 90.0%

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

      1. *-commutative 90.0%

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

      2. clear-num 90.0%

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

      3. frac-times 90.0%

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

      4. *-un-lft-identity 90.0%

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

   8. Applied egg-rr 90.0%

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

      1. associate-*r* 92.0%

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

      2. associate-*l/ 92.1%

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

      3. clear-num 91.7%

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

      4. associate-*l/ 91.7%

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

      5. associate-*r/ 91.7%

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

      6. associate-*r/ 91.7%

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

      7. associate-*l/ 91.7%

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

      8. clear-num 92.1%

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

      9. *-commutative 92.1%

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

   10. Applied egg-rr 92.1%

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

       1. times-frac 96.3%

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

   12. Applied egg-rr 96.3%

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

   if 4.00000000000000015e201 < (*.f64 l l)

   1. Initial program 34.1%

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

      1. associate-*l* 34.1%

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

      2. associate-*l* 34.1%

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

      3. associate-/r* 34.1%

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

      4. associate-/r/ 34.1%

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

      5. *-commutative 34.1%

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

      6. times-frac 35.3%

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

      7. +-commutative 35.3%

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

      8. associate--l+ 35.3%

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

      9. metadata-eval 35.3%

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

      10. +-rgt-identity 35.3%

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

      11. times-frac 35.3%

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

   3. Simplified 35.3%

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

   4. Taylor expanded in t around 0 63.0%

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

      1. unpow2 63.0%

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

   6. Simplified 63.0%

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

      1. associate-*l/ 63.0%

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

      2. associate-*l* 71.0%

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

   8. Applied egg-rr 71.0%

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

      1. associate-*l/ 70.9%

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

      2. associate-*r* 81.1%

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

      3. *-commutative 81.1%

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

      4. associate-*l/ 81.1%

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

      5. associate-*r/ 81.1%

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

   10. Simplified 81.1%

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

       1. clear-num 81.1%

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

       2. inv-pow 81.1%

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

   12. Applied egg-rr 81.1%

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

       1. unpow-1 81.1%

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

   14. Simplified 81.1%

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

   15. Taylor expanded in l around 0 62.9%

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

       1. associate-/l* 62.9%

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

       2. *-commutative 62.9%

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

       3. associate-/l* 63.1%

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

       4. *-commutative 63.1%

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

       5. unpow2 63.1%

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

       6. unpow2 63.1%

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

       7. times-frac 93.6%

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

       8. unpow2 93.6%

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

   17. Simplified 93.6%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{+201}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell}{\sin k \cdot \frac{\tan k}{\ell}}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t \cdot {\sin k}^{2}}{{\left(\frac{\ell}{k}\right)}^{2}}}\\ \end{array} \]

Alternative 2: 89.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+258}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell}{\sin k \cdot \frac{\tan k}{\ell}}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{1}{\frac{\sin k}{\ell \cdot 2}}}{k \cdot \left(k \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e+258)
   (* (/ 2.0 k) (/ (/ l (* (sin k) (/ (tan k) l))) (* k t)))
   (* (/ l (tan k)) (/ (/ 1.0 (/ (sin k) (* l 2.0))) (* k (* k t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+258) {
		tmp = (2.0 / k) * ((l / (sin(k) * (tan(k) / l))) / (k * t));
	} else {
		tmp = (l / tan(k)) * ((1.0 / (sin(k) / (l * 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) :: tmp
    if ((l * l) <= 1d+258) then
        tmp = (2.0d0 / k) * ((l / (sin(k) * (tan(k) / l))) / (k * t))
    else
        tmp = (l / tan(k)) * ((1.0d0 / (sin(k) / (l * 2.0d0))) / (k * (k * t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+258) {
		tmp = (2.0 / k) * ((l / (Math.sin(k) * (Math.tan(k) / l))) / (k * t));
	} else {
		tmp = (l / Math.tan(k)) * ((1.0 / (Math.sin(k) / (l * 2.0))) / (k * (k * t)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e+258:
		tmp = (2.0 / k) * ((l / (math.sin(k) * (math.tan(k) / l))) / (k * t))
	else:
		tmp = (l / math.tan(k)) * ((1.0 / (math.sin(k) / (l * 2.0))) / (k * (k * t)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e+258)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(l / Float64(sin(k) * Float64(tan(k) / l))) / Float64(k * t)));
	else
		tmp = Float64(Float64(l / tan(k)) * Float64(Float64(1.0 / Float64(sin(k) / Float64(l * 2.0))) / Float64(k * Float64(k * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e+258)
		tmp = (2.0 / k) * ((l / (sin(k) * (tan(k) / l))) / (k * t));
	else
		tmp = (l / tan(k)) * ((1.0 / (sin(k) / (l * 2.0))) / (k * (k * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e+258], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(l / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(N[Sin[k], $MachinePrecision] / N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.00000000000000006e258

   1. Initial program 36.0%

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

      1. associate-*l* 36.0%

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

      2. associate-*l* 36.0%

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

      3. associate-/r* 35.5%

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

      4. associate-/r/ 35.5%

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

      5. *-commutative 35.5%

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

      6. times-frac 37.5%

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

      7. +-commutative 37.5%

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

      8. associate--l+ 48.7%

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

      9. metadata-eval 48.7%

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

      10. +-rgt-identity 48.7%

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

      11. times-frac 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in t around 0 89.9%

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

      1. unpow2 89.9%

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

   6. Simplified 89.9%

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

      1. *-commutative 89.9%

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

      2. clear-num 89.9%

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

      3. frac-times 90.0%

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

      4. *-un-lft-identity 90.0%

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

   8. Applied egg-rr 90.0%

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

      1. associate-*r* 91.9%

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

      2. associate-*l/ 92.0%

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

      3. clear-num 91.6%

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

      4. associate-*l/ 91.6%

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

      5. associate-*r/ 91.6%

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

      6. associate-*r/ 91.6%

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

      7. associate-*l/ 91.6%

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

      8. clear-num 92.0%

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

      9. *-commutative 92.0%

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

   10. Applied egg-rr 92.0%

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

       1. times-frac 96.0%

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

   12. Applied egg-rr 96.0%

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

   if 1.00000000000000006e258 < (*.f64 l l)

   1. Initial program 32.6%

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

      1. associate-*l* 32.6%

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

      2. associate-*l* 32.6%

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

      3. associate-/r* 32.6%

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

      4. associate-/r/ 32.6%

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

      5. *-commutative 32.6%

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

      6. times-frac 32.5%

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

      7. +-commutative 32.5%

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

      8. associate--l+ 32.5%

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

      9. metadata-eval 32.5%

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

      10. +-rgt-identity 32.5%

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

      11. times-frac 32.5%

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

   3. Simplified 32.5%

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

   4. Taylor expanded in t around 0 59.5%

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

      1. unpow2 59.5%

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

   6. Simplified 59.5%

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

      1. associate-*l/ 59.5%

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

      2. associate-*l* 68.5%

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

   8. Applied egg-rr 68.5%

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

      1. associate-*l/ 68.5%

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

      2. associate-*r* 80.0%

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

      3. *-commutative 80.0%

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

      4. associate-*l/ 80.0%

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

      5. associate-*r/ 80.0%

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

   10. Simplified 80.0%

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

       1. clear-num 80.1%

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

       2. inv-pow 80.1%

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

   12. Applied egg-rr 80.1%

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

       1. unpow-1 80.1%

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

   14. Simplified 80.1%

       \[\leadsto \frac{\ell}{\tan k} \cdot \frac{\color{blue}{\frac{1}{\frac{\sin k}{2 \cdot \ell}}}}{k \cdot \left(k \cdot t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+258}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell}{\sin k \cdot \frac{\tan k}{\ell}}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{1}{\frac{\sin k}{\ell \cdot 2}}}{k \cdot \left(k \cdot t\right)}\\ \end{array} \]

Alternative 3: 89.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+258}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell}{\sin k \cdot \frac{\tan k}{\ell}}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(2 \cdot \frac{\ell}{\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e+258)
   (* (/ 2.0 k) (/ (/ l (* (sin k) (/ (tan k) l))) (* k t)))
   (* (/ l (tan k)) (* 2.0 (/ l (* (sin k) (* k (* k t))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+258) {
		tmp = (2.0 / k) * ((l / (sin(k) * (tan(k) / l))) / (k * t));
	} else {
		tmp = (l / tan(k)) * (2.0 * (l / (sin(k) * (k * (k * t)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d+258) then
        tmp = (2.0d0 / k) * ((l / (sin(k) * (tan(k) / l))) / (k * t))
    else
        tmp = (l / tan(k)) * (2.0d0 * (l / (sin(k) * (k * (k * t)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+258) {
		tmp = (2.0 / k) * ((l / (Math.sin(k) * (Math.tan(k) / l))) / (k * t));
	} else {
		tmp = (l / Math.tan(k)) * (2.0 * (l / (Math.sin(k) * (k * (k * t)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e+258:
		tmp = (2.0 / k) * ((l / (math.sin(k) * (math.tan(k) / l))) / (k * t))
	else:
		tmp = (l / math.tan(k)) * (2.0 * (l / (math.sin(k) * (k * (k * t)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e+258)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(l / Float64(sin(k) * Float64(tan(k) / l))) / Float64(k * t)));
	else
		tmp = Float64(Float64(l / tan(k)) * Float64(2.0 * Float64(l / Float64(sin(k) * Float64(k * Float64(k * t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e+258)
		tmp = (2.0 / k) * ((l / (sin(k) * (tan(k) / l))) / (k * t));
	else
		tmp = (l / tan(k)) * (2.0 * (l / (sin(k) * (k * (k * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e+258], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(l / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(l / N[(N[Sin[k], $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.00000000000000006e258

   1. Initial program 36.0%

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

      1. associate-*l* 36.0%

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

      2. associate-*l* 36.0%

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

      3. associate-/r* 35.5%

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

      4. associate-/r/ 35.5%

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

      5. *-commutative 35.5%

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

      6. times-frac 37.5%

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

      7. +-commutative 37.5%

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

      8. associate--l+ 48.7%

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

      9. metadata-eval 48.7%

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

      10. +-rgt-identity 48.7%

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

      11. times-frac 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in t around 0 89.9%

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

      1. unpow2 89.9%

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

   6. Simplified 89.9%

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

      1. *-commutative 89.9%

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

      2. clear-num 89.9%

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

      3. frac-times 90.0%

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

      4. *-un-lft-identity 90.0%

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

   8. Applied egg-rr 90.0%

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

      1. associate-*r* 91.9%

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

      2. associate-*l/ 92.0%

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

      3. clear-num 91.6%

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

      4. associate-*l/ 91.6%

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

      5. associate-*r/ 91.6%

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

      6. associate-*r/ 91.6%

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

      7. associate-*l/ 91.6%

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

      8. clear-num 92.0%

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

      9. *-commutative 92.0%

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

   10. Applied egg-rr 92.0%

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

       1. times-frac 96.0%

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

   12. Applied egg-rr 96.0%

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

   if 1.00000000000000006e258 < (*.f64 l l)

   1. Initial program 32.6%

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

      1. associate-*l* 32.6%

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

      2. associate-*l* 32.6%

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

      3. associate-/r* 32.6%

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

      4. associate-/r/ 32.6%

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

      5. *-commutative 32.6%

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

      6. times-frac 32.5%

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

      7. +-commutative 32.5%

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

      8. associate--l+ 32.5%

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

      9. metadata-eval 32.5%

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

      10. +-rgt-identity 32.5%

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

      11. times-frac 32.5%

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

   3. Simplified 32.5%

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

   4. Taylor expanded in t around 0 59.5%

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

      1. unpow2 59.5%

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

   6. Simplified 59.5%

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

      1. associate-*l/ 59.5%

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

      2. associate-*l* 68.5%

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

   8. Applied egg-rr 68.5%

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

      1. associate-*l/ 68.5%

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

      2. associate-*r* 80.0%

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

      3. *-commutative 80.0%

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

      4. associate-*l/ 80.0%

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

      5. associate-*r/ 80.0%

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

   10. Simplified 80.0%

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

   11. Taylor expanded in l around 0 68.5%

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

       1. *-commutative 68.5%

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

       2. associate-*l* 68.5%

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

       3. *-commutative 68.5%

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

       4. unpow2 68.5%

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

       5. associate-*r* 80.0%

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

   13. Simplified 80.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+258}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell}{\sin k \cdot \frac{\tan k}{\ell}}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(2 \cdot \frac{\ell}{\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)\\ \end{array} \]

Alternative 4: 89.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+258}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell}{\sin k \cdot \frac{\tan k}{\ell}}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\ell \cdot 2}{\sin k}}{k \cdot \left(k \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e+258)
   (* (/ 2.0 k) (/ (/ l (* (sin k) (/ (tan k) l))) (* k t)))
   (* (/ l (tan k)) (/ (/ (* l 2.0) (sin k)) (* k (* k t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+258) {
		tmp = (2.0 / k) * ((l / (sin(k) * (tan(k) / l))) / (k * t));
	} else {
		tmp = (l / tan(k)) * (((l * 2.0) / sin(k)) / (k * (k * t)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d+258) then
        tmp = (2.0d0 / k) * ((l / (sin(k) * (tan(k) / l))) / (k * t))
    else
        tmp = (l / tan(k)) * (((l * 2.0d0) / sin(k)) / (k * (k * t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+258) {
		tmp = (2.0 / k) * ((l / (Math.sin(k) * (Math.tan(k) / l))) / (k * t));
	} else {
		tmp = (l / Math.tan(k)) * (((l * 2.0) / Math.sin(k)) / (k * (k * t)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e+258:
		tmp = (2.0 / k) * ((l / (math.sin(k) * (math.tan(k) / l))) / (k * t))
	else:
		tmp = (l / math.tan(k)) * (((l * 2.0) / math.sin(k)) / (k * (k * t)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e+258)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(l / Float64(sin(k) * Float64(tan(k) / l))) / Float64(k * t)));
	else
		tmp = Float64(Float64(l / tan(k)) * Float64(Float64(Float64(l * 2.0) / sin(k)) / Float64(k * Float64(k * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e+258)
		tmp = (2.0 / k) * ((l / (sin(k) * (tan(k) / l))) / (k * t));
	else
		tmp = (l / tan(k)) * (((l * 2.0) / sin(k)) / (k * (k * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e+258], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(l / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * 2.0), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.00000000000000006e258

   1. Initial program 36.0%

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

      1. associate-*l* 36.0%

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

      2. associate-*l* 36.0%

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

      3. associate-/r* 35.5%

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

      4. associate-/r/ 35.5%

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

      5. *-commutative 35.5%

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

      6. times-frac 37.5%

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

      7. +-commutative 37.5%

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

      8. associate--l+ 48.7%

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

      9. metadata-eval 48.7%

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

      10. +-rgt-identity 48.7%

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

      11. times-frac 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in t around 0 89.9%

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

      1. unpow2 89.9%

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

   6. Simplified 89.9%

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

      1. *-commutative 89.9%

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

      2. clear-num 89.9%

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

      3. frac-times 90.0%

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

      4. *-un-lft-identity 90.0%

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

   8. Applied egg-rr 90.0%

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

      1. associate-*r* 91.9%

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

      2. associate-*l/ 92.0%

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

      3. clear-num 91.6%

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

      4. associate-*l/ 91.6%

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

      5. associate-*r/ 91.6%

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

      6. associate-*r/ 91.6%

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

      7. associate-*l/ 91.6%

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

      8. clear-num 92.0%

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

      9. *-commutative 92.0%

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

   10. Applied egg-rr 92.0%

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

       1. times-frac 96.0%

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

   12. Applied egg-rr 96.0%

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

   if 1.00000000000000006e258 < (*.f64 l l)

   1. Initial program 32.6%

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

      1. associate-*l* 32.6%

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

      2. associate-*l* 32.6%

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

      3. associate-/r* 32.6%

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

      4. associate-/r/ 32.6%

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

      5. *-commutative 32.6%

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

      6. times-frac 32.5%

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

      7. +-commutative 32.5%

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

      8. associate--l+ 32.5%

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

      9. metadata-eval 32.5%

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

      10. +-rgt-identity 32.5%

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

      11. times-frac 32.5%

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

   3. Simplified 32.5%

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

   4. Taylor expanded in t around 0 59.5%

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

      1. unpow2 59.5%

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

   6. Simplified 59.5%

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

      1. associate-*l/ 59.5%

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

      2. associate-*l* 68.5%

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

   8. Applied egg-rr 68.5%

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

      1. associate-*l/ 68.5%

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

      2. associate-*r* 80.0%

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

      3. *-commutative 80.0%

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

      4. associate-*l/ 80.0%

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

      5. associate-*r/ 80.0%

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

   10. Simplified 80.0%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+258}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell}{\sin k \cdot \frac{\tan k}{\ell}}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\ell \cdot 2}{\sin k}}{k \cdot \left(k \cdot t\right)}\\ \end{array} \]

Alternative 5: 88.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+192}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell}{\sin k \cdot \frac{\tan k}{\ell}}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{\sin k \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell \cdot 2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e+192)
   (* (/ 2.0 k) (/ (/ l (* (sin k) (/ (tan k) l))) (* k t)))
   (/ (/ l (tan k)) (* (sin k) (/ (* k (* k t)) (* l 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+192) {
		tmp = (2.0 / k) * ((l / (sin(k) * (tan(k) / l))) / (k * t));
	} else {
		tmp = (l / tan(k)) / (sin(k) * ((k * (k * t)) / (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d+192) then
        tmp = (2.0d0 / k) * ((l / (sin(k) * (tan(k) / l))) / (k * t))
    else
        tmp = (l / tan(k)) / (sin(k) * ((k * (k * t)) / (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+192) {
		tmp = (2.0 / k) * ((l / (Math.sin(k) * (Math.tan(k) / l))) / (k * t));
	} else {
		tmp = (l / Math.tan(k)) / (Math.sin(k) * ((k * (k * t)) / (l * 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e+192:
		tmp = (2.0 / k) * ((l / (math.sin(k) * (math.tan(k) / l))) / (k * t))
	else:
		tmp = (l / math.tan(k)) / (math.sin(k) * ((k * (k * t)) / (l * 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e+192)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(l / Float64(sin(k) * Float64(tan(k) / l))) / Float64(k * t)));
	else
		tmp = Float64(Float64(l / tan(k)) / Float64(sin(k) * Float64(Float64(k * Float64(k * t)) / Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e+192)
		tmp = (2.0 / k) * ((l / (sin(k) * (tan(k) / l))) / (k * t));
	else
		tmp = (l / tan(k)) / (sin(k) * ((k * (k * t)) / (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e+192], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(l / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] / N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.00000000000000004e192

   1. Initial program 35.2%

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

      1. associate-*l* 35.2%

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

      2. associate-*l* 35.2%

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

      3. associate-/r* 34.6%

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

      4. associate-/r/ 34.6%

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

      5. *-commutative 34.6%

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

      6. times-frac 36.2%

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

      7. +-commutative 36.2%

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

      8. associate--l+ 48.0%

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

      9. metadata-eval 48.0%

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

      10. +-rgt-identity 48.0%

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

      11. times-frac 55.3%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in t around 0 89.9%

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

      1. unpow2 89.9%

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

   6. Simplified 89.9%

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

      1. *-commutative 89.9%

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

      2. clear-num 89.9%

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

      3. frac-times 89.9%

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

      4. *-un-lft-identity 89.9%

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

   8. Applied egg-rr 89.9%

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

      1. associate-*r* 92.0%

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

      2. associate-*l/ 92.0%

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

      3. clear-num 91.6%

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

      4. associate-*l/ 91.6%

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

      5. associate-*r/ 91.6%

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

      6. associate-*r/ 91.6%

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

      7. associate-*l/ 91.6%

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

      8. clear-num 92.0%

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

      9. *-commutative 92.0%

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

   10. Applied egg-rr 92.0%

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

       1. times-frac 96.3%

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

   12. Applied egg-rr 96.3%

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

   if 1.00000000000000004e192 < (*.f64 l l)

   1. Initial program 34.9%

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

      1. associate-*l* 34.9%

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

      2. associate-*l* 35.0%

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

      3. associate-/r* 34.9%

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

      4. associate-/r/ 34.9%

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

      5. *-commutative 34.9%

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

      6. times-frac 36.1%

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

      7. +-commutative 36.1%

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

      8. associate--l+ 36.1%

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

      9. metadata-eval 36.1%

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

      10. +-rgt-identity 36.1%

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

      11. times-frac 36.1%

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

   3. Simplified 36.1%

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

   4. Taylor expanded in t around 0 63.4%

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

      1. unpow2 63.4%

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

   6. Simplified 63.4%

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

      1. associate-*l/ 63.5%

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

      2. associate-*l* 71.3%

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

   8. Applied egg-rr 71.3%

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

      1. associate-*l/ 71.3%

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

      2. associate-*r* 81.3%

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

      3. *-commutative 81.3%

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

      4. associate-*l/ 81.3%

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

      5. associate-*r/ 81.3%

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

   10. Simplified 81.3%

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

       1. expm1-log1p-u 35.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{\tan k} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}\right)\right)} \]

       2. expm1-udef 33.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\ell}{\tan k} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}\right)} - 1} \]

       3. *-commutative 33.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\tan k}}\right)} - 1 \]

   12. Applied egg-rr 33.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\tan k}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 35.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\tan k}\right)\right)} \]

       2. expm1-log1p 81.3%

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

       3. *-commutative 81.3%

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

       4. associate-*r/ 71.3%

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

       5. associate-/l* 81.4%

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

       6. associate-/r/ 81.3%

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

   14. Simplified 81.3%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+192}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell}{\sin k \cdot \frac{\tan k}{\ell}}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{\sin k \cdot \frac{k \cdot \left(k \cdot t\right)}{\ell \cdot 2}}\\ \end{array} \]

Alternative 6: 85.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(2.0 / k), $MachinePrecision] * N[(N[(l / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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* 34.7%

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

   4. associate-/r/ 34.7%

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

   5. *-commutative 34.7%

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

   6. times-frac 36.2%

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

   7. +-commutative 36.2%

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

   8. associate--l+ 44.4%

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

   9. metadata-eval 44.4%

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

   10. +-rgt-identity 44.4%

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

   11. times-frac 49.5%

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

3. Simplified 49.5%

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

4. Taylor expanded in t around 0 81.8%

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

   1. unpow2 81.8%

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

6. Simplified 81.8%

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

   1. *-commutative 81.8%

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

   2. clear-num 81.8%

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

   3. frac-times 81.9%

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

   4. *-un-lft-identity 81.9%

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

8. Applied egg-rr 81.9%

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

   1. associate-*r* 85.7%

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

   2. associate-*l/ 85.7%

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

   3. clear-num 85.5%

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

   4. associate-*l/ 85.5%

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

   5. associate-*r/ 85.5%

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

   6. associate-*r/ 85.5%

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

   7. associate-*l/ 85.5%

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

   8. clear-num 85.7%

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

   9. *-commutative 85.7%

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

10. Applied egg-rr 85.7%

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

    1. times-frac 88.4%

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

12. Applied egg-rr 88.4%

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

13. Final simplification 88.4%

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

Alternative 7: 72.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(k * N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t / N[Power[N[(l / k), $MachinePrecision], 2.0], $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. +-rgt-identity 25.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) + 0}} \]

   2. associate-*l* 25.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} + 0} \]

   3. mul0-rgt 18.6%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot 0}} \]

   4. distribute-lft-in 32.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) + 0\right)}} \]

   5. +-rgt-identity 35.1%

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

   6. sub-neg 35.1%

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

   7. +-commutative 35.1%

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

   8. associate-+l+ 42.2%

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

   9. metadata-eval 42.2%

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

   10. metadata-eval 42.2%

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

   11. +-rgt-identity 42.2%

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

3. Simplified 42.2%

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

   1. unpow3 42.2%

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

   2. *-un-lft-identity 42.2%

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

   3. times-frac 46.7%

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

5. Applied egg-rr 46.7%

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

6. Taylor expanded in t around 0 74.2%

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

   1. *-commutative 74.2%

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

   2. times-frac 75.6%

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

   3. unpow2 75.6%

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

   4. associate-/l* 75.7%

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

   5. associate-/r/ 75.6%

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

   6. associate-/l* 75.7%

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

   7. associate-/r/ 73.4%

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

   8. unpow2 73.4%

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

8. Simplified 73.4%

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

9. Taylor expanded in k around 0 66.2%

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

    1. *-commutative 66.2%

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

    2. associate-/l* 66.3%

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

    3. unpow2 66.3%

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

    4. unpow2 66.3%

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

    5. times-frac 74.2%

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

    6. unpow2 74.2%

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

11. Simplified 74.2%

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

12. Final simplification 74.2%

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

Alternative 8: 70.8% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / 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* 34.7%

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

   4. associate-/r/ 34.7%

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

   5. *-commutative 34.7%

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

   6. times-frac 36.2%

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

   7. +-commutative 36.2%

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

   8. associate--l+ 44.4%

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

   9. metadata-eval 44.4%

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

   10. +-rgt-identity 44.4%

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

   11. times-frac 49.5%

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

3. Simplified 49.5%

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

4. Taylor expanded in t around 0 81.8%

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

   1. unpow2 81.8%

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

6. Simplified 81.8%

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

7. Taylor expanded in k around 0 64.9%

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

   1. unpow2 64.9%

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

   2. unpow2 64.9%

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

9. Simplified 64.9%

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

    1. times-frac 72.1%

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

11. Applied egg-rr 72.1%

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

12. Taylor expanded in k around 0 72.1%

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

    1. unpow2 72.1%

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

    2. associate-*r* 72.1%

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

14. Simplified 72.1%

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

15. Final simplification 72.1%

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

Alternative 9: 70.9% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := N[(N[(2.0 * N[(l / N[(k * N[(k / l), $MachinePrecision]), $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* 34.7%

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

   4. associate-/r/ 34.7%

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

   5. *-commutative 34.7%

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

   6. times-frac 36.2%

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

   7. +-commutative 36.2%

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

   8. associate--l+ 44.4%

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

   9. metadata-eval 44.4%

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

   10. +-rgt-identity 44.4%

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

   11. times-frac 49.5%

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

3. Simplified 49.5%

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

4. Taylor expanded in t around 0 81.8%

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

   1. unpow2 81.8%

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

6. Simplified 81.8%

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

   1. *-commutative 81.8%

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

   2. clear-num 81.8%

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

   3. frac-times 81.9%

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

   4. *-un-lft-identity 81.9%

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

8. Applied egg-rr 81.9%

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

   1. associate-*r* 85.7%

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

   2. associate-*l/ 85.7%

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

   3. clear-num 85.5%

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

   4. associate-*l/ 85.5%

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

   5. associate-*r/ 85.5%

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

   6. associate-*r/ 85.5%

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

   7. associate-*l/ 85.5%

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

   8. clear-num 85.7%

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

   9. *-commutative 85.7%

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

10. Applied egg-rr 85.7%

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

11. Taylor expanded in k around 0 72.1%

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

    1. unpow2 72.1%

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

    2. associate-/l* 72.1%

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

    3. associate-/r/ 72.1%

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

13. Simplified 72.1%

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

14. Final simplification 72.1%

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

Reproduce

herbie shell --seed 2023242 
(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))))