Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.7% → 93.4%

Time: 18.5s

Alternatives: 10

Speedup: 3.8×

• 35.6% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 34.7% 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: 93.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.1%

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

   1. associate-*l* 36.1%

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

   2. associate-*l* 36.1%

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

   3. associate-/r* 36.1%

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

   4. associate-/r/ 35.7%

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

   5. *-commutative 35.7%

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

   6. times-frac 36.5%

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

   7. +-commutative 36.5%

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

   8. associate--l+ 42.9%

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

   9. metadata-eval 42.9%

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

   10. +-rgt-identity 42.9%

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

   11. times-frac 46.6%

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

3. Simplified 46.6%

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

4. Taylor expanded in t around 0 84.6%

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

   1. unpow2 84.6%

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

6. Simplified 84.6%

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

   1. associate-*l/ 84.7%

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

   2. associate-*l* 88.1%

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

8. Applied egg-rr 88.1%

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

   1. *-commutative 88.1%

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

   2. associate-*r/ 88.1%

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

   3. associate-*r* 84.6%

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

   4. unpow2 84.6%

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

   5. associate-/r* 84.7%

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

   6. unpow2 84.7%

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

   7. associate-*l* 88.0%

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

   8. unpow2 88.0%

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

   9. associate-/r* 87.8%

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

   10. unpow2 87.8%

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

   11. associate-*r* 92.7%

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

10. Simplified 92.7%

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

    1. associate-*l/ 92.7%

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

12. Applied egg-rr 92.7%

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

    1. frac-times 90.4%

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

14. Applied egg-rr 90.4%

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

    1. times-frac 92.7%

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

    2. associate-*r* 87.8%

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

    3. unpow2 87.8%

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

    4. associate-*r/ 87.8%

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

    5. unpow2 87.8%

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

    6. associate-*r* 92.7%

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

    7. times-frac 96.8%

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

    8. associate-/l* 96.4%

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

    9. associate-/r* 96.4%

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

16. Simplified 96.4%

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

17. Final simplification 96.4%

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

Alternative 2: 79.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(\frac{\ell}{k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell \cdot \frac{\ell}{\tan k}}{\sin k}}{k \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 8.2e-25)
   (* (/ l (sin k)) (* (/ l k) (/ 2.0 (* k (* k t)))))
   (* (/ 2.0 k) (/ (/ (* l (/ l (tan k))) (sin k)) (* k t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.2e-25) {
		tmp = (l / sin(k)) * ((l / k) * (2.0 / (k * (k * t))));
	} else {
		tmp = (2.0 / k) * (((l * (l / tan(k))) / sin(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 (k <= 8.2d-25) then
        tmp = (l / sin(k)) * ((l / k) * (2.0d0 / (k * (k * t))))
    else
        tmp = (2.0d0 / k) * (((l * (l / tan(k))) / sin(k)) / (k * t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.2e-25) {
		tmp = (l / Math.sin(k)) * ((l / k) * (2.0 / (k * (k * t))));
	} else {
		tmp = (2.0 / k) * (((l * (l / Math.tan(k))) / Math.sin(k)) / (k * t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 8.2e-25:
		tmp = (l / math.sin(k)) * ((l / k) * (2.0 / (k * (k * t))))
	else:
		tmp = (2.0 / k) * (((l * (l / math.tan(k))) / math.sin(k)) / (k * t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 8.2e-25)
		tmp = Float64(Float64(l / sin(k)) * Float64(Float64(l / k) * Float64(2.0 / Float64(k * Float64(k * t)))));
	else
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(Float64(l * Float64(l / tan(k))) / sin(k)) / Float64(k * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 8.2e-25)
		tmp = (l / sin(k)) * ((l / k) * (2.0 / (k * (k * t))));
	else
		tmp = (2.0 / k) * (((l * (l / tan(k))) / sin(k)) / (k * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 8.2e-25], N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.19999999999999974e-25

   1. Initial program 37.6%

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

      1. associate-*l* 37.6%

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

      2. associate-*l* 37.6%

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

      3. associate-/r* 37.6%

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

      4. associate-/r/ 37.6%

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

      5. *-commutative 37.6%

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

      6. times-frac 38.2%

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

      7. +-commutative 38.2%

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

      8. associate--l+ 43.5%

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

      9. metadata-eval 43.5%

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

      10. +-rgt-identity 43.5%

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

      11. times-frac 48.8%

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

   3. Simplified 48.8%

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

   4. Taylor expanded in t around 0 88.0%

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

      1. unpow2 88.0%

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

   6. Simplified 88.0%

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

      1. associate-*l/ 88.0%

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

      2. associate-*l* 91.4%

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

   8. Applied egg-rr 91.4%

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

      1. *-commutative 91.4%

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

      2. associate-*r/ 91.3%

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

      3. associate-*r* 88.0%

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

      4. unpow2 88.0%

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

      5. associate-/r* 88.0%

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

      6. unpow2 88.0%

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

      7. associate-*l* 91.6%

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

      8. unpow2 91.6%

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

      9. associate-/r* 91.6%

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

      10. unpow2 91.6%

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

      11. associate-*r* 96.5%

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

   10. Simplified 96.5%

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

   11. Taylor expanded in k around 0 85.1%

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

   if 8.19999999999999974e-25 < k

   1. Initial program 32.7%

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

      1. associate-*l* 32.7%

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

      2. associate-*l* 32.7%

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

      3. associate-/r* 32.7%

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

      4. associate-/r/ 31.4%

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

      5. *-commutative 31.4%

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

      6. times-frac 32.7%

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

      7. +-commutative 32.7%

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

      8. associate--l+ 41.5%

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

      9. metadata-eval 41.5%

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

      10. +-rgt-identity 41.5%

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

      11. times-frac 41.5%

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

   3. Simplified 41.5%

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

   4. Taylor expanded in t around 0 77.3%

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

      1. unpow2 77.3%

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

   6. Simplified 77.3%

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

      1. associate-*l/ 77.3%

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

      2. associate-*l* 81.0%

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

   8. Applied egg-rr 81.0%

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

      1. times-frac 85.8%

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

      2. associate-*l/ 85.8%

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

   10. Simplified 85.8%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(\frac{\ell}{k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell \cdot \frac{\ell}{\tan k}}{\sin k}}{k \cdot t}\\ \end{array} \]

Alternative 3: 88.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.1%

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

   1. associate-*l* 36.1%

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

   2. associate-*l* 36.1%

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

   3. associate-/r* 36.1%

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

   4. associate-/r/ 35.7%

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

   5. *-commutative 35.7%

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

   6. times-frac 36.5%

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

   7. +-commutative 36.5%

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

   8. associate--l+ 42.9%

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

   9. metadata-eval 42.9%

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

   10. +-rgt-identity 42.9%

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

   11. times-frac 46.6%

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

3. Simplified 46.6%

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

4. Taylor expanded in t around 0 84.6%

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

   1. unpow2 84.6%

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

6. Simplified 84.6%

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

   1. associate-*l/ 84.7%

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

   2. associate-*l* 88.1%

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

8. Applied egg-rr 88.1%

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

   1. *-commutative 88.1%

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

   2. associate-*r/ 88.1%

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

   3. associate-*r* 84.6%

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

   4. unpow2 84.6%

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

   5. associate-/r* 84.7%

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

   6. unpow2 84.7%

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

   7. associate-*l* 88.0%

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

   8. unpow2 88.0%

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

   9. associate-/r* 87.8%

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

   10. unpow2 87.8%

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

   11. associate-*r* 92.7%

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

10. Simplified 92.7%

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

11. Final simplification 92.7%

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

Alternative 4: 73.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{+68}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(\frac{\ell}{k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell}{\frac{t}{\frac{\ell}{{k}^{4}}}} - \frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.35e+68)
   (* (/ l (sin k)) (* (/ l k) (/ 2.0 (* k (* k t)))))
   (- (* 2.0 (/ l (/ t (/ l (pow k 4.0))))) (* (/ l k) (/ l (* k t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.35e+68) {
		tmp = (l / sin(k)) * ((l / k) * (2.0 / (k * (k * t))));
	} else {
		tmp = (2.0 * (l / (t / (l / pow(k, 4.0))))) - ((l / k) * (l / (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 (k <= 1.35d+68) then
        tmp = (l / sin(k)) * ((l / k) * (2.0d0 / (k * (k * t))))
    else
        tmp = (2.0d0 * (l / (t / (l / (k ** 4.0d0))))) - ((l / k) * (l / (k * t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.35e+68) {
		tmp = (l / Math.sin(k)) * ((l / k) * (2.0 / (k * (k * t))));
	} else {
		tmp = (2.0 * (l / (t / (l / Math.pow(k, 4.0))))) - ((l / k) * (l / (k * t)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.35e+68:
		tmp = (l / math.sin(k)) * ((l / k) * (2.0 / (k * (k * t))))
	else:
		tmp = (2.0 * (l / (t / (l / math.pow(k, 4.0))))) - ((l / k) * (l / (k * t)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.35e+68)
		tmp = Float64(Float64(l / sin(k)) * Float64(Float64(l / k) * Float64(2.0 / Float64(k * Float64(k * t)))));
	else
		tmp = Float64(Float64(2.0 * Float64(l / Float64(t / Float64(l / (k ^ 4.0))))) - Float64(Float64(l / k) * Float64(l / Float64(k * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.35e+68)
		tmp = (l / sin(k)) * ((l / k) * (2.0 / (k * (k * t))));
	else
		tmp = (2.0 * (l / (t / (l / (k ^ 4.0))))) - ((l / k) * (l / (k * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.35e+68], N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l / N[(t / N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.34999999999999995e68

   1. Initial program 36.2%

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

      1. associate-*l* 36.2%

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

      2. associate-*l* 36.2%

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

      3. associate-/r* 36.2%

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

      4. associate-/r/ 36.2%

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

      5. *-commutative 36.2%

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

      6. times-frac 36.7%

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

      7. +-commutative 36.7%

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

      8. associate--l+ 42.6%

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

      9. metadata-eval 42.6%

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

      10. +-rgt-identity 42.6%

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

      11. times-frac 47.4%

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

   3. Simplified 47.4%

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

   4. Taylor expanded in t around 0 88.6%

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

      1. unpow2 88.6%

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

   6. Simplified 88.6%

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

      1. associate-*l/ 88.6%

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

      2. associate-*l* 91.6%

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

   8. Applied egg-rr 91.6%

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

      1. *-commutative 91.6%

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

      2. associate-*r/ 91.6%

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

      3. associate-*r* 88.6%

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

      4. unpow2 88.6%

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

      5. associate-/r* 88.6%

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

      6. unpow2 88.6%

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

      7. associate-*l* 92.1%

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

      8. unpow2 92.1%

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

      9. associate-/r* 91.9%

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

      10. unpow2 91.9%

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

      11. associate-*r* 96.3%

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

   10. Simplified 96.3%

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

   11. Taylor expanded in k around 0 84.0%

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

   if 1.34999999999999995e68 < k

   1. Initial program 35.7%

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

      1. +-rgt-identity 35.6%

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

      2. associate-*l* 35.6%

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

      3. mul0-rgt 9.9%

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

      4. distribute-lft-in 35.7%

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

      5. +-rgt-identity 35.7%

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

      6. sub-neg 35.7%

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

      7. +-commutative 35.7%

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

      8. associate-+l+ 43.9%

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

      9. metadata-eval 43.9%

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

      10. metadata-eval 43.9%

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

      11. +-rgt-identity 43.9%

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

   3. Simplified 43.9%

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

   4. Taylor expanded in t around 0 72.3%

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

      1. unpow2 72.3%

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

      2. times-frac 71.0%

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

      3. unpow2 71.0%

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

      4. *-commutative 71.0%

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

   6. Simplified 71.0%

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

   7. Taylor expanded in k around 0 58.6%

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

      1. *-commutative 58.6%

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

      2. unpow2 58.6%

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

      3. times-frac 59.1%

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

      4. unpow2 59.1%

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

   9. Simplified 59.1%

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

   10. Taylor expanded in k around 0 55.3%

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

       1. +-commutative 55.3%

          \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} + -1 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]

       2. mul-1-neg 55.3%

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

       3. unsub-neg 55.3%

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

       4. unpow2 55.3%

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

       5. associate-/l* 58.6%

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

       6. *-commutative 58.6%

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

       7. associate-/l* 58.6%

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

       8. unpow2 58.6%

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

       9. unpow2 58.6%

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

       10. associate-*r* 59.1%

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

       11. *-commutative 59.1%

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

       12. times-frac 60.6%

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

   12. Simplified 60.6%

       \[\leadsto \color{blue}{2 \cdot \frac{\ell}{\frac{t}{\frac{\ell}{{k}^{4}}}} - \frac{\ell}{k \cdot t} \cdot \frac{\ell}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{+68}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(\frac{\ell}{k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell}{\frac{t}{\frac{\ell}{{k}^{4}}}} - \frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\\ \end{array} \]

Alternative 5: 74.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(\frac{\ell}{k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{t \cdot \frac{k \cdot k}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 0.0)
   (* (/ l (sin k)) (* (/ l k) (/ 2.0 (* k (* k t)))))
   (/ 2.0 (* (/ (* k k) (cos k)) (/ (* t (/ (* k k) l)) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (l / sin(k)) * ((l / k) * (2.0 / (k * (k * t))));
	} else {
		tmp = 2.0 / (((k * k) / cos(k)) * ((t * ((k * k) / l)) / l));
	}
	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) <= 0.0d0) then
        tmp = (l / sin(k)) * ((l / k) * (2.0d0 / (k * (k * t))))
    else
        tmp = 2.0d0 / (((k * k) / cos(k)) * ((t * ((k * k) / l)) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (l / Math.sin(k)) * ((l / k) * (2.0 / (k * (k * t))));
	} else {
		tmp = 2.0 / (((k * k) / Math.cos(k)) * ((t * ((k * k) / l)) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = (l / math.sin(k)) * ((l / k) * (2.0 / (k * (k * t))))
	else:
		tmp = 2.0 / (((k * k) / math.cos(k)) * ((t * ((k * k) / l)) / l))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 22.5%

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

      1. associate-*l* 22.5%

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

      2. associate-*l* 22.5%

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

      3. associate-/r* 22.5%

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

      4. associate-/r/ 22.5%

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

      5. *-commutative 22.5%

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

      6. times-frac 22.5%

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

      7. +-commutative 22.5%

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

      8. associate--l+ 34.9%

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

      9. metadata-eval 34.9%

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

      10. +-rgt-identity 34.9%

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

      11. times-frac 49.8%

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

   3. Simplified 49.8%

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

   4. Taylor expanded in t around 0 93.2%

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

      1. unpow2 93.2%

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

   6. Simplified 93.2%

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

      1. associate-*l/ 93.2%

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

      2. associate-*l* 94.9%

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

   8. Applied egg-rr 94.9%

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

      1. *-commutative 94.9%

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

      2. associate-*r/ 94.8%

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

      3. associate-*r* 93.2%

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

      4. unpow2 93.2%

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

      5. associate-/r* 93.2%

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

      6. unpow2 93.2%

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

      7. associate-*l* 96.5%

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

      8. unpow2 96.5%

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

      9. associate-/r* 96.5%

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

      10. unpow2 96.5%

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

      11. associate-*r* 98.1%

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

   10. Simplified 98.1%

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

   11. Taylor expanded in k around 0 97.4%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 40.0%

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

      1. +-rgt-identity 20.8%

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

      2. associate-*l* 20.8%

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

      3. mul0-rgt 23.3%

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

      4. distribute-lft-in 34.0%

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

      5. +-rgt-identity 40.0%

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

      6. sub-neg 40.0%

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

      7. +-commutative 40.0%

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

      8. associate-+l+ 44.7%

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

      9. metadata-eval 44.7%

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

      10. metadata-eval 44.7%

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

      11. +-rgt-identity 44.7%

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

   3. Simplified 44.7%

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

   4. Taylor expanded in t around 0 82.0%

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

      1. unpow2 82.0%

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

      2. times-frac 82.7%

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

      3. unpow2 82.7%

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

      4. *-commutative 82.7%

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

   6. Simplified 82.7%

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

   7. Taylor expanded in k around 0 71.7%

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

      1. *-commutative 71.7%

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

      2. unpow2 71.7%

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

      3. times-frac 71.3%

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

      4. unpow2 71.3%

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

   9. Simplified 71.3%

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

       1. associate-*l/ 73.4%

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

   11. Applied egg-rr 73.4%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(\frac{\ell}{k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{t \cdot \frac{k \cdot k}{\ell}}{\ell}}\\ \end{array} \]

Alternative 6: 73.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-37}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(\frac{\ell}{k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.7e-37)
   (* (/ l (sin k)) (* (/ l k) (/ 2.0 (* k (* k t)))))
   (/ 2.0 (* (/ (* k k) (cos k)) (* (/ (* k k) l) (/ t l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e-37) {
		tmp = (l / sin(k)) * ((l / k) * (2.0 / (k * (k * t))));
	} else {
		tmp = 2.0 / (((k * k) / cos(k)) * (((k * k) / l) * (t / l)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.7d-37) then
        tmp = (l / sin(k)) * ((l / k) * (2.0d0 / (k * (k * t))))
    else
        tmp = 2.0d0 / (((k * k) / cos(k)) * (((k * k) / l) * (t / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e-37) {
		tmp = (l / Math.sin(k)) * ((l / k) * (2.0 / (k * (k * t))));
	} else {
		tmp = 2.0 / (((k * k) / Math.cos(k)) * (((k * k) / l) * (t / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.7e-37:
		tmp = (l / math.sin(k)) * ((l / k) * (2.0 / (k * (k * t))))
	else:
		tmp = 2.0 / (((k * k) / math.cos(k)) * (((k * k) / l) * (t / l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.7e-37)
		tmp = Float64(Float64(l / sin(k)) * Float64(Float64(l / k) * Float64(2.0 / Float64(k * Float64(k * t)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / cos(k)) * Float64(Float64(Float64(k * k) / l) * Float64(t / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.7e-37)
		tmp = (l / sin(k)) * ((l / k) * (2.0 / (k * (k * t))));
	else
		tmp = 2.0 / (((k * k) / cos(k)) * (((k * k) / l) * (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.7e-37], N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.70000000000000016e-37

   1. Initial program 37.6%

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

      1. associate-*l* 37.6%

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

      2. associate-*l* 37.6%

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

      3. associate-/r* 37.6%

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

      4. associate-/r/ 37.6%

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

      5. *-commutative 37.6%

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

      6. times-frac 38.2%

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

      7. +-commutative 38.2%

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

      8. associate--l+ 43.5%

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

      9. metadata-eval 43.5%

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

      10. +-rgt-identity 43.5%

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

      11. times-frac 48.8%

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

   3. Simplified 48.8%

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

   4. Taylor expanded in t around 0 88.0%

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

      1. unpow2 88.0%

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

   6. Simplified 88.0%

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

      1. associate-*l/ 88.0%

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

      2. associate-*l* 91.4%

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

   8. Applied egg-rr 91.4%

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

      1. *-commutative 91.4%

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

      2. associate-*r/ 91.3%

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

      3. associate-*r* 88.0%

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

      4. unpow2 88.0%

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

      5. associate-/r* 88.0%

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

      6. unpow2 88.0%

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

      7. associate-*l* 91.6%

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

      8. unpow2 91.6%

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

      9. associate-/r* 91.6%

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

      10. unpow2 91.6%

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

      11. associate-*r* 96.5%

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

   10. Simplified 96.5%

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

   11. Taylor expanded in k around 0 85.1%

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

   if 2.70000000000000016e-37 < k

   1. Initial program 32.7%

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

      1. +-rgt-identity 30.1%

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

      2. associate-*l* 30.1%

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

      3. mul0-rgt 10.2%

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

      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 32.7%

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

      6. sub-neg 32.7%

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

      7. +-commutative 32.7%

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

      8. associate-+l+ 41.5%

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

      9. metadata-eval 41.5%

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

      10. metadata-eval 41.5%

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

      11. +-rgt-identity 41.5%

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

   3. Simplified 41.5%

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

   4. Taylor expanded in t around 0 77.2%

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

      1. unpow2 77.2%

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

      2. times-frac 76.3%

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

      3. unpow2 76.3%

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

      4. *-commutative 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in k around 0 62.0%

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

      1. *-commutative 62.0%

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

      2. unpow2 62.0%

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

      3. times-frac 62.4%

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

      4. unpow2 62.4%

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

   9. Simplified 62.4%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-37}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(\frac{\ell}{k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{k \cdot k}{\ell} \cdot \frac{t}{\ell}\right)}\\ \end{array} \]

Alternative 7: 70.0% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.1%

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

   1. associate-*l* 36.1%

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

   2. associate-*l* 36.1%

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

   3. associate-/r* 36.1%

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

   4. associate-/r/ 35.7%

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

   5. *-commutative 35.7%

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

   6. times-frac 36.5%

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

   7. +-commutative 36.5%

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

   8. associate--l+ 42.9%

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

   9. metadata-eval 42.9%

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

   10. +-rgt-identity 42.9%

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

   11. times-frac 46.6%

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

3. Simplified 46.6%

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

4. Taylor expanded in t around 0 84.6%

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

   1. unpow2 84.6%

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

6. Simplified 84.6%

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

7. Taylor expanded in k around 0 75.5%

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

8. Final simplification 75.5%

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

Alternative 8: 72.3% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.1%

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

   1. associate-*l* 36.1%

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

   2. associate-*l* 36.1%

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

   3. associate-/r* 36.1%

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

   4. associate-/r/ 35.7%

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

   5. *-commutative 35.7%

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

   6. times-frac 36.5%

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

   7. +-commutative 36.5%

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

   8. associate--l+ 42.9%

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

   9. metadata-eval 42.9%

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

   10. +-rgt-identity 42.9%

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

   11. times-frac 46.6%

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

3. Simplified 46.6%

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

4. Taylor expanded in t around 0 84.6%

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

   1. unpow2 84.6%

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

6. Simplified 84.6%

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

   1. associate-*l/ 84.7%

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

   2. associate-*l* 88.1%

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

8. Applied egg-rr 88.1%

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

   1. *-commutative 88.1%

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

   2. associate-*r/ 88.1%

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

   3. associate-*r* 84.6%

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

   4. unpow2 84.6%

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

   5. associate-/r* 84.7%

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

   6. unpow2 84.7%

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

   7. associate-*l* 88.0%

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

   8. unpow2 88.0%

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

   9. associate-/r* 87.8%

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

   10. unpow2 87.8%

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

   11. associate-*r* 92.7%

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

10. Simplified 92.7%

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

11. Taylor expanded in k around 0 77.2%

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

12. Final simplification 77.2%

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

Alternative 9: 66.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (* (/ l (pow k 4.0)) (/ l t))))

C

double code(double t, double l, double k) {
	return 2.0 * ((l / pow(k, 4.0)) * (l / 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 ** 4.0d0)) * (l / t))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l / Math.pow(k, 4.0)) * (l / t));
}

Python

def code(t, l, k):
	return 2.0 * ((l / math.pow(k, 4.0)) * (l / t))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / (k ^ 4.0)) * Float64(l / t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l / (k ^ 4.0)) * (l / t));
end

Wolfram

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

Derivation

1. Initial program 36.1%

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

   1. associate-*l* 36.1%

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

   2. associate-*l* 36.1%

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

   3. associate-/r* 36.1%

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

   4. associate-/r/ 35.7%

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

   5. *-commutative 35.7%

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

   6. times-frac 36.5%

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

   7. +-commutative 36.5%

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

   8. associate--l+ 42.9%

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

   9. metadata-eval 42.9%

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

   10. +-rgt-identity 42.9%

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

   11. times-frac 46.6%

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

3. Simplified 46.6%

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

4. Taylor expanded in t around 0 84.6%

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

   1. unpow2 84.6%

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

6. Simplified 84.6%

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

7. Taylor expanded in k around 0 67.5%

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

   1. *-commutative 67.5%

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

   2. associate-/r* 66.7%

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

   3. unpow2 66.7%

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

9. Simplified 66.7%

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

10. Taylor expanded in l around 0 67.5%

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

    1. unpow2 67.5%

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

    2. times-frac 70.1%

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

12. Simplified 70.1%

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

13. Final simplification 70.1%

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

Alternative 10: 67.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{\ell}{\frac{t}{\frac{\ell}{{k}^{4}}}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (/ l (/ t (/ l (pow k 4.0))))))

C

double code(double t, double l, double k) {
	return 2.0 * (l / (t / (l / pow(k, 4.0))));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (l / (t / (l / (k ** 4.0d0))))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (l / (t / (l / Math.pow(k, 4.0))));
}

Python

def code(t, l, k):
	return 2.0 * (l / (t / (l / math.pow(k, 4.0))))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(l / Float64(t / Float64(l / (k ^ 4.0)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (l / (t / (l / (k ^ 4.0))));
end

Wolfram

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

Derivation

1. Initial program 36.1%

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

   1. associate-*l* 36.1%

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

   2. associate-*l* 36.1%

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

   3. associate-/r* 36.1%

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

   4. associate-/r/ 35.7%

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

   5. *-commutative 35.7%

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

   6. times-frac 36.5%

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

   7. +-commutative 36.5%

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

   8. associate--l+ 42.9%

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

   9. metadata-eval 42.9%

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

   10. +-rgt-identity 42.9%

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

   11. times-frac 46.6%

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

3. Simplified 46.6%

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

4. Taylor expanded in t around 0 84.6%

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

   1. unpow2 84.6%

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

6. Simplified 84.6%

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

   1. associate-*l/ 84.7%

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

   2. associate-*l* 88.1%

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

8. Applied egg-rr 88.1%

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

   1. *-commutative 88.1%

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

   2. associate-*r/ 88.1%

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

   3. associate-*r* 84.6%

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

   4. unpow2 84.6%

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

   5. associate-/r* 84.7%

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

   6. unpow2 84.7%

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

   7. associate-*l* 88.0%

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

   8. unpow2 88.0%

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

   9. associate-/r* 87.8%

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

   10. unpow2 87.8%

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

   11. associate-*r* 92.7%

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

10. Simplified 92.7%

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

    1. associate-*l/ 92.7%

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

12. Applied egg-rr 92.7%

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

13. Taylor expanded in k around 0 67.5%

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

    1. unpow2 67.5%

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

    2. associate-/l* 72.1%

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

    3. *-commutative 72.1%

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

    4. associate-/l* 72.2%

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

15. Simplified 72.2%

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

16. Final simplification 72.2%

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

Reproduce

herbie shell --seed 2023213 
(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))))