Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.0% → 95.2%

Time: 20.5s

Alternatives: 11

Speedup: 28.1×

• 34.8% 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: 34.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+223}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\frac{k}{\frac{\ell}{\tan k}}} \cdot \frac{2}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e+223)
   (* (/ (/ l (sin k)) (/ k (/ l (tan k)))) (/ 2.0 (* k t)))
   (* 2.0 (* (/ (pow (/ l k) 2.0) t) (/ (cos k) (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+223) {
		tmp = ((l / sin(k)) / (k / (l / tan(k)))) * (2.0 / (k * t));
	} else {
		tmp = 2.0 * ((pow((l / k), 2.0) / t) * (cos(k) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d+223) then
        tmp = ((l / sin(k)) / (k / (l / tan(k)))) * (2.0d0 / (k * t))
    else
        tmp = 2.0d0 * ((((l / k) ** 2.0d0) / t) * (cos(k) / (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e+223) {
		tmp = ((l / Math.sin(k)) / (k / (l / Math.tan(k)))) * (2.0 / (k * t));
	} else {
		tmp = 2.0 * ((Math.pow((l / k), 2.0) / t) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e+223:
		tmp = ((l / math.sin(k)) / (k / (l / math.tan(k)))) * (2.0 / (k * t))
	else:
		tmp = 2.0 * ((math.pow((l / k), 2.0) / t) * (math.cos(k) / math.pow(math.sin(k), 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e+223)
		tmp = Float64(Float64(Float64(l / sin(k)) / Float64(k / Float64(l / tan(k)))) * Float64(2.0 / Float64(k * t)));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k) ^ 2.0) / t) * Float64(cos(k) / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e+223)
		tmp = ((l / sin(k)) / (k / (l / tan(k)))) * (2.0 / (k * t));
	else
		tmp = 2.0 * ((((l / k) ^ 2.0) / t) * (cos(k) / (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e+223], N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(k / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.00000000000000005e223

   1. Initial program 34.1%

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

      1. associate-*l* 34.1%

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

      2. associate-*l* 34.1%

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

      3. associate-/r* 34.1%

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

      4. associate-/r/ 33.6%

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

      5. *-commutative 33.6%

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

      6. times-frac 33.7%

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

      7. +-commutative 33.7%

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

      8. associate--l+ 43.0%

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

      9. metadata-eval 43.0%

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

      10. +-rgt-identity 43.0%

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

      11. times-frac 49.9%

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

   3. Simplified 49.9%

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

   4. Taylor expanded in t around 0 88.5%

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

      1. unpow2 88.5%

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

   6. Simplified 88.5%

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

      1. associate-*l/ 88.5%

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

      2. associate-*l* 93.5%

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

   8. Applied egg-rr 93.5%

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

      1. associate-*l/ 93.5%

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

      2. *-commutative 93.5%

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

      3. associate-*l/ 91.0%

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

      4. associate-/l* 93.5%

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

   10. Simplified 93.5%

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

       1. associate-*r/ 93.5%

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

       2. associate-/r/ 93.5%

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

       3. *-commutative 93.5%

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

   12. Applied egg-rr 93.5%

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

       1. times-frac 97.0%

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

       2. *-commutative 97.0%

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

   14. Simplified 97.0%

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

       1. *-un-lft-identity 97.0%

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

       2. associate-/l* 97.5%

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

   16. Applied egg-rr 97.5%

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

   if 1.00000000000000005e223 < (*.f64 l l)

   1. Initial program 36.3%

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

      1. associate-*l* 36.3%

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

      2. associate-*l* 36.3%

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

      3. associate-/r* 36.3%

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

      4. associate-/r/ 36.3%

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

      5. *-commutative 36.3%

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

      6. times-frac 36.2%

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

      7. +-commutative 36.2%

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

      8. associate--l+ 36.3%

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

      9. metadata-eval 36.3%

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

      10. +-rgt-identity 36.3%

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

      11. times-frac 36.3%

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

   3. Simplified 36.3%

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

   4. Taylor expanded in t around 0 60.9%

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

      1. associate-/r* 63.5%

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

      2. unpow2 63.5%

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

      3. unpow2 63.5%

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

      4. *-commutative 63.5%

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

   6. Simplified 63.5%

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

   7. Taylor expanded in k around inf 63.5%

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

      1. associate-/l* 63.5%

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

      2. unpow2 63.5%

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

      3. unpow2 63.5%

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

      4. times-frac 96.1%

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

   9. Simplified 96.1%

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

   10. Taylor expanded in k around inf 60.9%

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

       1. *-commutative 60.9%

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

       2. associate-*r* 60.9%

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

       3. unpow2 60.9%

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

       4. associate-*r* 64.8%

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

       5. *-commutative 64.8%

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

       6. times-frac 64.8%

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

       7. associate-*r* 60.9%

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

       8. unpow2 60.9%

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

       9. associate-/r* 63.5%

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

       10. unpow2 63.5%

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

       11. unpow2 63.5%

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

       12. times-frac 96.1%

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

       13. unpow2 96.1%

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

   12. Simplified 96.1%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+223}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\frac{k}{\frac{\ell}{\tan k}}} \cdot \frac{2}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 2: 95.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+233}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\frac{k}{\frac{\ell}{\tan k}}} \cdot \frac{2}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{t \cdot {\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 2e+233)
   (* (/ (/ l (sin k)) (/ k (/ l (tan k)))) (/ 2.0 (* k t)))
   (* 2.0 (/ (cos k) (* t (pow (* (sin k) (/ k l)) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e+233) {
		tmp = ((l / sin(k)) / (k / (l / tan(k)))) * (2.0 / (k * t));
	} else {
		tmp = 2.0 * (cos(k) / (t * pow((sin(k) * (k / l)), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 2d+233) then
        tmp = ((l / sin(k)) / (k / (l / tan(k)))) * (2.0d0 / (k * t))
    else
        tmp = 2.0d0 * (cos(k) / (t * ((sin(k) * (k / l)) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 2e+233) {
		tmp = ((l / Math.sin(k)) / (k / (l / Math.tan(k)))) * (2.0 / (k * t));
	} else {
		tmp = 2.0 * (Math.cos(k) / (t * Math.pow((Math.sin(k) * (k / l)), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 2e+233:
		tmp = ((l / math.sin(k)) / (k / (l / math.tan(k)))) * (2.0 / (k * t))
	else:
		tmp = 2.0 * (math.cos(k) / (t * math.pow((math.sin(k) * (k / l)), 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 2e+233)
		tmp = Float64(Float64(Float64(l / sin(k)) / Float64(k / Float64(l / tan(k)))) * Float64(2.0 / Float64(k * t)));
	else
		tmp = Float64(2.0 * Float64(cos(k) / Float64(t * (Float64(sin(k) * Float64(k / l)) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 2e+233)
		tmp = ((l / sin(k)) / (k / (l / tan(k)))) * (2.0 / (k * t));
	else
		tmp = 2.0 * (cos(k) / (t * ((sin(k) * (k / l)) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 2e+233], N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(k / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[(N[Sin[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999995e233

   1. Initial program 33.9%

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

      1. associate-*l* 33.9%

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

      2. associate-*l* 33.9%

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

      3. associate-/r* 33.9%

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

      4. associate-/r/ 33.4%

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

      5. *-commutative 33.4%

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

      6. times-frac 33.5%

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

      7. +-commutative 33.5%

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

      8. associate--l+ 42.8%

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

      9. metadata-eval 42.8%

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

      10. +-rgt-identity 42.8%

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

      11. times-frac 49.6%

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

   3. Simplified 49.6%

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

   4. Taylor expanded in t around 0 88.5%

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

      1. unpow2 88.5%

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

   6. Simplified 88.5%

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

      1. associate-*l/ 88.5%

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

      2. associate-*l* 93.5%

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

   8. Applied egg-rr 93.5%

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

      1. associate-*l/ 93.5%

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

      2. *-commutative 93.5%

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

      3. associate-*l/ 91.0%

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

      4. associate-/l* 93.5%

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

   10. Simplified 93.5%

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

       1. associate-*r/ 93.6%

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

       2. associate-/r/ 93.5%

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

       3. *-commutative 93.5%

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

   12. Applied egg-rr 93.5%

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

       1. times-frac 97.0%

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

       2. *-commutative 97.0%

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

   14. Simplified 97.0%

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

       1. *-un-lft-identity 97.0%

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

       2. associate-/l* 97.6%

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

   16. Applied egg-rr 97.6%

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

   if 1.99999999999999995e233 < (*.f64 l l)

   1. Initial program 36.8%

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

      1. associate-*l* 36.8%

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

      2. associate-*l* 36.8%

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

      3. associate-/r* 36.8%

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

      4. associate-/r/ 36.8%

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

      5. *-commutative 36.8%

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

      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+ 36.8%

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

      9. metadata-eval 36.8%

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

      10. +-rgt-identity 36.8%

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

      11. times-frac 36.8%

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

   3. Simplified 36.8%

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

   4. Taylor expanded in t around 0 60.3%

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

      1. associate-/r* 63.0%

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

      2. unpow2 63.0%

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

      3. unpow2 63.0%

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

      4. *-commutative 63.0%

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

   6. Simplified 63.0%

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

   7. Taylor expanded in k around inf 63.0%

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

      1. associate-/l* 63.0%

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

      2. unpow2 63.0%

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

      3. unpow2 63.0%

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

      4. times-frac 96.0%

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

   9. Simplified 96.0%

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

       1. expm1-log1p-u 49.6%

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

       2. expm1-udef 34.9%

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

       3. associate-/l/ 34.9%

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

       4. pow2 34.9%

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

   11. Applied egg-rr 34.9%

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

       1. expm1-def 49.6%

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

       2. expm1-log1p 96.0%

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

       3. associate-*l* 96.0%

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

   13. Simplified 96.0%

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

       1. *-commutative 96.0%

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

       2. pow-prod-down 96.0%

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

   15. Applied egg-rr 96.0%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+233}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{\frac{k}{\frac{\ell}{\tan k}}} \cdot \frac{2}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{t \cdot {\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 3: 94.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{k \cdot t}\\ \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;t_1 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{\ell}{k}}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot \sin k} \cdot \left(\frac{\ell}{\tan k} \cdot t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* k t))))
   (if (<= (* l l) 0.0)
     (* t_1 (* (/ l (sin k)) (/ (/ l k) (tan k))))
     (* (/ l (* k (sin k))) (* (/ l (tan k)) t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 / (k * t);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = t_1 * ((l / sin(k)) * ((l / k) / tan(k)));
	} else {
		tmp = (l / (k * sin(k))) * ((l / tan(k)) * t_1);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 / (k * t)
    if ((l * l) <= 0.0d0) then
        tmp = t_1 * ((l / sin(k)) * ((l / k) / tan(k)))
    else
        tmp = (l / (k * sin(k))) * ((l / tan(k)) * t_1)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 / (k * t);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = t_1 * ((l / Math.sin(k)) * ((l / k) / Math.tan(k)));
	} else {
		tmp = (l / (k * Math.sin(k))) * ((l / Math.tan(k)) * t_1);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 2.0 / (k * t)
	tmp = 0
	if (l * l) <= 0.0:
		tmp = t_1 * ((l / math.sin(k)) * ((l / k) / math.tan(k)))
	else:
		tmp = (l / (k * math.sin(k))) * ((l / math.tan(k)) * t_1)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 / N[(k * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(t$95$1 * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 15.5%

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

      1. associate-*l* 15.5%

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

      2. associate-*l* 15.5%

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

      3. associate-/r* 15.5%

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

      4. associate-/r/ 15.5%

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

      5. *-commutative 15.5%

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

      6. times-frac 15.5%

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

      7. +-commutative 15.5%

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

      8. associate--l+ 21.9%

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

      9. metadata-eval 21.9%

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

      10. +-rgt-identity 21.9%

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

      11. times-frac 37.5%

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

   3. Simplified 37.5%

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

   4. Taylor expanded in t around 0 86.0%

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

      1. unpow2 86.0%

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

   6. Simplified 86.0%

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

      1. associate-*l/ 85.9%

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

      2. associate-*l* 93.7%

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

   8. Applied egg-rr 93.7%

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

      1. associate-*l/ 93.8%

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

      2. *-commutative 93.8%

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

      3. associate-*l/ 88.0%

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

      4. associate-/l* 93.8%

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

   10. Simplified 93.8%

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

       1. associate-*r/ 93.7%

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

       2. associate-/r/ 93.7%

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

       3. *-commutative 93.7%

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

   12. Applied egg-rr 93.7%

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

       1. times-frac 94.9%

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

       2. *-commutative 94.9%

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

   14. Simplified 94.9%

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

       1. expm1-log1p-u 89.3%

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

       2. expm1-udef 85.6%

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

       3. associate-/l* 85.6%

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

   16. Applied egg-rr 85.6%

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

       1. expm1-def 90.5%

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

       2. expm1-log1p 96.1%

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

       3. associate-/r/ 96.0%

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

       4. associate-*l/ 94.9%

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

       5. times-frac 62.0%

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

       6. *-rgt-identity 62.0%

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

       7. associate-*r/ 62.0%

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

       8. unpow2 62.0%

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

       9. associate-*l/ 62.0%

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

       10. unpow2 62.0%

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

       11. associate-*l* 83.1%

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

       12. times-frac 96.0%

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

       13. associate-*r/ 96.0%

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

       14. *-rgt-identity 96.0%

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

   18. Simplified 96.0%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 43.2%

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

      1. associate-*l* 43.2%

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

      2. associate-*l* 43.2%

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

      3. associate-/r* 43.2%

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

      4. associate-/r/ 42.7%

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

      5. *-commutative 42.7%

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

      6. times-frac 42.8%

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

      7. +-commutative 42.8%

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

      8. associate--l+ 49.3%

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

      9. metadata-eval 49.3%

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

      10. +-rgt-identity 49.3%

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

      11. times-frac 49.4%

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

   3. Simplified 49.4%

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

   4. Taylor expanded in t around 0 77.5%

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

      1. unpow2 77.5%

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

   6. Simplified 77.5%

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

      1. associate-*l/ 77.5%

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

      2. associate-*l* 80.8%

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

   8. Applied egg-rr 80.8%

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

      1. associate-*l/ 80.8%

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

      2. *-commutative 80.8%

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

      3. associate-*l/ 80.8%

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

      4. associate-/l* 80.8%

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

   10. Simplified 80.8%

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

       1. associate-*r/ 80.8%

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

       2. associate-/r/ 80.8%

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

       3. *-commutative 80.8%

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

   12. Applied egg-rr 80.8%

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

       1. times-frac 84.7%

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

       2. *-commutative 84.7%

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

   14. Simplified 84.7%

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

       1. expm1-log1p-u 52.0%

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

       2. expm1-udef 40.5%

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

       3. associate-/l* 43.7%

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

       4. *-commutative 43.7%

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

   16. Applied egg-rr 43.7%

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

       1. expm1-def 56.8%

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

       2. expm1-log1p 90.1%

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

       3. associate-/r/ 90.1%

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

       4. associate-*l* 94.9%

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

       5. associate-/l/ 94.9%

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

       6. *-commutative 94.9%

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

   18. Simplified 94.9%

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

4. Final simplification 95.2%

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

Alternative 4: 94.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{\tan k}\\ \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-196}:\\ \;\;\;\;t_1 \cdot \frac{\ell \cdot \frac{2}{k \cdot \left(k \cdot t\right)}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot \sin k} \cdot \left(t_1 \cdot \frac{2}{k \cdot t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (tan k))))
   (if (<= (* l l) 2e-196)
     (* t_1 (/ (* l (/ 2.0 (* k (* k t)))) (sin k)))
     (* (/ l (* k (sin k))) (* t_1 (/ 2.0 (* k t)))))))

C

double code(double t, double l, double k) {
	double t_1 = l / tan(k);
	double tmp;
	if ((l * l) <= 2e-196) {
		tmp = t_1 * ((l * (2.0 / (k * (k * t)))) / sin(k));
	} else {
		tmp = (l / (k * sin(k))) * (t_1 * (2.0 / (k * t)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l / tan(k)
    if ((l * l) <= 2d-196) then
        tmp = t_1 * ((l * (2.0d0 / (k * (k * t)))) / sin(k))
    else
        tmp = (l / (k * sin(k))) * (t_1 * (2.0d0 / (k * t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l / Math.tan(k);
	double tmp;
	if ((l * l) <= 2e-196) {
		tmp = t_1 * ((l * (2.0 / (k * (k * t)))) / Math.sin(k));
	} else {
		tmp = (l / (k * Math.sin(k))) * (t_1 * (2.0 / (k * t)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l / math.tan(k)
	tmp = 0
	if (l * l) <= 2e-196:
		tmp = t_1 * ((l * (2.0 / (k * (k * t)))) / math.sin(k))
	else:
		tmp = (l / (k * math.sin(k))) * (t_1 * (2.0 / (k * t)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / tan(k))
	tmp = 0.0
	if (Float64(l * l) <= 2e-196)
		tmp = Float64(t_1 * Float64(Float64(l * Float64(2.0 / Float64(k * Float64(k * t)))) / sin(k)));
	else
		tmp = Float64(Float64(l / Float64(k * sin(k))) * Float64(t_1 * Float64(2.0 / Float64(k * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l / tan(k);
	tmp = 0.0;
	if ((l * l) <= 2e-196)
		tmp = t_1 * ((l * (2.0 / (k * (k * t)))) / sin(k));
	else
		tmp = (l / (k * sin(k))) * (t_1 * (2.0 / (k * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 2e-196], N[(t$95$1 * N[(N[(l * N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(2.0 / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 2.0000000000000001e-196

   1. Initial program 23.2%

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

      1. associate-*l* 23.2%

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

      2. associate-*l* 23.2%

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

      3. associate-/r* 23.2%

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

      4. associate-/r/ 22.3%

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

      5. *-commutative 22.3%

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

      6. times-frac 22.3%

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

      7. +-commutative 22.3%

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

      8. associate--l+ 30.4%

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

      9. metadata-eval 30.4%

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

      10. +-rgt-identity 30.4%

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

      11. times-frac 42.7%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in t around 0 88.9%

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

      1. unpow2 88.9%

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

   6. Simplified 88.9%

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

      1. associate-*l/ 88.8%

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

      2. associate-*l* 95.0%

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

   8. Applied egg-rr 95.0%

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

      1. associate-*l/ 95.1%

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

      2. associate-*r* 96.9%

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

      3. associate-*r/ 97.0%

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

   10. Simplified 97.0%

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

   if 2.0000000000000001e-196 < (*.f64 l l)

   1. Initial program 42.1%

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

      1. associate-*l* 42.1%

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

      2. associate-*l* 42.0%

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

      3. associate-/r* 42.1%

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

      4. associate-/r/ 42.1%

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

      5. *-commutative 42.1%

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

      6. times-frac 42.1%

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

      7. +-commutative 42.1%

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

      8. associate--l+ 47.6%

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

      9. metadata-eval 47.6%

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

      10. +-rgt-identity 47.6%

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

      11. times-frac 47.7%

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

   3. Simplified 47.7%

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

   4. Taylor expanded in t around 0 74.5%

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

      1. unpow2 74.5%

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

   6. Simplified 74.5%

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

      1. associate-*l/ 74.5%

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

      2. associate-*l* 78.3%

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

   8. Applied egg-rr 78.3%

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

      1. associate-*l/ 78.3%

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

      2. *-commutative 78.3%

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

      3. associate-*l/ 78.2%

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

      4. associate-/l* 78.2%

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

   10. Simplified 78.2%

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

       1. associate-*r/ 78.3%

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

       2. associate-/r/ 78.3%

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

       3. *-commutative 78.3%

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

   12. Applied egg-rr 78.3%

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

       1. times-frac 83.3%

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

       2. *-commutative 83.3%

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

   14. Simplified 83.3%

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

       1. expm1-log1p-u 47.5%

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

       2. expm1-udef 36.9%

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

       3. associate-/l* 40.4%

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

       4. *-commutative 40.4%

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

   16. Applied egg-rr 40.4%

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

       1. expm1-def 52.9%

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

       2. expm1-log1p 89.5%

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

       3. associate-/r/ 89.4%

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

       4. associate-*l* 94.3%

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

       5. associate-/l/ 94.2%

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

       6. *-commutative 94.2%

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

   18. Simplified 94.2%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-196}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\ell \cdot \frac{2}{k \cdot \left(k \cdot t\right)}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot \sin k} \cdot \left(\frac{\ell}{\tan k} \cdot \frac{2}{k \cdot t}\right)\\ \end{array} \]

Alternative 5: 90.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.8%

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

   1. associate-*l* 34.8%

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

   2. associate-*l* 34.8%

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

   3. associate-/r* 34.8%

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

   4. associate-/r/ 34.4%

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

   5. *-commutative 34.4%

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

   6. times-frac 34.5%

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

   7. +-commutative 34.5%

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

   8. associate--l+ 41.0%

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

   9. metadata-eval 41.0%

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

   10. +-rgt-identity 41.0%

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

   11. times-frac 45.7%

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

3. Simplified 45.7%

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

4. Taylor expanded in t around 0 80.0%

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

   1. unpow2 80.0%

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

6. Simplified 80.0%

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

   1. associate-*l/ 80.0%

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

   2. associate-*l* 84.8%

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

8. Applied egg-rr 84.8%

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

   1. associate-*l/ 84.8%

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

   2. *-commutative 84.8%

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

   3. associate-*l/ 83.0%

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

   4. associate-/l* 84.8%

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

10. Simplified 84.8%

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

    1. associate-*r/ 84.8%

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

    2. associate-/r/ 84.8%

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

    3. *-commutative 84.8%

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

12. Applied egg-rr 84.8%

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

    1. times-frac 87.8%

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

    2. *-commutative 87.8%

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

14. Simplified 87.8%

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

    1. expm1-log1p-u 69.3%

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

    2. expm1-udef 60.5%

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

    3. associate-/l* 62.6%

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

16. Applied egg-rr 62.6%

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

    1. expm1-def 71.7%

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

    2. expm1-log1p 92.0%

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

    3. associate-/r/ 91.9%

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

    4. associate-*l/ 87.8%

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

    5. times-frac 77.8%

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

    6. *-rgt-identity 77.8%

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

    7. associate-*r/ 77.7%

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

    8. unpow2 77.7%

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

    9. associate-*l/ 77.8%

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

    10. unpow2 77.8%

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

    11. associate-*l* 88.0%

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

    12. times-frac 91.9%

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

    13. associate-*r/ 91.9%

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

    14. *-rgt-identity 91.9%

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

18. Simplified 91.9%

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

19. Final simplification 91.9%

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

Alternative 6: 72.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := k \cdot \left(k \cdot t\right)\\ \mathbf{if}\;k \leq 6.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{2}{t_1} \cdot \frac{\ell}{k \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{t_1} + \frac{\ell \cdot \ell}{t} \cdot 0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (* k t))))
   (if (<= k 6.8e-66)
     (* (/ 2.0 t_1) (/ l (* k (/ k l))))
     (*
      2.0
      (*
       (/ (cos k) (* k k))
       (+ (/ (* l l) t_1) (* (/ (* l l) t) 0.3333333333333333)))))))

C

double code(double t, double l, double k) {
	double t_1 = k * (k * t);
	double tmp;
	if (k <= 6.8e-66) {
		tmp = (2.0 / t_1) * (l / (k * (k / l)));
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * (((l * l) / t_1) + (((l * l) / t) * 0.3333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = k * (k * t)
    if (k <= 6.8d-66) then
        tmp = (2.0d0 / t_1) * (l / (k * (k / l)))
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * (((l * l) / t_1) + (((l * l) / t) * 0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = k * (k * t);
	double tmp;
	if (k <= 6.8e-66) {
		tmp = (2.0 / t_1) * (l / (k * (k / l)));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * (((l * l) / t_1) + (((l * l) / t) * 0.3333333333333333)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = k * (k * t)
	tmp = 0
	if k <= 6.8e-66:
		tmp = (2.0 / t_1) * (l / (k * (k / l)))
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * (((l * l) / t_1) + (((l * l) / t) * 0.3333333333333333)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(k * t))
	tmp = 0.0
	if (k <= 6.8e-66)
		tmp = Float64(Float64(2.0 / t_1) * Float64(l / Float64(k * Float64(k / l))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(Float64(l * l) / t_1) + Float64(Float64(Float64(l * l) / t) * 0.3333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = k * (k * t);
	tmp = 0.0;
	if (k <= 6.8e-66)
		tmp = (2.0 / t_1) * (l / (k * (k / l)));
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * (((l * l) / t_1) + (((l * l) / t) * 0.3333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 6.8e-66], N[(N[(2.0 / t$95$1), $MachinePrecision] * N[(l / N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 6.79999999999999994e-66

   1. Initial program 37.8%

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

      1. associate-*l* 37.8%

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

      2. associate-*l* 37.8%

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

      3. associate-/r* 37.8%

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

      4. associate-/r/ 37.3%

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

      5. *-commutative 37.3%

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

      6. times-frac 36.8%

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

      7. +-commutative 36.8%

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

      8. associate--l+ 40.8%

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

      9. metadata-eval 40.8%

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

      10. +-rgt-identity 40.8%

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

      11. times-frac 48.0%

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

   3. Simplified 48.0%

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

   4. Taylor expanded in t around 0 84.2%

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

      1. unpow2 84.2%

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

   6. Simplified 84.2%

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

      1. associate-*l/ 84.2%

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

      2. associate-*l* 88.4%

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

   8. Applied egg-rr 88.4%

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

      1. associate-*l/ 88.4%

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

      2. *-commutative 88.4%

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

      3. associate-*l/ 85.7%

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

      4. associate-/l* 88.4%

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

   10. Simplified 88.4%

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

   11. Taylor expanded in k around 0 77.0%

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

       1. unpow2 77.0%

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

       2. associate-*r/ 80.6%

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

   13. Simplified 80.6%

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

   if 6.79999999999999994e-66 < k

   1. Initial program 28.9%

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

      1. associate-*l* 28.9%

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

      2. associate-*l* 28.9%

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

      3. associate-/r* 29.0%

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

      4. associate-/r/ 28.9%

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

      5. *-commutative 28.9%

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

      6. times-frac 30.0%

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

      7. +-commutative 30.0%

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

      8. associate--l+ 41.3%

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

      9. metadata-eval 41.3%

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

      10. +-rgt-identity 41.3%

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

      11. times-frac 41.4%

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

   3. Simplified 41.4%

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

   4. Taylor expanded in t around 0 72.2%

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

      1. unpow2 72.2%

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

      2. times-frac 74.4%

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

      3. unpow2 74.4%

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

      4. *-commutative 74.4%

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

   6. Simplified 74.4%

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

   7. Taylor expanded in k around 0 66.7%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. unpow2 66.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      2. unpow2 66.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      3. associate-*r* 66.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

      4. *-commutative 66.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \color{blue}{\frac{{\ell}^{2}}{t} \cdot 0.3333333333333333}\right)\right) \]

      5. unpow2 66.7%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\color{blue}{\ell \cdot \ell}}{t} \cdot 0.3333333333333333\right)\right) \]

   9. Simplified 66.7%

      \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\ell \cdot \ell}{t} \cdot 0.3333333333333333\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{k \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\ell \cdot \ell}{t} \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 7: 71.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k}{\ell}\right)}^{-2}}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot \left(k \cdot k\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.2e-46)
   (/ (* 2.0 (pow (/ k l) -2.0)) (* k (* k t)))
   (* 2.0 (* (/ (cos k) (* k k)) (/ (* l l) (* t (* k k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.2e-46) {
		tmp = (2.0 * pow((k / l), -2.0)) / (k * (k * t));
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l * l) / (t * (k * k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.2d-46) then
        tmp = (2.0d0 * ((k / l) ** (-2.0d0))) / (k * (k * t))
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * ((l * l) / (t * (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.2e-46) {
		tmp = (2.0 * Math.pow((k / l), -2.0)) / (k * (k * t));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * ((l * l) / (t * (k * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.2e-46:
		tmp = (2.0 * math.pow((k / l), -2.0)) / (k * (k * t))
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * ((l * l) / (t * (k * k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.2e-46)
		tmp = Float64(Float64(2.0 * (Float64(k / l) ^ -2.0)) / Float64(k * Float64(k * t)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(l * l) / Float64(t * Float64(k * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.2e-46)
		tmp = (2.0 * ((k / l) ^ -2.0)) / (k * (k * t));
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l * l) / (t * (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.2e-46], N[(N[(2.0 * N[Power[N[(k / l), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.2000000000000001e-46

   1. Initial program 38.5%

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

      1. associate-*l* 38.5%

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

      2. associate-*l* 38.5%

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

      3. associate-/r* 38.5%

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

      4. associate-/r/ 38.0%

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

      5. *-commutative 38.0%

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

      6. times-frac 37.5%

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

      7. +-commutative 37.5%

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

      8. associate--l+ 41.9%

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

      9. metadata-eval 41.9%

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

      10. +-rgt-identity 41.9%

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

      11. times-frac 49.0%

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

   3. Simplified 49.0%

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

   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 66.1%

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

      1. unpow2 66.1%

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

      2. unpow2 66.1%

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

   9. Simplified 66.1%

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

       1. associate-*r* 66.1%

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

       2. associate-*l/ 66.1%

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

       3. clear-num 66.1%

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

       4. frac-times 81.1%

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

       5. pow2 81.1%

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

       6. pow-flip 81.2%

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

       7. metadata-eval 81.2%

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

   11. Applied egg-rr 81.2%

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

   if 2.2000000000000001e-46 < k

   1. Initial program 27.0%

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

      1. associate-*l* 27.0%

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

      2. associate-*l* 27.0%

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

      3. associate-/r* 27.0%

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

      4. associate-/r/ 27.0%

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

      5. *-commutative 27.0%

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

      6. times-frac 28.1%

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

      7. +-commutative 28.1%

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

      8. associate--l+ 39.0%

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

      9. metadata-eval 39.0%

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

      10. +-rgt-identity 39.0%

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

      11. times-frac 39.1%

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

   3. Simplified 39.1%

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

   4. Taylor expanded in t around 0 70.5%

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

      1. unpow2 70.5%

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

      2. times-frac 72.8%

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

      3. unpow2 72.8%

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

      4. *-commutative 72.8%

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

   6. Simplified 72.8%

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

   7. Taylor expanded in k around 0 63.9%

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

      1. unpow2 63.9%

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

   9. Simplified 63.9%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k}{\ell}\right)}^{-2}}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot \left(k \cdot k\right)}\right)\\ \end{array} \]

Alternative 8: 71.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k}{\ell}\right)}^{-2}}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{k \cdot k}}{t \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.2e-46)
   (/ (* 2.0 (pow (/ k l) -2.0)) (* k (* k t)))
   (* 2.0 (/ (/ (* (* l l) (cos k)) (* k k)) (* t (* k k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.2e-46) {
		tmp = (2.0 * pow((k / l), -2.0)) / (k * (k * t));
	} else {
		tmp = 2.0 * ((((l * l) * cos(k)) / (k * k)) / (t * (k * k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.2d-46) then
        tmp = (2.0d0 * ((k / l) ** (-2.0d0))) / (k * (k * t))
    else
        tmp = 2.0d0 * ((((l * l) * cos(k)) / (k * k)) / (t * (k * k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.2e-46) {
		tmp = (2.0 * Math.pow((k / l), -2.0)) / (k * (k * t));
	} else {
		tmp = 2.0 * ((((l * l) * Math.cos(k)) / (k * k)) / (t * (k * k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.2e-46:
		tmp = (2.0 * math.pow((k / l), -2.0)) / (k * (k * t))
	else:
		tmp = 2.0 * ((((l * l) * math.cos(k)) / (k * k)) / (t * (k * k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.2e-46)
		tmp = Float64(Float64(2.0 * (Float64(k / l) ^ -2.0)) / Float64(k * Float64(k * t)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(Float64(l * l) * cos(k)) / Float64(k * k)) / Float64(t * Float64(k * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.2e-46)
		tmp = (2.0 * ((k / l) ^ -2.0)) / (k * (k * t));
	else
		tmp = 2.0 * ((((l * l) * cos(k)) / (k * k)) / (t * (k * k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.2e-46], N[(N[(2.0 * N[Power[N[(k / l), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(N[(l * l), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.2000000000000001e-46

   1. Initial program 38.5%

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

      1. associate-*l* 38.5%

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

      2. associate-*l* 38.5%

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

      3. associate-/r* 38.5%

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

      4. associate-/r/ 38.0%

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

      5. *-commutative 38.0%

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

      6. times-frac 37.5%

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

      7. +-commutative 37.5%

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

      8. associate--l+ 41.9%

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

      9. metadata-eval 41.9%

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

      10. +-rgt-identity 41.9%

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

      11. times-frac 49.0%

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

   3. Simplified 49.0%

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

   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 66.1%

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

      1. unpow2 66.1%

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

      2. unpow2 66.1%

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

   9. Simplified 66.1%

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

       1. associate-*r* 66.1%

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

       2. associate-*l/ 66.1%

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

       3. clear-num 66.1%

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

       4. frac-times 81.1%

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

       5. pow2 81.1%

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

       6. pow-flip 81.2%

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

       7. metadata-eval 81.2%

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

   11. Applied egg-rr 81.2%

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

   if 2.2000000000000001e-46 < k

   1. Initial program 27.0%

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

      1. associate-*l* 27.0%

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

      2. associate-*l* 27.0%

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

      3. associate-/r* 27.0%

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

      4. associate-/r/ 27.0%

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

      5. *-commutative 27.0%

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

      6. times-frac 28.1%

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

      7. +-commutative 28.1%

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

      8. associate--l+ 39.0%

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

      9. metadata-eval 39.0%

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

      10. +-rgt-identity 39.0%

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

      11. times-frac 39.1%

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

   3. Simplified 39.1%

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

   4. Taylor expanded in t around 0 70.5%

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

      1. associate-/r* 73.0%

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

      2. unpow2 73.0%

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

      3. unpow2 73.0%

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

      4. *-commutative 73.0%

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

   6. Simplified 73.0%

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

   7. Taylor expanded in k around 0 63.9%

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

      1. unpow2 63.9%

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

   9. Simplified 63.9%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{k}{\ell}\right)}^{-2}}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{k \cdot k}}{t \cdot \left(k \cdot k\right)}\\ \end{array} \]

Alternative 9: 70.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.8%

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

   1. associate-*l* 34.8%

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

   2. associate-*l* 34.8%

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

   3. associate-/r* 34.8%

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

   4. associate-/r/ 34.4%

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

   5. *-commutative 34.4%

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

   6. times-frac 34.5%

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

   7. +-commutative 34.5%

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

   8. associate--l+ 41.0%

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

   9. metadata-eval 41.0%

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

   10. +-rgt-identity 41.0%

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

   11. times-frac 45.7%

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

3. Simplified 45.7%

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

4. Taylor expanded in t around 0 80.0%

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

   1. unpow2 80.0%

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

6. Simplified 80.0%

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

7. Taylor expanded in k around 0 64.9%

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

   1. unpow2 64.9%

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

   2. unpow2 64.9%

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

9. Simplified 64.9%

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

    1. associate-*r* 64.9%

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

    2. associate-*l/ 64.9%

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

    3. clear-num 64.9%

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

    4. frac-times 74.8%

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

    5. pow2 74.8%

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

    6. pow-flip 74.8%

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

    7. metadata-eval 74.8%

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

11. Applied egg-rr 74.8%

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

12. Final simplification 74.8%

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

Alternative 10: 69.5% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.8%

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

   1. associate-*l* 34.8%

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

   2. associate-*l* 34.8%

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

   3. associate-/r* 34.8%

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

   4. associate-/r/ 34.4%

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

   5. *-commutative 34.4%

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

   6. times-frac 34.5%

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

   7. +-commutative 34.5%

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

   8. associate--l+ 41.0%

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

   9. metadata-eval 41.0%

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

   10. +-rgt-identity 41.0%

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

   11. times-frac 45.7%

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

3. Simplified 45.7%

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

4. Taylor expanded in t around 0 80.0%

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

   1. unpow2 80.0%

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

6. Simplified 80.0%

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

7. Taylor expanded in k around 0 64.9%

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

   1. unpow2 64.9%

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

   2. unpow2 64.9%

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

9. Simplified 64.9%

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

    1. times-frac 72.4%

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

11. Applied egg-rr 72.4%

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

12. Final simplification 72.4%

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

Alternative 11: 70.6% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.8%

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

   1. associate-*l* 34.8%

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

   2. associate-*l* 34.8%

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

   3. associate-/r* 34.8%

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

   4. associate-/r/ 34.4%

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

   5. *-commutative 34.4%

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

   6. times-frac 34.5%

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

   7. +-commutative 34.5%

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

   8. associate--l+ 41.0%

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

   9. metadata-eval 41.0%

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

   10. +-rgt-identity 41.0%

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

   11. times-frac 45.7%

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

3. Simplified 45.7%

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

4. Taylor expanded in t around 0 80.0%

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

   1. unpow2 80.0%

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

6. Simplified 80.0%

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

   1. associate-*l/ 80.0%

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

   2. associate-*l* 84.8%

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

8. Applied egg-rr 84.8%

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

   1. associate-*l/ 84.8%

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

   2. *-commutative 84.8%

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

   3. associate-*l/ 83.0%

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

   4. associate-/l* 84.8%

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

10. Simplified 84.8%

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

11. Taylor expanded in k around 0 72.4%

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

    1. unpow2 72.4%

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

    2. associate-*r/ 74.8%

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

13. Simplified 74.8%

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

14. Final simplification 74.8%

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

Reproduce

herbie shell --seed 2023224 
(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))))