Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.0% → 85.8%

Time: 21.0s

Alternatives: 21

Speedup: 22.2×

• 27.0% of points produce a very large (infinite) output. You may want to add a precondition.

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: 54.0% 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: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 10^{+242}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\ell}{\tan k} \cdot \frac{\frac{\ell}{{t}^{3}}}{\sin k}\right)}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{t}{\frac{\ell}{k}} \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ 1.0 (+ 1.0 t_1))))
        1e+242)
     (/ (* 2.0 (* (/ l (tan k)) (/ (/ l (pow t 3.0)) (sin k)))) (+ 2.0 t_1))
     (/ 2.0 (* (/ k l) (* (/ t (/ l k)) (* (sin k) (tan k))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 1e+242) {
		tmp = (2.0 * ((l / tan(k)) * ((l / pow(t, 3.0)) / sin(k)))) / (2.0 + t_1);
	} else {
		tmp = 2.0 / ((k / l) * ((t / (l / k)) * (sin(k) * tan(k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / t) ** 2.0d0
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * (1.0d0 + (1.0d0 + t_1)))) <= 1d+242) then
        tmp = (2.0d0 * ((l / tan(k)) * ((l / (t ** 3.0d0)) / sin(k)))) / (2.0d0 + t_1)
    else
        tmp = 2.0d0 / ((k / l) * ((t / (l / k)) * (sin(k) * tan(k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + t_1)))) <= 1e+242) {
		tmp = (2.0 * ((l / Math.tan(k)) * ((l / Math.pow(t, 3.0)) / Math.sin(k)))) / (2.0 + t_1);
	} else {
		tmp = 2.0 / ((k / l) * ((t / (l / k)) * (Math.sin(k) * Math.tan(k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	tmp = 0
	if (2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * (1.0 + (1.0 + t_1)))) <= 1e+242:
		tmp = (2.0 * ((l / math.tan(k)) * ((l / math.pow(t, 3.0)) / math.sin(k)))) / (2.0 + t_1)
	else:
		tmp = 2.0 / ((k / l) * ((t / (l / k)) * (math.sin(k) * math.tan(k))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1)))) <= 1e+242)
		tmp = Float64(Float64(2.0 * Float64(Float64(l / tan(k)) * Float64(Float64(l / (t ^ 3.0)) / sin(k)))) / Float64(2.0 + t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(t / Float64(l / k)) * Float64(sin(k) * tan(k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 1e+242)
		tmp = (2.0 * ((l / tan(k)) * ((l / (t ^ 3.0)) / sin(k)))) / (2.0 + t_1);
	else
		tmp = 2.0 / ((k / l) * ((t / (l / k)) * (sin(k) * tan(k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[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[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+242], N[(N[(2.0 * N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 1.00000000000000005e242

   1. Initial program 81.4%

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

      1. associate-*l* 81.4%

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

      2. associate-/l/ 81.4%

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

      3. *-commutative 81.4%

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

      4. associate-*r/ 82.4%

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

      5. associate-/l* 82.4%

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

      6. associate-/r/ 73.1%

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

   3. Simplified 75.0%

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

      1. expm1-log1p-u 58.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)\right)\right)} \]

      2. expm1-udef 53.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)\right)} - 1} \]

      3. associate-*l/ 53.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{2 \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}\right)} - 1 \]

      4. *-commutative 53.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{t}^{3}}\right)}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\right)} - 1 \]

   5. Applied egg-rr 53.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \frac{\ell}{{t}^{3}}\right)}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 58.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\ell \cdot \frac{\ell}{{t}^{3}}\right)}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\right)\right)} \]

      2. expm1-log1p 75.0%

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

      3. times-frac 75.0%

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

      4. associate-*l/ 75.0%

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

      5. associate-*r/ 73.1%

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

      6. unpow2 73.1%

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

      7. associate-/l/ 73.3%

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

   7. Simplified 86.6%

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

   if 1.00000000000000005e242 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 17.5%

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

      1. *-commutative 17.5%

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

      2. associate-*l* 17.5%

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

      3. associate-*r* 17.5%

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

      4. +-commutative 17.5%

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

      5. associate-+r+ 17.5%

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

      6. metadata-eval 17.5%

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

   3. Simplified 17.5%

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

   4. Taylor expanded in k around inf 58.7%

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

      1. associate-/l* 55.2%

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

      2. associate-/r/ 55.3%

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

      3. unpow2 55.3%

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

      4. unpow2 55.3%

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

   6. Simplified 55.3%

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

      1. pow1 55.3%

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

      2. associate-*l* 55.2%

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

      3. times-frac 77.2%

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

   8. Applied egg-rr 77.2%

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

      1. unpow1 77.2%

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

      2. associate-*r* 78.8%

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

      3. *-commutative 78.8%

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

      4. associate-*r* 89.9%

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

      5. *-commutative 89.9%

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

   10. Simplified 89.9%

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

       1. pow1 89.9%

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

   12. Applied egg-rr 89.9%

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

       1. unpow1 89.9%

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

       2. associate-*l* 89.9%

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

       3. associate-*r/ 86.5%

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

       4. associate-/l* 89.9%

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

   14. Simplified 89.9%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 10^{+242}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\ell}{\tan k} \cdot \frac{\frac{\ell}{{t}^{3}}}{\sin k}\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{t}{\frac{\ell}{k}} \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \end{array} \]

Alternative 2: 69.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\frac{\ell}{k}}\\ t_2 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq 1.7 \cdot 10^{-131}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \frac{{t}^{3} \cdot \frac{k}{\ell}}{\ell}\right)}\\ \mathbf{elif}\;k \leq 3.3 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\frac{k}{\ell} \cdot \sqrt[3]{t_1 \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot t_1\right)\right)}\right)}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{+170}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\frac{k}{\ell} \cdot \frac{t \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (/ l k))) (t_2 (* (sin k) (tan k))))
   (if (<= k 1.7e-131)
     (/
      2.0
      (*
       (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))
       (* (tan k) (/ (* (pow t 3.0) (/ k l)) l))))
     (if (<= k 3.3e+27)
       (/ 2.0 (* t_2 (* (/ k l) (cbrt (* t_1 (* t (* (/ k l) t_1)))))))
       (if (<= k 5.5e+170)
         (/ 2.0 (* t_2 (* (/ k l) (/ (* t k) l))))
         (/ 2.0 (* t_2 (* (/ k l) (* t (/ k l))))))))))

C

double code(double t, double l, double k) {
	double t_1 = t / (l / k);
	double t_2 = sin(k) * tan(k);
	double tmp;
	if (k <= 1.7e-131) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t), 2.0))) * (tan(k) * ((pow(t, 3.0) * (k / l)) / l)));
	} else if (k <= 3.3e+27) {
		tmp = 2.0 / (t_2 * ((k / l) * cbrt((t_1 * (t * ((k / l) * t_1))))));
	} else if (k <= 5.5e+170) {
		tmp = 2.0 / (t_2 * ((k / l) * ((t * k) / l)));
	} else {
		tmp = 2.0 / (t_2 * ((k / l) * (t * (k / l))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = t / (l / k);
	double t_2 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (k <= 1.7e-131) {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t), 2.0))) * (Math.tan(k) * ((Math.pow(t, 3.0) * (k / l)) / l)));
	} else if (k <= 3.3e+27) {
		tmp = 2.0 / (t_2 * ((k / l) * Math.cbrt((t_1 * (t * ((k / l) * t_1))))));
	} else if (k <= 5.5e+170) {
		tmp = 2.0 / (t_2 * ((k / l) * ((t * k) / l)));
	} else {
		tmp = 2.0 / (t_2 * ((k / l) * (t * (k / l))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(t / Float64(l / k))
	t_2 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (k <= 1.7e-131)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))) * Float64(tan(k) * Float64(Float64((t ^ 3.0) * Float64(k / l)) / l))));
	elseif (k <= 3.3e+27)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(k / l) * cbrt(Float64(t_1 * Float64(t * Float64(Float64(k / l) * t_1)))))));
	elseif (k <= 5.5e+170)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(k / l) * Float64(Float64(t * k) / l))));
	else
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(k / l) * Float64(t * Float64(k / l)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.7e-131], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.3e+27], N[(2.0 / N[(t$95$2 * N[(N[(k / l), $MachinePrecision] * N[Power[N[(t$95$1 * N[(t * N[(N[(k / l), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.5e+170], N[(2.0 / N[(t$95$2 * N[(N[(k / l), $MachinePrecision] * N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[(N[(k / l), $MachinePrecision] * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 1.69999999999999998e-131

   1. Initial program 53.1%

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

   2. Taylor expanded in k around 0 52.8%

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

      1. *-commutative 52.8%

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

      2. unpow2 52.8%

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

      3. times-frac 63.5%

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

   4. Simplified 63.5%

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

      1. associate-*l/ 63.5%

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

   6. Applied egg-rr 63.5%

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

   if 1.69999999999999998e-131 < k < 3.2999999999999998e27

   1. Initial program 66.4%

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

      1. *-commutative 66.4%

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

      2. associate-*l* 66.3%

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

      3. associate-*r* 66.2%

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

      4. +-commutative 66.2%

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

      5. associate-+r+ 66.2%

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

      6. metadata-eval 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in k around inf 66.1%

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

      1. associate-/l* 72.5%

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

      2. associate-/r/ 66.1%

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

      3. unpow2 66.1%

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

      4. unpow2 66.1%

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

   6. Simplified 66.1%

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

      1. pow1 66.1%

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

      2. associate-*l* 66.0%

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

      3. times-frac 66.3%

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

   8. Applied egg-rr 66.3%

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

      1. unpow1 66.3%

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

      2. associate-*r* 66.3%

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

      3. *-commutative 66.3%

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

      4. associate-*r* 69.4%

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

      5. *-commutative 69.4%

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

   10. Simplified 69.4%

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

       1. add-cbrt-cube 82.3%

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

   12. Applied egg-rr 82.3%

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

       1. associate-*l* 82.3%

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

       2. associate-*r/ 82.2%

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

       3. associate-/l* 82.3%

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

       4. associate-*l* 82.3%

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

       5. associate-*r/ 82.2%

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

       6. associate-/l* 82.3%

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

   14. Simplified 82.3%

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

   if 3.2999999999999998e27 < k < 5.4999999999999999e170

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

      2. associate-*l* 51.8%

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

      3. associate-*r* 51.7%

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

      4. +-commutative 51.7%

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

      5. associate-+r+ 51.7%

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

      6. metadata-eval 51.7%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in k around inf 85.2%

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

      1. associate-/l* 78.5%

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

      2. associate-/r/ 78.3%

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

      3. unpow2 78.3%

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

      4. unpow2 78.3%

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

   6. Simplified 78.3%

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

      1. pow1 78.3%

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

      2. associate-*l* 78.3%

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

      3. times-frac 78.4%

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

   8. Applied egg-rr 78.4%

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

      1. unpow1 78.4%

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

      2. associate-*r* 78.4%

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

      3. *-commutative 78.4%

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

      4. associate-*r* 88.9%

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

      5. *-commutative 88.9%

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

   10. Simplified 88.9%

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

       1. associate-*r/ 96.1%

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

   12. Applied egg-rr 96.1%

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

   if 5.4999999999999999e170 < k

   1. Initial program 50.2%

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

      1. *-commutative 50.2%

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

      2. associate-*l* 50.2%

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

      3. associate-*r* 50.2%

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

      4. +-commutative 50.2%

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

      5. associate-+r+ 50.2%

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

      6. metadata-eval 50.2%

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

   3. Simplified 50.2%

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

   4. Taylor expanded in k around inf 66.1%

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

      1. associate-/l* 65.7%

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

      2. associate-/r/ 66.1%

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

      3. unpow2 66.1%

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

      4. unpow2 66.1%

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

   6. Simplified 66.1%

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

      1. pow1 66.1%

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

      2. associate-*l* 66.1%

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

      3. times-frac 92.4%

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

   8. Applied egg-rr 92.4%

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

      1. unpow1 92.4%

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

      2. associate-*r* 92.2%

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

      3. *-commutative 92.2%

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

      4. associate-*r* 99.5%

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

      5. *-commutative 99.5%

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

   10. Simplified 99.5%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-131}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \frac{{t}^{3} \cdot \frac{k}{\ell}}{\ell}\right)}\\ \mathbf{elif}\;k \leq 3.3 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \sqrt[3]{\frac{t}{\frac{\ell}{k}} \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right)\right)}\right)}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{+170}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{t \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternative 3: 68.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq 1.8 \cdot 10^{-130}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 1.75 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{{\sin k}^{2}}{\frac{\ell}{\frac{t}{\ell}}}}\\ \mathbf{elif}\;k \leq 290:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+170}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\frac{k}{\ell} \cdot \frac{t \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= k 1.8e-130)
     (/
      2.0
      (*
       (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))
       (* (tan k) (* (/ k l) (/ (pow t 3.0) l)))))
     (if (<= k 1.75e-45)
       (/ 2.0 (* (/ k (/ (cos k) k)) (/ (pow (sin k) 2.0) (/ l (/ t l)))))
       (if (<= k 290.0)
         (* (/ l (pow t 3.0)) (/ l (* k k)))
         (if (<= k 1.8e+170)
           (/ 2.0 (* t_1 (* (/ k l) (/ (* t k) l))))
           (/ 2.0 (* t_1 (* (/ k l) (* t (/ k l)))))))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (k <= 1.8e-130) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t), 2.0))) * (tan(k) * ((k / l) * (pow(t, 3.0) / l))));
	} else if (k <= 1.75e-45) {
		tmp = 2.0 / ((k / (cos(k) / k)) * (pow(sin(k), 2.0) / (l / (t / l))));
	} else if (k <= 290.0) {
		tmp = (l / pow(t, 3.0)) * (l / (k * k));
	} else if (k <= 1.8e+170) {
		tmp = 2.0 / (t_1 * ((k / l) * ((t * k) / l)));
	} else {
		tmp = 2.0 / (t_1 * ((k / l) * (t * (k / l))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) * tan(k)
    if (k <= 1.8d-130) then
        tmp = 2.0d0 / ((1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))) * (tan(k) * ((k / l) * ((t ** 3.0d0) / l))))
    else if (k <= 1.75d-45) then
        tmp = 2.0d0 / ((k / (cos(k) / k)) * ((sin(k) ** 2.0d0) / (l / (t / l))))
    else if (k <= 290.0d0) then
        tmp = (l / (t ** 3.0d0)) * (l / (k * k))
    else if (k <= 1.8d+170) then
        tmp = 2.0d0 / (t_1 * ((k / l) * ((t * k) / l)))
    else
        tmp = 2.0d0 / (t_1 * ((k / l) * (t * (k / l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (k <= 1.8e-130) {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t), 2.0))) * (Math.tan(k) * ((k / l) * (Math.pow(t, 3.0) / l))));
	} else if (k <= 1.75e-45) {
		tmp = 2.0 / ((k / (Math.cos(k) / k)) * (Math.pow(Math.sin(k), 2.0) / (l / (t / l))));
	} else if (k <= 290.0) {
		tmp = (l / Math.pow(t, 3.0)) * (l / (k * k));
	} else if (k <= 1.8e+170) {
		tmp = 2.0 / (t_1 * ((k / l) * ((t * k) / l)));
	} else {
		tmp = 2.0 / (t_1 * ((k / l) * (t * (k / l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(k) * math.tan(k)
	tmp = 0
	if k <= 1.8e-130:
		tmp = 2.0 / ((1.0 + (1.0 + math.pow((k / t), 2.0))) * (math.tan(k) * ((k / l) * (math.pow(t, 3.0) / l))))
	elif k <= 1.75e-45:
		tmp = 2.0 / ((k / (math.cos(k) / k)) * (math.pow(math.sin(k), 2.0) / (l / (t / l))))
	elif k <= 290.0:
		tmp = (l / math.pow(t, 3.0)) * (l / (k * k))
	elif k <= 1.8e+170:
		tmp = 2.0 / (t_1 * ((k / l) * ((t * k) / l)))
	else:
		tmp = 2.0 / (t_1 * ((k / l) * (t * (k / l))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (k <= 1.8e-130)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))) * Float64(tan(k) * Float64(Float64(k / l) * Float64((t ^ 3.0) / l)))));
	elseif (k <= 1.75e-45)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(cos(k) / k)) * Float64((sin(k) ^ 2.0) / Float64(l / Float64(t / l)))));
	elseif (k <= 290.0)
		tmp = Float64(Float64(l / (t ^ 3.0)) * Float64(l / Float64(k * k)));
	elseif (k <= 1.8e+170)
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(k / l) * Float64(Float64(t * k) / l))));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(k / l) * Float64(t * Float64(k / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) * tan(k);
	tmp = 0.0;
	if (k <= 1.8e-130)
		tmp = 2.0 / ((1.0 + (1.0 + ((k / t) ^ 2.0))) * (tan(k) * ((k / l) * ((t ^ 3.0) / l))));
	elseif (k <= 1.75e-45)
		tmp = 2.0 / ((k / (cos(k) / k)) * ((sin(k) ^ 2.0) / (l / (t / l))));
	elseif (k <= 290.0)
		tmp = (l / (t ^ 3.0)) * (l / (k * k));
	elseif (k <= 1.8e+170)
		tmp = 2.0 / (t_1 * ((k / l) * ((t * k) / l)));
	else
		tmp = 2.0 / (t_1 * ((k / l) * (t * (k / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.8e-130], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.75e-45], N[(2.0 / N[(N[(k / N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 290.0], N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.8e+170], N[(2.0 / N[(t$95$1 * N[(N[(k / l), $MachinePrecision] * N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[(k / l), $MachinePrecision] * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if k < 1.8000000000000001e-130

   1. Initial program 53.1%

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

   2. Taylor expanded in k around 0 52.8%

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

      1. *-commutative 52.8%

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

      2. unpow2 52.8%

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

      3. times-frac 63.5%

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

   4. Simplified 63.5%

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

   if 1.8000000000000001e-130 < k < 1.75e-45

   1. Initial program 51.9%

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

      1. *-commutative 51.9%

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

      2. associate-*l* 51.8%

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

      3. associate-*r* 51.8%

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

      4. +-commutative 51.8%

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

      5. associate-+r+ 51.8%

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

      6. metadata-eval 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in k around inf 44.2%

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

      1. times-frac 59.6%

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

      2. unpow2 59.6%

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

      3. associate-/l* 59.5%

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

      4. associate-/l* 75.6%

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

      5. unpow2 75.6%

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

      6. associate-/l* 76.1%

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

   6. Simplified 76.1%

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

   if 1.75e-45 < k < 290

   1. Initial program 72.4%

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

      1. associate-*l* 72.4%

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

      2. associate-/l/ 72.4%

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

      3. *-commutative 72.4%

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

      4. associate-*r/ 72.4%

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

      5. associate-/l* 72.4%

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

      6. associate-/r/ 72.4%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in k around 0 81.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 81.8%

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

      2. unpow2 81.8%

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

   6. Simplified 81.8%

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

      1. times-frac 82.0%

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   8. Applied egg-rr 82.0%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   if 290 < k < 1.8e170

   1. Initial program 57.5%

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

      1. *-commutative 57.5%

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

      2. associate-*l* 57.5%

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

      3. associate-*r* 57.4%

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

      4. +-commutative 57.4%

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

      5. associate-+r+ 57.4%

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

      6. metadata-eval 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in k around inf 87.9%

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

      1. associate-/l* 82.4%

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

      2. associate-/r/ 82.2%

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

      3. unpow2 82.2%

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

      4. unpow2 82.2%

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

   6. Simplified 82.2%

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

      1. pow1 82.2%

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

      2. associate-*l* 82.2%

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

      3. times-frac 82.3%

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

   8. Applied egg-rr 82.3%

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

      1. unpow1 82.3%

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

      2. associate-*r* 82.3%

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

      3. *-commutative 82.3%

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

      4. associate-*r* 90.9%

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

      5. *-commutative 90.9%

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

   10. Simplified 90.9%

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

       1. associate-*r/ 96.8%

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

   12. Applied egg-rr 96.8%

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

   if 1.8e170 < k

   1. Initial program 50.2%

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

      1. *-commutative 50.2%

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

      2. associate-*l* 50.2%

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

      3. associate-*r* 50.2%

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

      4. +-commutative 50.2%

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

      5. associate-+r+ 50.2%

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

      6. metadata-eval 50.2%

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

   3. Simplified 50.2%

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

   4. Taylor expanded in k around inf 66.1%

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

      1. associate-/l* 65.7%

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

      2. associate-/r/ 66.1%

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

      3. unpow2 66.1%

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

      4. unpow2 66.1%

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

   6. Simplified 66.1%

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

      1. pow1 66.1%

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

      2. associate-*l* 66.1%

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

      3. times-frac 92.4%

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

   8. Applied egg-rr 92.4%

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

      1. unpow1 92.4%

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

      2. associate-*r* 92.2%

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

      3. *-commutative 92.2%

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

      4. associate-*r* 99.5%

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

      5. *-commutative 99.5%

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

   10. Simplified 99.5%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-130}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 1.75 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{{\sin k}^{2}}{\frac{\ell}{\frac{t}{\ell}}}}\\ \mathbf{elif}\;k \leq 290:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+170}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{t \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternative 4: 69.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq 1.7 \cdot 10^{-130}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \frac{{t}^{3} \cdot \frac{k}{\ell}}{\ell}\right)}\\ \mathbf{elif}\;k \leq 8.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{{\sin k}^{2}}{\frac{\ell}{\frac{t}{\ell}}}}\\ \mathbf{elif}\;k \leq 210:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+170}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\frac{k}{\ell} \cdot \frac{t \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= k 1.7e-130)
     (/
      2.0
      (*
       (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))
       (* (tan k) (/ (* (pow t 3.0) (/ k l)) l))))
     (if (<= k 8.8e-45)
       (/ 2.0 (* (/ k (/ (cos k) k)) (/ (pow (sin k) 2.0) (/ l (/ t l)))))
       (if (<= k 210.0)
         (* (/ l (pow t 3.0)) (/ l (* k k)))
         (if (<= k 3.8e+170)
           (/ 2.0 (* t_1 (* (/ k l) (/ (* t k) l))))
           (/ 2.0 (* t_1 (* (/ k l) (* t (/ k l)))))))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (k <= 1.7e-130) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t), 2.0))) * (tan(k) * ((pow(t, 3.0) * (k / l)) / l)));
	} else if (k <= 8.8e-45) {
		tmp = 2.0 / ((k / (cos(k) / k)) * (pow(sin(k), 2.0) / (l / (t / l))));
	} else if (k <= 210.0) {
		tmp = (l / pow(t, 3.0)) * (l / (k * k));
	} else if (k <= 3.8e+170) {
		tmp = 2.0 / (t_1 * ((k / l) * ((t * k) / l)));
	} else {
		tmp = 2.0 / (t_1 * ((k / l) * (t * (k / l))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) * tan(k)
    if (k <= 1.7d-130) then
        tmp = 2.0d0 / ((1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))) * (tan(k) * (((t ** 3.0d0) * (k / l)) / l)))
    else if (k <= 8.8d-45) then
        tmp = 2.0d0 / ((k / (cos(k) / k)) * ((sin(k) ** 2.0d0) / (l / (t / l))))
    else if (k <= 210.0d0) then
        tmp = (l / (t ** 3.0d0)) * (l / (k * k))
    else if (k <= 3.8d+170) then
        tmp = 2.0d0 / (t_1 * ((k / l) * ((t * k) / l)))
    else
        tmp = 2.0d0 / (t_1 * ((k / l) * (t * (k / l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (k <= 1.7e-130) {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t), 2.0))) * (Math.tan(k) * ((Math.pow(t, 3.0) * (k / l)) / l)));
	} else if (k <= 8.8e-45) {
		tmp = 2.0 / ((k / (Math.cos(k) / k)) * (Math.pow(Math.sin(k), 2.0) / (l / (t / l))));
	} else if (k <= 210.0) {
		tmp = (l / Math.pow(t, 3.0)) * (l / (k * k));
	} else if (k <= 3.8e+170) {
		tmp = 2.0 / (t_1 * ((k / l) * ((t * k) / l)));
	} else {
		tmp = 2.0 / (t_1 * ((k / l) * (t * (k / l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(k) * math.tan(k)
	tmp = 0
	if k <= 1.7e-130:
		tmp = 2.0 / ((1.0 + (1.0 + math.pow((k / t), 2.0))) * (math.tan(k) * ((math.pow(t, 3.0) * (k / l)) / l)))
	elif k <= 8.8e-45:
		tmp = 2.0 / ((k / (math.cos(k) / k)) * (math.pow(math.sin(k), 2.0) / (l / (t / l))))
	elif k <= 210.0:
		tmp = (l / math.pow(t, 3.0)) * (l / (k * k))
	elif k <= 3.8e+170:
		tmp = 2.0 / (t_1 * ((k / l) * ((t * k) / l)))
	else:
		tmp = 2.0 / (t_1 * ((k / l) * (t * (k / l))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (k <= 1.7e-130)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))) * Float64(tan(k) * Float64(Float64((t ^ 3.0) * Float64(k / l)) / l))));
	elseif (k <= 8.8e-45)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(cos(k) / k)) * Float64((sin(k) ^ 2.0) / Float64(l / Float64(t / l)))));
	elseif (k <= 210.0)
		tmp = Float64(Float64(l / (t ^ 3.0)) * Float64(l / Float64(k * k)));
	elseif (k <= 3.8e+170)
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(k / l) * Float64(Float64(t * k) / l))));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(k / l) * Float64(t * Float64(k / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) * tan(k);
	tmp = 0.0;
	if (k <= 1.7e-130)
		tmp = 2.0 / ((1.0 + (1.0 + ((k / t) ^ 2.0))) * (tan(k) * (((t ^ 3.0) * (k / l)) / l)));
	elseif (k <= 8.8e-45)
		tmp = 2.0 / ((k / (cos(k) / k)) * ((sin(k) ^ 2.0) / (l / (t / l))));
	elseif (k <= 210.0)
		tmp = (l / (t ^ 3.0)) * (l / (k * k));
	elseif (k <= 3.8e+170)
		tmp = 2.0 / (t_1 * ((k / l) * ((t * k) / l)));
	else
		tmp = 2.0 / (t_1 * ((k / l) * (t * (k / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.7e-130], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 8.8e-45], N[(2.0 / N[(N[(k / N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 210.0], N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.8e+170], N[(2.0 / N[(t$95$1 * N[(N[(k / l), $MachinePrecision] * N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[(k / l), $MachinePrecision] * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if k < 1.70000000000000003e-130

   1. Initial program 53.1%

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

   2. Taylor expanded in k around 0 52.8%

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

      1. *-commutative 52.8%

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

      2. unpow2 52.8%

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

      3. times-frac 63.5%

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

   4. Simplified 63.5%

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

      1. associate-*l/ 63.5%

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

   6. Applied egg-rr 63.5%

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

   if 1.70000000000000003e-130 < k < 8.79999999999999974e-45

   1. Initial program 51.9%

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

      1. *-commutative 51.9%

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

      2. associate-*l* 51.8%

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

      3. associate-*r* 51.8%

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

      4. +-commutative 51.8%

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

      5. associate-+r+ 51.8%

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

      6. metadata-eval 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in k around inf 44.2%

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

      1. times-frac 59.6%

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

      2. unpow2 59.6%

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

      3. associate-/l* 59.5%

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

      4. associate-/l* 75.6%

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

      5. unpow2 75.6%

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

      6. associate-/l* 76.1%

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

   6. Simplified 76.1%

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

   if 8.79999999999999974e-45 < k < 210

   1. Initial program 72.4%

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

      1. associate-*l* 72.4%

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

      2. associate-/l/ 72.4%

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

      3. *-commutative 72.4%

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

      4. associate-*r/ 72.4%

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

      5. associate-/l* 72.4%

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

      6. associate-/r/ 72.4%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in k around 0 81.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 81.8%

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

      2. unpow2 81.8%

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

   6. Simplified 81.8%

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

      1. times-frac 82.0%

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   8. Applied egg-rr 82.0%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   if 210 < k < 3.7999999999999998e170

   1. Initial program 57.5%

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

      1. *-commutative 57.5%

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

      2. associate-*l* 57.5%

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

      3. associate-*r* 57.4%

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

      4. +-commutative 57.4%

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

      5. associate-+r+ 57.4%

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

      6. metadata-eval 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in k around inf 87.9%

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

      1. associate-/l* 82.4%

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

      2. associate-/r/ 82.2%

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

      3. unpow2 82.2%

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

      4. unpow2 82.2%

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

   6. Simplified 82.2%

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

      1. pow1 82.2%

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

      2. associate-*l* 82.2%

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

      3. times-frac 82.3%

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

   8. Applied egg-rr 82.3%

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

      1. unpow1 82.3%

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

      2. associate-*r* 82.3%

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

      3. *-commutative 82.3%

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

      4. associate-*r* 90.9%

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

      5. *-commutative 90.9%

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

   10. Simplified 90.9%

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

       1. associate-*r/ 96.8%

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

   12. Applied egg-rr 96.8%

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

   if 3.7999999999999998e170 < k

   1. Initial program 50.2%

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

      1. *-commutative 50.2%

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

      2. associate-*l* 50.2%

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

      3. associate-*r* 50.2%

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

      4. +-commutative 50.2%

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

      5. associate-+r+ 50.2%

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

      6. metadata-eval 50.2%

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

   3. Simplified 50.2%

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

   4. Taylor expanded in k around inf 66.1%

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

      1. associate-/l* 65.7%

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

      2. associate-/r/ 66.1%

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

      3. unpow2 66.1%

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

      4. unpow2 66.1%

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

   6. Simplified 66.1%

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

      1. pow1 66.1%

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

      2. associate-*l* 66.1%

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

      3. times-frac 92.4%

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

   8. Applied egg-rr 92.4%

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

      1. unpow1 92.4%

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

      2. associate-*r* 92.2%

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

      3. *-commutative 92.2%

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

      4. associate-*r* 99.5%

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

      5. *-commutative 99.5%

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

   10. Simplified 99.5%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-130}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \frac{{t}^{3} \cdot \frac{k}{\ell}}{\ell}\right)}\\ \mathbf{elif}\;k \leq 8.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{{\sin k}^{2}}{\frac{\ell}{\frac{t}{\ell}}}}\\ \mathbf{elif}\;k \leq 210:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+170}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{t \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternative 5: 67.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.26 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k}{\frac{\frac{\ell}{\frac{{t}^{3}}{\ell}}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{t}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.26e-69)
   (/ 2.0 (* 2.0 (/ k (/ (/ l (/ (pow t 3.0) l)) k))))
   (/ 2.0 (* (sin k) (* (tan k) (* (/ k l) (* k (/ t l))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.26e-69) {
		tmp = 2.0 / (2.0 * (k / ((l / (pow(t, 3.0) / l)) / k)));
	} else {
		tmp = 2.0 / (sin(k) * (tan(k) * ((k / l) * (k * (t / l)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.26d-69) then
        tmp = 2.0d0 / (2.0d0 * (k / ((l / ((t ** 3.0d0) / l)) / k)))
    else
        tmp = 2.0d0 / (sin(k) * (tan(k) * ((k / l) * (k * (t / l)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.26e-69) {
		tmp = 2.0 / (2.0 * (k / ((l / (Math.pow(t, 3.0) / l)) / k)));
	} else {
		tmp = 2.0 / (Math.sin(k) * (Math.tan(k) * ((k / l) * (k * (t / l)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.26e-69:
		tmp = 2.0 / (2.0 * (k / ((l / (math.pow(t, 3.0) / l)) / k)))
	else:
		tmp = 2.0 / (math.sin(k) * (math.tan(k) * ((k / l) * (k * (t / l)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.26e-69)
		tmp = Float64(2.0 / Float64(2.0 * Float64(k / Float64(Float64(l / Float64((t ^ 3.0) / l)) / k))));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(Float64(k / l) * Float64(k * Float64(t / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.26e-69)
		tmp = 2.0 / (2.0 * (k / ((l / ((t ^ 3.0) / l)) / k)));
	else
		tmp = 2.0 / (sin(k) * (tan(k) * ((k / l) * (k * (t / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.26e-69], N[(2.0 / N[(2.0 * N[(k / N[(N[(l / N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.2599999999999999e-69

   1. Initial program 53.5%

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

      1. *-commutative 53.5%

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

      2. associate-*l* 46.8%

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

      3. associate-*r* 46.8%

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

      4. +-commutative 46.8%

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

      5. associate-+r+ 46.8%

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

      6. metadata-eval 46.8%

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

   3. Simplified 46.8%

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

   4. Taylor expanded in k around 0 47.7%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 47.2%

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

      2. unpow2 47.2%

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

      3. unpow2 47.2%

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

      4. associate-*r/ 56.2%

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

      5. associate-/l* 63.8%

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

      6. associate-*r/ 54.8%

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

      7. associate-/l* 63.8%

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

   6. Simplified 63.8%

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

   if 2.2599999999999999e-69 < k

   1. Initial program 55.9%

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

      1. *-commutative 55.9%

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

      2. associate-*l* 55.9%

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

      3. associate-*r* 55.9%

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

      4. +-commutative 55.9%

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

      5. associate-+r+ 55.9%

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

      6. metadata-eval 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in k around inf 75.1%

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

      1. associate-/l* 72.5%

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

      2. associate-/r/ 72.6%

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

      3. unpow2 72.6%

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

      4. unpow2 72.6%

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

   6. Simplified 72.6%

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

      1. pow1 72.6%

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

      2. associate-*l* 72.6%

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

      3. times-frac 82.0%

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

   8. Applied egg-rr 82.0%

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

      1. unpow1 82.0%

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

      2. associate-*r* 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-*r* 89.7%

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

      5. *-commutative 89.7%

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

   10. Simplified 89.7%

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

       1. pow1 89.7%

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

   12. Applied egg-rr 89.7%

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

       1. unpow1 89.7%

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

       2. *-commutative 89.7%

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

       3. associate-*l* 89.8%

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

       4. associate-*r/ 88.6%

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

       5. associate-/l* 89.8%

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

       6. associate-/r/ 87.9%

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

   14. Simplified 87.9%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.26 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k}{\frac{\frac{\ell}{\frac{{t}^{3}}{\ell}}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{t}{\ell}\right)\right)\right)}\\ \end{array} \]

Alternative 6: 68.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.42 \cdot 10^{-68}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k}{\frac{\frac{\ell}{\frac{{t}^{3}}{\ell}}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.42e-68)
   (/ 2.0 (* 2.0 (/ k (/ (/ l (/ (pow t 3.0) l)) k))))
   (/ 2.0 (* (* (sin k) (tan k)) (* (/ k l) (* t (/ k l)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.42e-68) {
		tmp = 2.0 / (2.0 * (k / ((l / (pow(t, 3.0) / l)) / k)));
	} else {
		tmp = 2.0 / ((sin(k) * tan(k)) * ((k / l) * (t * (k / l))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.42d-68) then
        tmp = 2.0d0 / (2.0d0 * (k / ((l / ((t ** 3.0d0) / l)) / k)))
    else
        tmp = 2.0d0 / ((sin(k) * tan(k)) * ((k / l) * (t * (k / l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.42e-68) {
		tmp = 2.0 / (2.0 * (k / ((l / (Math.pow(t, 3.0) / l)) / k)));
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.tan(k)) * ((k / l) * (t * (k / l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.42e-68:
		tmp = 2.0 / (2.0 * (k / ((l / (math.pow(t, 3.0) / l)) / k)))
	else:
		tmp = 2.0 / ((math.sin(k) * math.tan(k)) * ((k / l) * (t * (k / l))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.42e-68)
		tmp = Float64(2.0 / Float64(2.0 * Float64(k / Float64(Float64(l / Float64((t ^ 3.0) / l)) / k))));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * tan(k)) * Float64(Float64(k / l) * Float64(t * Float64(k / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.42e-68)
		tmp = 2.0 / (2.0 * (k / ((l / ((t ^ 3.0) / l)) / k)));
	else
		tmp = 2.0 / ((sin(k) * tan(k)) * ((k / l) * (t * (k / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.42e-68], N[(2.0 / N[(2.0 * N[(k / N[(N[(l / N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.42e-68

   1. Initial program 53.5%

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

      1. *-commutative 53.5%

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

      2. associate-*l* 46.8%

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

      3. associate-*r* 46.8%

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

      4. +-commutative 46.8%

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

      5. associate-+r+ 46.8%

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

      6. metadata-eval 46.8%

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

   3. Simplified 46.8%

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

   4. Taylor expanded in k around 0 47.7%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 47.2%

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

      2. unpow2 47.2%

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

      3. unpow2 47.2%

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

      4. associate-*r/ 56.2%

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

      5. associate-/l* 63.8%

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

      6. associate-*r/ 54.8%

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

      7. associate-/l* 63.8%

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

   6. Simplified 63.8%

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

   if 1.42e-68 < k

   1. Initial program 55.9%

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

      1. *-commutative 55.9%

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

      2. associate-*l* 55.9%

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

      3. associate-*r* 55.9%

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

      4. +-commutative 55.9%

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

      5. associate-+r+ 55.9%

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

      6. metadata-eval 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in k around inf 75.1%

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

      1. associate-/l* 72.5%

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

      2. associate-/r/ 72.6%

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

      3. unpow2 72.6%

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

      4. unpow2 72.6%

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

   6. Simplified 72.6%

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

      1. pow1 72.6%

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

      2. associate-*l* 72.6%

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

      3. times-frac 82.0%

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

   8. Applied egg-rr 82.0%

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

      1. unpow1 82.0%

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

      2. associate-*r* 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-*r* 89.7%

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

      5. *-commutative 89.7%

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

   10. Simplified 89.7%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.42 \cdot 10^{-68}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k}{\frac{\frac{\ell}{\frac{{t}^{3}}{\ell}}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternative 7: 68.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k}{\frac{\frac{\ell}{\frac{{t}^{3}}{\ell}}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{t}{\frac{\ell}{k}} \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 9.6e-69)
   (/ 2.0 (* 2.0 (/ k (/ (/ l (/ (pow t 3.0) l)) k))))
   (/ 2.0 (* (/ k l) (* (/ t (/ l k)) (* (sin k) (tan k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.6e-69) {
		tmp = 2.0 / (2.0 * (k / ((l / (pow(t, 3.0) / l)) / k)));
	} else {
		tmp = 2.0 / ((k / l) * ((t / (l / k)) * (sin(k) * tan(k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9.6d-69) then
        tmp = 2.0d0 / (2.0d0 * (k / ((l / ((t ** 3.0d0) / l)) / k)))
    else
        tmp = 2.0d0 / ((k / l) * ((t / (l / k)) * (sin(k) * tan(k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.6e-69) {
		tmp = 2.0 / (2.0 * (k / ((l / (Math.pow(t, 3.0) / l)) / k)));
	} else {
		tmp = 2.0 / ((k / l) * ((t / (l / k)) * (Math.sin(k) * Math.tan(k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 9.6e-69:
		tmp = 2.0 / (2.0 * (k / ((l / (math.pow(t, 3.0) / l)) / k)))
	else:
		tmp = 2.0 / ((k / l) * ((t / (l / k)) * (math.sin(k) * math.tan(k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 9.6e-69)
		tmp = Float64(2.0 / Float64(2.0 * Float64(k / Float64(Float64(l / Float64((t ^ 3.0) / l)) / k))));
	else
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(t / Float64(l / k)) * Float64(sin(k) * tan(k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 9.6e-69)
		tmp = 2.0 / (2.0 * (k / ((l / ((t ^ 3.0) / l)) / k)));
	else
		tmp = 2.0 / ((k / l) * ((t / (l / k)) * (sin(k) * tan(k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 9.6e-69], N[(2.0 / N[(2.0 * N[(k / N[(N[(l / N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.6000000000000005e-69

   1. Initial program 53.5%

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

      1. *-commutative 53.5%

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

      2. associate-*l* 46.8%

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

      3. associate-*r* 46.8%

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

      4. +-commutative 46.8%

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

      5. associate-+r+ 46.8%

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

      6. metadata-eval 46.8%

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

   3. Simplified 46.8%

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

   4. Taylor expanded in k around 0 47.7%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 47.2%

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

      2. unpow2 47.2%

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

      3. unpow2 47.2%

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

      4. associate-*r/ 56.2%

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

      5. associate-/l* 63.8%

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

      6. associate-*r/ 54.8%

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

      7. associate-/l* 63.8%

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

   6. Simplified 63.8%

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

   if 9.6000000000000005e-69 < k

   1. Initial program 55.9%

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

      1. *-commutative 55.9%

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

      2. associate-*l* 55.9%

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

      3. associate-*r* 55.9%

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

      4. +-commutative 55.9%

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

      5. associate-+r+ 55.9%

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

      6. metadata-eval 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in k around inf 75.1%

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

      1. associate-/l* 72.5%

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

      2. associate-/r/ 72.6%

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

      3. unpow2 72.6%

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

      4. unpow2 72.6%

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

   6. Simplified 72.6%

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

      1. pow1 72.6%

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

      2. associate-*l* 72.6%

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

      3. times-frac 82.0%

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

   8. Applied egg-rr 82.0%

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

      1. unpow1 82.0%

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

      2. associate-*r* 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-*r* 89.7%

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

      5. *-commutative 89.7%

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

   10. Simplified 89.7%

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

       1. pow1 89.7%

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

   12. Applied egg-rr 89.7%

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

       1. unpow1 89.7%

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

       2. associate-*l* 89.8%

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

       3. associate-*r/ 88.5%

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

       4. associate-/l* 89.8%

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

   14. Simplified 89.8%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k}{\frac{\frac{\ell}{\frac{{t}^{3}}{\ell}}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{t}{\frac{\ell}{k}} \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \end{array} \]

Alternative 8: 67.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k}{\frac{\frac{\ell}{\frac{{t}^{3}}{\ell}}}{k}}}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{k}{\frac{\cos k}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left({t}^{3} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -1.5e-38)
   (/ 2.0 (* 2.0 (/ k (/ (/ l (/ (pow t 3.0) l)) k))))
   (if (<= t 4.6e-98)
     (/ 2.0 (* (* (/ k l) (* t (/ k l))) (/ k (/ (cos k) k))))
     (/ 2.0 (* 2.0 (* (pow t 3.0) (* (/ k l) (/ k l))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.5e-38) {
		tmp = 2.0 / (2.0 * (k / ((l / (pow(t, 3.0) / l)) / k)));
	} else if (t <= 4.6e-98) {
		tmp = 2.0 / (((k / l) * (t * (k / l))) * (k / (cos(k) / k)));
	} else {
		tmp = 2.0 / (2.0 * (pow(t, 3.0) * ((k / l) * (k / l))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-1.5d-38)) then
        tmp = 2.0d0 / (2.0d0 * (k / ((l / ((t ** 3.0d0) / l)) / k)))
    else if (t <= 4.6d-98) then
        tmp = 2.0d0 / (((k / l) * (t * (k / l))) * (k / (cos(k) / k)))
    else
        tmp = 2.0d0 / (2.0d0 * ((t ** 3.0d0) * ((k / l) * (k / l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.5e-38) {
		tmp = 2.0 / (2.0 * (k / ((l / (Math.pow(t, 3.0) / l)) / k)));
	} else if (t <= 4.6e-98) {
		tmp = 2.0 / (((k / l) * (t * (k / l))) * (k / (Math.cos(k) / k)));
	} else {
		tmp = 2.0 / (2.0 * (Math.pow(t, 3.0) * ((k / l) * (k / l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -1.5e-38:
		tmp = 2.0 / (2.0 * (k / ((l / (math.pow(t, 3.0) / l)) / k)))
	elif t <= 4.6e-98:
		tmp = 2.0 / (((k / l) * (t * (k / l))) * (k / (math.cos(k) / k)))
	else:
		tmp = 2.0 / (2.0 * (math.pow(t, 3.0) * ((k / l) * (k / l))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -1.5e-38)
		tmp = Float64(2.0 / Float64(2.0 * Float64(k / Float64(Float64(l / Float64((t ^ 3.0) / l)) / k))));
	elseif (t <= 4.6e-98)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t * Float64(k / l))) * Float64(k / Float64(cos(k) / k))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64((t ^ 3.0) * Float64(Float64(k / l) * Float64(k / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -1.5e-38)
		tmp = 2.0 / (2.0 * (k / ((l / ((t ^ 3.0) / l)) / k)));
	elseif (t <= 4.6e-98)
		tmp = 2.0 / (((k / l) * (t * (k / l))) * (k / (cos(k) / k)));
	else
		tmp = 2.0 / (2.0 * ((t ^ 3.0) * ((k / l) * (k / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -1.5e-38], N[(2.0 / N[(2.0 * N[(k / N[(N[(l / N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.6e-98], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k / N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[Power[t, 3.0], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.49999999999999994e-38

   1. Initial program 69.0%

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

      1. *-commutative 69.0%

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

      2. associate-*l* 59.9%

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

      3. associate-*r* 59.9%

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

      4. +-commutative 59.9%

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

      5. associate-+r+ 59.9%

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

      6. metadata-eval 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in k around 0 57.2%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 57.1%

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

      2. unpow2 57.1%

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

      3. unpow2 57.1%

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

      4. associate-*r/ 60.5%

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

      5. associate-/l* 69.7%

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

      6. associate-*r/ 66.3%

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

      7. associate-/l* 69.6%

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

   6. Simplified 69.6%

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

   if -1.49999999999999994e-38 < t < 4.60000000000000001e-98

   1. Initial program 26.5%

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

      1. *-commutative 26.5%

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

      2. associate-*l* 26.5%

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

      3. associate-*r* 26.5%

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

      4. +-commutative 26.5%

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

      5. associate-+r+ 26.5%

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

      6. metadata-eval 26.5%

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

   3. Simplified 26.5%

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

   4. Taylor expanded in k around inf 68.5%

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

      1. times-frac 69.1%

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

      2. unpow2 69.1%

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

      3. associate-/l* 69.1%

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

      4. associate-/l* 69.1%

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

      5. unpow2 69.1%

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

      6. associate-/l* 82.0%

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

   6. Simplified 82.0%

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

   7. Taylor expanded in k around 0 60.6%

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

      1. unpow2 60.6%

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

      2. unpow2 60.6%

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

      3. associate-*l* 60.6%

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

      4. times-frac 74.0%

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

      5. *-commutative 74.0%

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

      6. associate-*r/ 74.9%

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

   9. Simplified 74.9%

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

   if 4.60000000000000001e-98 < t

   1. Initial program 73.1%

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

      1. *-commutative 73.1%

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

      2. associate-*l* 66.3%

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

      3. associate-*r* 66.3%

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

      4. +-commutative 66.3%

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

      5. associate-+r+ 66.3%

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

      6. metadata-eval 66.3%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in l around 0 58.6%

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

      1. associate-/l* 57.4%

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

      2. unpow2 57.4%

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

      3. unpow2 57.4%

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

      4. unpow2 57.4%

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

   6. Simplified 57.4%

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

   7. Taylor expanded in k around 0 62.2%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   8. Step-by-step derivation

      1. associate-/l* 61.2%

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

      2. associate-/r/ 62.2%

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

      3. unpow2 62.2%

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

      4. unpow2 62.2%

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

      5. times-frac 73.2%

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

   9. Simplified 73.2%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k}{\frac{\frac{\ell}{\frac{{t}^{3}}{\ell}}}{k}}}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{k}{\frac{\cos k}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left({t}^{3} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternative 9: 66.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.52 \cdot 10^{-37} \lor \neg \left(t \leq 3 \cdot 10^{-100}\right):\\ \;\;\;\;\frac{2}{2 \cdot \left({t}^{3} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right) \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.52e-37) (not (<= t 3e-100)))
   (/ 2.0 (* 2.0 (* (pow t 3.0) (* (/ k l) (/ k l)))))
   (*
    2.0
    (* (* (/ l (* k k)) (/ l t)) (+ -0.16666666666666666 (/ 1.0 (* k k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.52e-37) || !(t <= 3e-100)) {
		tmp = 2.0 / (2.0 * (pow(t, 3.0) * ((k / l) * (k / l))));
	} else {
		tmp = 2.0 * (((l / (k * k)) * (l / t)) * (-0.16666666666666666 + (1.0 / (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 ((t <= (-1.52d-37)) .or. (.not. (t <= 3d-100))) then
        tmp = 2.0d0 / (2.0d0 * ((t ** 3.0d0) * ((k / l) * (k / l))))
    else
        tmp = 2.0d0 * (((l / (k * k)) * (l / t)) * ((-0.16666666666666666d0) + (1.0d0 / (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.52e-37) || !(t <= 3e-100)) {
		tmp = 2.0 / (2.0 * (Math.pow(t, 3.0) * ((k / l) * (k / l))));
	} else {
		tmp = 2.0 * (((l / (k * k)) * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -1.52e-37) or not (t <= 3e-100):
		tmp = 2.0 / (2.0 * (math.pow(t, 3.0) * ((k / l) * (k / l))))
	else:
		tmp = 2.0 * (((l / (k * k)) * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.52e-37) || !(t <= 3e-100))
		tmp = Float64(2.0 / Float64(2.0 * Float64((t ^ 3.0) * Float64(Float64(k / l) * Float64(k / l)))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / Float64(k * k)) * Float64(l / t)) * Float64(-0.16666666666666666 + Float64(1.0 / Float64(k * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.52e-37) || ~((t <= 3e-100)))
		tmp = 2.0 / (2.0 * ((t ^ 3.0) * ((k / l) * (k / l))));
	else
		tmp = 2.0 * (((l / (k * k)) * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -1.52e-37], N[Not[LessEqual[t, 3e-100]], $MachinePrecision]], N[(2.0 / N[(2.0 * N[(N[Power[t, 3.0], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 + N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.52e-37 or 3.0000000000000001e-100 < t

   1. Initial program 71.4%

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

      1. *-commutative 71.4%

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

      2. associate-*l* 63.6%

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

      3. associate-*r* 63.6%

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

      4. +-commutative 63.6%

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

      5. associate-+r+ 63.6%

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

      6. metadata-eval 63.6%

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

   3. Simplified 63.6%

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

   4. Taylor expanded in l around 0 56.7%

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

      1. associate-/l* 56.6%

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

      2. unpow2 56.6%

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

      3. unpow2 56.6%

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

      4. unpow2 56.6%

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

   6. Simplified 56.6%

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

   7. Taylor expanded in k around 0 60.1%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   8. Step-by-step derivation

      1. associate-/l* 59.5%

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

      2. associate-/r/ 58.8%

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

      3. unpow2 58.8%

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

      4. unpow2 58.8%

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

      5. times-frac 70.9%

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

   9. Simplified 70.9%

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

   if -1.52e-37 < t < 3.0000000000000001e-100

   1. Initial program 26.5%

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

      1. associate-/l/ 26.5%

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

      2. associate-*l/ 25.8%

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

      3. associate-*l/ 25.8%

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

      4. associate-/r/ 25.8%

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

      5. *-commutative 25.8%

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

      6. associate-/l/ 25.8%

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

      7. associate-*r* 25.8%

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

      8. *-commutative 25.8%

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

      9. associate-*r* 25.8%

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

      10. *-commutative 25.8%

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

   3. Simplified 25.8%

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

   4. Taylor expanded in k around inf 68.2%

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

      1. associate-/l* 68.2%

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

      2. unpow2 68.2%

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

      3. associate-/r* 68.2%

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

   6. Simplified 68.2%

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

   7. Taylor expanded in k around 0 56.3%

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

   8. Taylor expanded in l around 0 59.7%

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

      1. unpow2 59.7%

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

      2. associate-*l/ 59.8%

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

      3. unpow2 59.8%

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

      4. associate-/r* 59.8%

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

      5. associate-*r/ 59.8%

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

      6. metadata-eval 59.8%

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

      7. div-sub 59.8%

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

      8. unpow2 59.8%

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

      9. times-frac 60.0%

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

      10. *-commutative 60.0%

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

      11. associate-/l* 56.5%

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

      12. associate-/r/ 60.0%

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

      13. unpow2 60.0%

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

      14. times-frac 74.1%

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

   10. Simplified 74.1%

       \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.52 \cdot 10^{-37} \lor \neg \left(t \leq 3 \cdot 10^{-100}\right):\\ \;\;\;\;\frac{2}{2 \cdot \left({t}^{3} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right) \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)\right)\\ \end{array} \]

Alternative 10: 66.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k}{\frac{\frac{\ell}{\frac{{t}^{3}}{\ell}}}{k}}}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-95}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right) \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left({t}^{3} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -3.6e-39)
   (/ 2.0 (* 2.0 (/ k (/ (/ l (/ (pow t 3.0) l)) k))))
   (if (<= t 2.4e-95)
     (*
      2.0
      (* (* (/ l (* k k)) (/ l t)) (+ -0.16666666666666666 (/ 1.0 (* k k)))))
     (/ 2.0 (* 2.0 (* (pow t 3.0) (* (/ k l) (/ k l))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.6e-39) {
		tmp = 2.0 / (2.0 * (k / ((l / (pow(t, 3.0) / l)) / k)));
	} else if (t <= 2.4e-95) {
		tmp = 2.0 * (((l / (k * k)) * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))));
	} else {
		tmp = 2.0 / (2.0 * (pow(t, 3.0) * ((k / l) * (k / l))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-3.6d-39)) then
        tmp = 2.0d0 / (2.0d0 * (k / ((l / ((t ** 3.0d0) / l)) / k)))
    else if (t <= 2.4d-95) then
        tmp = 2.0d0 * (((l / (k * k)) * (l / t)) * ((-0.16666666666666666d0) + (1.0d0 / (k * k))))
    else
        tmp = 2.0d0 / (2.0d0 * ((t ** 3.0d0) * ((k / l) * (k / l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.6e-39) {
		tmp = 2.0 / (2.0 * (k / ((l / (Math.pow(t, 3.0) / l)) / k)));
	} else if (t <= 2.4e-95) {
		tmp = 2.0 * (((l / (k * k)) * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))));
	} else {
		tmp = 2.0 / (2.0 * (Math.pow(t, 3.0) * ((k / l) * (k / l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -3.6e-39:
		tmp = 2.0 / (2.0 * (k / ((l / (math.pow(t, 3.0) / l)) / k)))
	elif t <= 2.4e-95:
		tmp = 2.0 * (((l / (k * k)) * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))))
	else:
		tmp = 2.0 / (2.0 * (math.pow(t, 3.0) * ((k / l) * (k / l))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -3.6e-39)
		tmp = Float64(2.0 / Float64(2.0 * Float64(k / Float64(Float64(l / Float64((t ^ 3.0) / l)) / k))));
	elseif (t <= 2.4e-95)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / Float64(k * k)) * Float64(l / t)) * Float64(-0.16666666666666666 + Float64(1.0 / Float64(k * k)))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64((t ^ 3.0) * Float64(Float64(k / l) * Float64(k / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -3.6e-39)
		tmp = 2.0 / (2.0 * (k / ((l / ((t ^ 3.0) / l)) / k)));
	elseif (t <= 2.4e-95)
		tmp = 2.0 * (((l / (k * k)) * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))));
	else
		tmp = 2.0 / (2.0 * ((t ^ 3.0) * ((k / l) * (k / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -3.6e-39], N[(2.0 / N[(2.0 * N[(k / N[(N[(l / N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.4e-95], N[(2.0 * N[(N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 + N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[Power[t, 3.0], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.6000000000000001e-39

   1. Initial program 69.0%

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

      1. *-commutative 69.0%

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

      2. associate-*l* 59.9%

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

      3. associate-*r* 59.9%

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

      4. +-commutative 59.9%

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

      5. associate-+r+ 59.9%

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

      6. metadata-eval 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in k around 0 57.2%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 57.1%

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

      2. unpow2 57.1%

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

      3. unpow2 57.1%

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

      4. associate-*r/ 60.5%

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

      5. associate-/l* 69.7%

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

      6. associate-*r/ 66.3%

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

      7. associate-/l* 69.6%

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

   6. Simplified 69.6%

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

   if -3.6000000000000001e-39 < t < 2.4e-95

   1. Initial program 26.5%

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

      1. associate-/l/ 26.5%

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

      2. associate-*l/ 25.8%

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

      3. associate-*l/ 25.8%

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

      4. associate-/r/ 25.8%

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

      5. *-commutative 25.8%

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

      6. associate-/l/ 25.8%

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

      7. associate-*r* 25.8%

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

      8. *-commutative 25.8%

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

      9. associate-*r* 25.8%

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

      10. *-commutative 25.8%

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

   3. Simplified 25.8%

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

   4. Taylor expanded in k around inf 68.2%

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

      1. associate-/l* 68.2%

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

      2. unpow2 68.2%

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

      3. associate-/r* 68.2%

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

   6. Simplified 68.2%

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

   7. Taylor expanded in k around 0 56.3%

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

   8. Taylor expanded in l around 0 59.7%

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

      1. unpow2 59.7%

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

      2. associate-*l/ 59.8%

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

      3. unpow2 59.8%

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

      4. associate-/r* 59.8%

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

      5. associate-*r/ 59.8%

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

      6. metadata-eval 59.8%

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

      7. div-sub 59.8%

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

      8. unpow2 59.8%

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

      9. times-frac 60.0%

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

      10. *-commutative 60.0%

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

      11. associate-/l* 56.5%

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

      12. associate-/r/ 60.0%

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

      13. unpow2 60.0%

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

      14. times-frac 74.1%

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

   10. Simplified 74.1%

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

   if 2.4e-95 < t

   1. Initial program 73.1%

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

      1. *-commutative 73.1%

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

      2. associate-*l* 66.3%

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

      3. associate-*r* 66.3%

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

      4. +-commutative 66.3%

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

      5. associate-+r+ 66.3%

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

      6. metadata-eval 66.3%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in l around 0 58.6%

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

      1. associate-/l* 57.4%

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

      2. unpow2 57.4%

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

      3. unpow2 57.4%

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

      4. unpow2 57.4%

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

   6. Simplified 57.4%

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

   7. Taylor expanded in k around 0 62.2%

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{{\ell}^{2}}}} \]
   8. Step-by-step derivation

      1. associate-/l* 61.2%

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

      2. associate-/r/ 62.2%

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

      3. unpow2 62.2%

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

      4. unpow2 62.2%

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

      5. times-frac 73.2%

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

   9. Simplified 73.2%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k}{\frac{\frac{\ell}{\frac{{t}^{3}}{\ell}}}{k}}}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-95}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right) \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left({t}^{3} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternative 11: 65.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{-39} \lor \neg \left(t \leq 2.6 \cdot 10^{-97}\right):\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right) \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -4.6e-39) (not (<= t 2.6e-97)))
   (* (/ l (pow t 3.0)) (/ (/ l k) k))
   (*
    2.0
    (* (* (/ l (* k k)) (/ l t)) (+ -0.16666666666666666 (/ 1.0 (* k k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -4.6e-39) || !(t <= 2.6e-97)) {
		tmp = (l / pow(t, 3.0)) * ((l / k) / k);
	} else {
		tmp = 2.0 * (((l / (k * k)) * (l / t)) * (-0.16666666666666666 + (1.0 / (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 ((t <= (-4.6d-39)) .or. (.not. (t <= 2.6d-97))) then
        tmp = (l / (t ** 3.0d0)) * ((l / k) / k)
    else
        tmp = 2.0d0 * (((l / (k * k)) * (l / t)) * ((-0.16666666666666666d0) + (1.0d0 / (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -4.6e-39) || !(t <= 2.6e-97)) {
		tmp = (l / Math.pow(t, 3.0)) * ((l / k) / k);
	} else {
		tmp = 2.0 * (((l / (k * k)) * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -4.6e-39) or not (t <= 2.6e-97):
		tmp = (l / math.pow(t, 3.0)) * ((l / k) / k)
	else:
		tmp = 2.0 * (((l / (k * k)) * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -4.6e-39) || !(t <= 2.6e-97))
		tmp = Float64(Float64(l / (t ^ 3.0)) * Float64(Float64(l / k) / k));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / Float64(k * k)) * Float64(l / t)) * Float64(-0.16666666666666666 + Float64(1.0 / Float64(k * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -4.6e-39) || ~((t <= 2.6e-97)))
		tmp = (l / (t ^ 3.0)) * ((l / k) / k);
	else
		tmp = 2.0 * (((l / (k * k)) * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -4.6e-39], N[Not[LessEqual[t, 2.6e-97]], $MachinePrecision]], N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 + N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.60000000000000016e-39 or 2.60000000000000007e-97 < t

   1. Initial program 71.4%

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

      1. associate-*l* 71.3%

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

      2. associate-/l/ 71.3%

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

      3. *-commutative 71.3%

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

      4. associate-*r/ 72.7%

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

      5. associate-/l* 72.3%

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

      6. associate-/r/ 63.6%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in k around 0 60.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 60.1%

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

      2. unpow2 60.1%

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

   6. Simplified 60.1%

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

   7. Taylor expanded in l around 0 60.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   8. Step-by-step derivation

      1. unpow2 60.1%

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

      2. times-frac 62.3%

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

      3. unpow2 62.3%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

      4. associate-/r* 69.1%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{k}} \cdot \frac{\ell}{{t}^{3}} \]

   9. Simplified 69.1%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{{t}^{3}}} \]

   if -4.60000000000000016e-39 < t < 2.60000000000000007e-97

   1. Initial program 26.5%

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

      1. associate-/l/ 26.5%

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

      2. associate-*l/ 25.8%

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

      3. associate-*l/ 25.8%

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

      4. associate-/r/ 25.8%

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

      5. *-commutative 25.8%

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

      6. associate-/l/ 25.8%

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

      7. associate-*r* 25.8%

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

      8. *-commutative 25.8%

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

      9. associate-*r* 25.8%

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

      10. *-commutative 25.8%

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

   3. Simplified 25.8%

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

   4. Taylor expanded in k around inf 68.2%

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

      1. associate-/l* 68.2%

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

      2. unpow2 68.2%

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

      3. associate-/r* 68.2%

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

   6. Simplified 68.2%

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

   7. Taylor expanded in k around 0 56.3%

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

   8. Taylor expanded in l around 0 59.7%

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

      1. unpow2 59.7%

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

      2. associate-*l/ 59.8%

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

      3. unpow2 59.8%

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

      4. associate-/r* 59.8%

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

      5. associate-*r/ 59.8%

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

      6. metadata-eval 59.8%

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

      7. div-sub 59.8%

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

      8. unpow2 59.8%

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

      9. times-frac 60.0%

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

      10. *-commutative 60.0%

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

      11. associate-/l* 56.5%

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

      12. associate-/r/ 60.0%

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

      13. unpow2 60.0%

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

      14. times-frac 74.1%

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

   10. Simplified 74.1%

       \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{-39} \lor \neg \left(t \leq 2.6 \cdot 10^{-97}\right):\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\frac{\ell}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right) \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)\right)\\ \end{array} \]

Alternative 12: 65.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left({t}^{3} \cdot k\right)}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-95}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right) \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\frac{\ell}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -1.25e-5)
   (/ (* l l) (* k (* (pow t 3.0) k)))
   (if (<= t 9e-95)
     (*
      2.0
      (* (* (/ l (* k k)) (/ l t)) (+ -0.16666666666666666 (/ 1.0 (* k k)))))
     (* (/ l (pow t 3.0)) (/ (/ l k) k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.25e-5) {
		tmp = (l * l) / (k * (pow(t, 3.0) * k));
	} else if (t <= 9e-95) {
		tmp = 2.0 * (((l / (k * k)) * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))));
	} else {
		tmp = (l / pow(t, 3.0)) * ((l / 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 (t <= (-1.25d-5)) then
        tmp = (l * l) / (k * ((t ** 3.0d0) * k))
    else if (t <= 9d-95) then
        tmp = 2.0d0 * (((l / (k * k)) * (l / t)) * ((-0.16666666666666666d0) + (1.0d0 / (k * k))))
    else
        tmp = (l / (t ** 3.0d0)) * ((l / k) / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.25e-5) {
		tmp = (l * l) / (k * (Math.pow(t, 3.0) * k));
	} else if (t <= 9e-95) {
		tmp = 2.0 * (((l / (k * k)) * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))));
	} else {
		tmp = (l / Math.pow(t, 3.0)) * ((l / k) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -1.25e-5:
		tmp = (l * l) / (k * (math.pow(t, 3.0) * k))
	elif t <= 9e-95:
		tmp = 2.0 * (((l / (k * k)) * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))))
	else:
		tmp = (l / math.pow(t, 3.0)) * ((l / k) / k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -1.25e-5)
		tmp = Float64(Float64(l * l) / Float64(k * Float64((t ^ 3.0) * k)));
	elseif (t <= 9e-95)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / Float64(k * k)) * Float64(l / t)) * Float64(-0.16666666666666666 + Float64(1.0 / Float64(k * k)))));
	else
		tmp = Float64(Float64(l / (t ^ 3.0)) * Float64(Float64(l / k) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -1.25e-5)
		tmp = (l * l) / (k * ((t ^ 3.0) * k));
	elseif (t <= 9e-95)
		tmp = 2.0 * (((l / (k * k)) * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))));
	else
		tmp = (l / (t ^ 3.0)) * ((l / k) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -1.25e-5], N[(N[(l * l), $MachinePrecision] / N[(k * N[(N[Power[t, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9e-95], N[(2.0 * N[(N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 + N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.25000000000000006e-5

   1. Initial program 67.6%

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

      1. associate-*l* 67.6%

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

      2. associate-/l/ 67.6%

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

      3. *-commutative 67.6%

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

      4. associate-*r/ 67.6%

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

      5. associate-/l* 67.6%

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

      6. associate-/r/ 59.1%

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

   3. Simplified 62.8%

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

   4. Taylor expanded in k around 0 57.5%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 57.5%

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

      2. unpow2 57.5%

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

   6. Simplified 57.5%

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

   7. Taylor expanded in l around 0 57.5%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   8. Step-by-step derivation

      1. unpow2 57.5%

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

      2. unpow2 57.5%

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

      3. associate-*r* 65.9%

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

   9. Simplified 65.9%

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

   if -1.25000000000000006e-5 < t < 9e-95

   1. Initial program 29.8%

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

      1. associate-/l/ 29.7%

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

      2. associate-*l/ 29.1%

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

      3. associate-*l/ 28.2%

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

      4. associate-/r/ 28.2%

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

      5. *-commutative 28.2%

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

      6. associate-/l/ 28.2%

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

      7. associate-*r* 28.2%

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

      8. *-commutative 28.2%

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

      9. associate-*r* 28.2%

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

      10. *-commutative 28.2%

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

   3. Simplified 28.2%

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

   4. Taylor expanded in k around inf 68.6%

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

      1. associate-/l* 68.6%

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

      2. unpow2 68.6%

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

      3. associate-/r* 68.6%

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

   6. Simplified 68.6%

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

   7. Taylor expanded in k around 0 56.4%

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

   8. Taylor expanded in l around 0 59.7%

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

      1. unpow2 59.7%

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

      2. associate-*l/ 59.7%

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

      3. unpow2 59.7%

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

      4. associate-/r* 59.7%

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

      5. associate-*r/ 59.7%

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

      6. metadata-eval 59.7%

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

      7. div-sub 59.7%

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

      8. unpow2 59.7%

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

      9. times-frac 59.9%

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

      10. *-commutative 59.9%

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

      11. associate-/l* 56.6%

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

      12. associate-/r/ 59.9%

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

      13. unpow2 59.9%

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

      14. times-frac 73.2%

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

   10. Simplified 73.2%

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

   if 9e-95 < t

   1. Initial program 73.1%

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

      1. associate-*l* 73.1%

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

      2. associate-/l/ 73.0%

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

      3. *-commutative 73.0%

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

      4. associate-*r/ 75.5%

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

      5. associate-/l* 74.7%

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

      6. associate-/r/ 66.4%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in k around 0 62.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 62.2%

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

      2. unpow2 62.2%

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

   6. Simplified 62.2%

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

   7. Taylor expanded in l around 0 62.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   8. Step-by-step derivation

      1. unpow2 62.2%

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

      2. times-frac 64.8%

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

      3. unpow2 64.8%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

      4. associate-/r* 72.3%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{k}} \cdot \frac{\ell}{{t}^{3}} \]

   9. Simplified 72.3%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left({t}^{3} \cdot k\right)}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-95}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right) \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\frac{\ell}{k}}{k}\\ \end{array} \]

Alternative 13: 59.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(t_1 \cdot \frac{\ell}{t}\right) \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (* k k))))
   (if (<= t -1e-9)
     (* (/ l (pow t 3.0)) t_1)
     (* 2.0 (* (* t_1 (/ l t)) (+ -0.16666666666666666 (/ 1.0 (* k k))))))))

C

double code(double t, double l, double k) {
	double t_1 = l / (k * k);
	double tmp;
	if (t <= -1e-9) {
		tmp = (l / pow(t, 3.0)) * t_1;
	} else {
		tmp = 2.0 * ((t_1 * (l / t)) * (-0.16666666666666666 + (1.0 / (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) :: t_1
    real(8) :: tmp
    t_1 = l / (k * k)
    if (t <= (-1d-9)) then
        tmp = (l / (t ** 3.0d0)) * t_1
    else
        tmp = 2.0d0 * ((t_1 * (l / t)) * ((-0.16666666666666666d0) + (1.0d0 / (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l / (k * k);
	double tmp;
	if (t <= -1e-9) {
		tmp = (l / Math.pow(t, 3.0)) * t_1;
	} else {
		tmp = 2.0 * ((t_1 * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l / (k * k)
	tmp = 0
	if t <= -1e-9:
		tmp = (l / math.pow(t, 3.0)) * t_1
	else:
		tmp = 2.0 * ((t_1 * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / Float64(k * k))
	tmp = 0.0
	if (t <= -1e-9)
		tmp = Float64(Float64(l / (t ^ 3.0)) * t_1);
	else
		tmp = Float64(2.0 * Float64(Float64(t_1 * Float64(l / t)) * Float64(-0.16666666666666666 + Float64(1.0 / Float64(k * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l / (k * k);
	tmp = 0.0;
	if (t <= -1e-9)
		tmp = (l / (t ^ 3.0)) * t_1;
	else
		tmp = 2.0 * ((t_1 * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1e-9], N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(2.0 * N[(N[(t$95$1 * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 + N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.00000000000000006e-9

   1. Initial program 67.6%

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

      1. associate-*l* 67.6%

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

      2. associate-/l/ 67.6%

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

      3. *-commutative 67.6%

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

      4. associate-*r/ 67.6%

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

      5. associate-/l* 67.6%

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

      6. associate-/r/ 59.1%

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

   3. Simplified 62.8%

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

   4. Taylor expanded in k around 0 57.5%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 57.5%

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

      2. unpow2 57.5%

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

   6. Simplified 57.5%

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

      1. times-frac 59.5%

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   8. Applied egg-rr 59.5%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   if -1.00000000000000006e-9 < t

   1. Initial program 50.0%

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

      1. associate-/l/ 50.0%

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

      2. associate-*l/ 50.7%

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

      3. associate-*l/ 49.1%

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

      4. associate-/r/ 49.2%

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

      5. *-commutative 49.2%

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

      6. associate-/l/ 49.2%

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

      7. associate-*r* 49.2%

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

      8. *-commutative 49.2%

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

      9. associate-*r* 49.2%

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

      10. *-commutative 49.2%

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

   3. Simplified 49.2%

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

   4. Taylor expanded in k around inf 65.4%

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

      1. associate-/l* 65.4%

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

      2. unpow2 65.4%

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

      3. associate-/r* 65.4%

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

   6. Simplified 65.4%

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

   7. Taylor expanded in k around 0 57.4%

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

   8. Taylor expanded in l around 0 60.2%

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

      1. unpow2 60.2%

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

      2. associate-*l/ 59.8%

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

      3. unpow2 59.8%

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

      4. associate-/r* 59.8%

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

      5. associate-*r/ 59.8%

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

      6. metadata-eval 59.8%

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

      7. div-sub 59.8%

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

      8. unpow2 59.8%

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

      9. times-frac 60.3%

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

      10. *-commutative 60.3%

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

      11. associate-/l* 58.0%

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

      12. associate-/r/ 60.8%

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

      13. unpow2 60.8%

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

      14. times-frac 68.1%

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

   10. Simplified 68.1%

       \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right) \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)\right)\\ \end{array} \]

Alternative 14: 54.6% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-240}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot -6 + -36\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{k \cdot k} \cdot \frac{\frac{\frac{1}{t}}{k}}{k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e-240)
   (* (* l l) (/ 2.0 (* t (+ (* (* k k) -6.0) -36.0))))
   (* (* l l) (* (/ 2.0 (* k k)) (/ (/ (/ 1.0 t) k) k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-240) {
		tmp = (l * l) * (2.0 / (t * (((k * k) * -6.0) + -36.0)));
	} else {
		tmp = (l * l) * ((2.0 / (k * k)) * (((1.0 / 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 ((l * l) <= 1d-240) then
        tmp = (l * l) * (2.0d0 / (t * (((k * k) * (-6.0d0)) + (-36.0d0))))
    else
        tmp = (l * l) * ((2.0d0 / (k * k)) * (((1.0d0 / t) / k) / k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-240) {
		tmp = (l * l) * (2.0 / (t * (((k * k) * -6.0) + -36.0)));
	} else {
		tmp = (l * l) * ((2.0 / (k * k)) * (((1.0 / t) / k) / k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e-240:
		tmp = (l * l) * (2.0 / (t * (((k * k) * -6.0) + -36.0)))
	else:
		tmp = (l * l) * ((2.0 / (k * k)) * (((1.0 / t) / k) / k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-240)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t * Float64(Float64(Float64(k * k) * -6.0) + -36.0))));
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 / Float64(k * k)) * Float64(Float64(Float64(1.0 / t) / k) / k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-240)
		tmp = (l * l) * (2.0 / (t * (((k * k) * -6.0) + -36.0)));
	else
		tmp = (l * l) * ((2.0 / (k * k)) * (((1.0 / t) / k) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e-240], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t * N[(N[(N[(k * k), $MachinePrecision] * -6.0), $MachinePrecision] + -36.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / t), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.9999999999999997e-241

   1. Initial program 54.1%

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

      1. associate-/l/ 54.0%

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

      2. associate-*l/ 54.6%

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

      3. associate-*l/ 53.2%

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

      4. associate-/r/ 53.2%

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

      5. *-commutative 53.2%

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

      6. associate-/l/ 53.2%

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

      7. associate-*r* 53.2%

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

      8. *-commutative 53.2%

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

      9. associate-*r* 53.2%

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

      10. *-commutative 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in k around inf 51.4%

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

      1. associate-/l* 51.4%

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

      2. unpow2 51.4%

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

      3. associate-/r* 51.4%

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

   6. Simplified 51.4%

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

   7. Taylor expanded in k around 0 48.4%

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

   8. Taylor expanded in k around inf 64.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{-36 \cdot t + -6 \cdot \left({k}^{2} \cdot t\right)}} \]
   9. Step-by-step derivation

      1. +-commutative 64.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{-6 \cdot \left({k}^{2} \cdot t\right) + -36 \cdot t}} \]

      2. unpow2 64.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{-6 \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) + -36 \cdot t} \]

      3. associate-*r* 64.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(-6 \cdot \left(k \cdot k\right)\right) \cdot t} + -36 \cdot t} \]

      4. distribute-rgt-out 64.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{t \cdot \left(-6 \cdot \left(k \cdot k\right) + -36\right)}} \]

   10. Simplified 64.3%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{t \cdot \left(-6 \cdot \left(k \cdot k\right) + -36\right)}} \]

   if 9.9999999999999997e-241 < (*.f64 l l)

   1. Initial program 54.2%

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

      1. associate-/l/ 54.2%

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

      2. associate-*l/ 54.8%

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

      3. associate-*l/ 53.7%

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

      4. associate-/r/ 53.2%

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

      5. *-commutative 53.2%

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

      6. associate-/l/ 53.2%

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

      7. associate-*r* 53.2%

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

      8. *-commutative 53.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 53.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 53.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 53.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 63.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k}}} \]
   5. Step-by-step derivation

      1. associate-/l* 63.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}}} \]

      2. unpow2 63.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}} \]

      3. associate-/r* 63.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

   6. Simplified 63.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

   7. Taylor expanded in k around 0 52.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{1}{{k}^{2} \cdot t}}}} \]
   8. Step-by-step derivation

      1. *-commutative 52.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\frac{1}{\color{blue}{t \cdot {k}^{2}}}}} \]

      2. associate-/r* 52.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\frac{1}{t}}{{k}^{2}}}}} \]

      3. unpow2 52.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\frac{\frac{1}{t}}{\color{blue}{k \cdot k}}}} \]

   9. Simplified 52.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\frac{1}{t}}{k \cdot k}}}} \]
   10. Step-by-step derivation

       1. associate-/r/ 52.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{2}{k \cdot k} \cdot \frac{\frac{1}{t}}{k \cdot k}\right)} \]

       2. associate-/r* 52.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{k \cdot k} \cdot \color{blue}{\frac{\frac{\frac{1}{t}}{k}}{k}}\right) \]

   11. Applied egg-rr 52.2%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{2}{k \cdot k} \cdot \frac{\frac{\frac{1}{t}}{k}}{k}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-240}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot -6 + -36\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{k \cdot k} \cdot \frac{\frac{\frac{1}{t}}{k}}{k}\right)\\ \end{array} \]

Alternative 15: 54.6% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-240}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot -6 + -36\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\frac{\frac{1}{t}}{k \cdot k}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e-240)
   (* (* l l) (/ 2.0 (* t (+ (* (* k k) -6.0) -36.0))))
   (* (* l l) (/ 2.0 (/ (* k k) (/ (/ 1.0 t) (* k k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-240) {
		tmp = (l * l) * (2.0 / (t * (((k * k) * -6.0) + -36.0)));
	} else {
		tmp = (l * l) * (2.0 / ((k * k) / ((1.0 / 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 ((l * l) <= 1d-240) then
        tmp = (l * l) * (2.0d0 / (t * (((k * k) * (-6.0d0)) + (-36.0d0))))
    else
        tmp = (l * l) * (2.0d0 / ((k * k) / ((1.0d0 / t) / (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-240) {
		tmp = (l * l) * (2.0 / (t * (((k * k) * -6.0) + -36.0)));
	} else {
		tmp = (l * l) * (2.0 / ((k * k) / ((1.0 / t) / (k * k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e-240:
		tmp = (l * l) * (2.0 / (t * (((k * k) * -6.0) + -36.0)))
	else:
		tmp = (l * l) * (2.0 / ((k * k) / ((1.0 / t) / (k * k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-240)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t * Float64(Float64(Float64(k * k) * -6.0) + -36.0))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(Float64(k * k) / Float64(Float64(1.0 / t) / Float64(k * k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-240)
		tmp = (l * l) * (2.0 / (t * (((k * k) * -6.0) + -36.0)));
	else
		tmp = (l * l) * (2.0 / ((k * k) / ((1.0 / t) / (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e-240], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t * N[(N[(N[(k * k), $MachinePrecision] * -6.0), $MachinePrecision] + -36.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[(k * k), $MachinePrecision] / N[(N[(1.0 / t), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.9999999999999997e-241

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 54.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 54.6%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 53.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 53.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 53.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 53.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 53.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 53.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 53.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 53.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 53.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 51.4%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k}}} \]
   5. Step-by-step derivation

      1. associate-/l* 51.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}}} \]

      2. unpow2 51.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}} \]

      3. associate-/r* 51.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

   6. Simplified 51.4%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

   7. Taylor expanded in k around 0 48.4%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}}}} \]

   8. Taylor expanded in k around inf 64.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{-36 \cdot t + -6 \cdot \left({k}^{2} \cdot t\right)}} \]
   9. Step-by-step derivation

      1. +-commutative 64.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{-6 \cdot \left({k}^{2} \cdot t\right) + -36 \cdot t}} \]

      2. unpow2 64.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{-6 \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) + -36 \cdot t} \]

      3. associate-*r* 64.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(-6 \cdot \left(k \cdot k\right)\right) \cdot t} + -36 \cdot t} \]

      4. distribute-rgt-out 64.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{t \cdot \left(-6 \cdot \left(k \cdot k\right) + -36\right)}} \]

   10. Simplified 64.3%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{t \cdot \left(-6 \cdot \left(k \cdot k\right) + -36\right)}} \]

   if 9.9999999999999997e-241 < (*.f64 l l)

   1. Initial program 54.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 54.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 54.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 53.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 53.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 53.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 53.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 53.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 53.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 53.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 53.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 53.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 63.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k}}} \]
   5. Step-by-step derivation

      1. associate-/l* 63.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}}} \]

      2. unpow2 63.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}} \]

      3. associate-/r* 63.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

   6. Simplified 63.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

   7. Taylor expanded in k around 0 52.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{1}{{k}^{2} \cdot t}}}} \]
   8. Step-by-step derivation

      1. *-commutative 52.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\frac{1}{\color{blue}{t \cdot {k}^{2}}}}} \]

      2. associate-/r* 52.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\frac{1}{t}}{{k}^{2}}}}} \]

      3. unpow2 52.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\frac{\frac{1}{t}}{\color{blue}{k \cdot k}}}} \]

   9. Simplified 52.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\frac{1}{t}}{k \cdot k}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-240}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot -6 + -36\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\frac{\frac{1}{t}}{k \cdot k}}}\\ \end{array} \]

Alternative 16: 54.6% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-240}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot -6 + -36\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \frac{k \cdot k}{\frac{1}{t}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 1e-240)
   (* (* l l) (/ 2.0 (* t (+ (* (* k k) -6.0) -36.0))))
   (/ (* 2.0 (* l l)) (* (* k k) (/ (* k k) (/ 1.0 t))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-240) {
		tmp = (l * l) * (2.0 / (t * (((k * k) * -6.0) + -36.0)));
	} else {
		tmp = (2.0 * (l * l)) / ((k * k) * ((k * k) / (1.0 / t)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d-240) then
        tmp = (l * l) * (2.0d0 / (t * (((k * k) * (-6.0d0)) + (-36.0d0))))
    else
        tmp = (2.0d0 * (l * l)) / ((k * k) * ((k * k) / (1.0d0 / t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-240) {
		tmp = (l * l) * (2.0 / (t * (((k * k) * -6.0) + -36.0)));
	} else {
		tmp = (2.0 * (l * l)) / ((k * k) * ((k * k) / (1.0 / t)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 1e-240:
		tmp = (l * l) * (2.0 / (t * (((k * k) * -6.0) + -36.0)))
	else:
		tmp = (2.0 * (l * l)) / ((k * k) * ((k * k) / (1.0 / t)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-240)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t * Float64(Float64(Float64(k * k) * -6.0) + -36.0))));
	else
		tmp = Float64(Float64(2.0 * Float64(l * l)) / Float64(Float64(k * k) * Float64(Float64(k * k) / Float64(1.0 / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-240)
		tmp = (l * l) * (2.0 / (t * (((k * k) * -6.0) + -36.0)));
	else
		tmp = (2.0 * (l * l)) / ((k * k) * ((k * k) / (1.0 / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 1e-240], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t * N[(N[(N[(k * k), $MachinePrecision] * -6.0), $MachinePrecision] + -36.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.9999999999999997e-241

   1. Initial program 54.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 54.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 54.6%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 53.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 53.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 53.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 53.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 53.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 53.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 53.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 53.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 53.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 51.4%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k}}} \]
   5. Step-by-step derivation

      1. associate-/l* 51.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}}} \]

      2. unpow2 51.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}} \]

      3. associate-/r* 51.4%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

   6. Simplified 51.4%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

   7. Taylor expanded in k around 0 48.4%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}}}} \]

   8. Taylor expanded in k around inf 64.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{-36 \cdot t + -6 \cdot \left({k}^{2} \cdot t\right)}} \]
   9. Step-by-step derivation

      1. +-commutative 64.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{-6 \cdot \left({k}^{2} \cdot t\right) + -36 \cdot t}} \]

      2. unpow2 64.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{-6 \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) + -36 \cdot t} \]

      3. associate-*r* 64.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(-6 \cdot \left(k \cdot k\right)\right) \cdot t} + -36 \cdot t} \]

      4. distribute-rgt-out 64.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{t \cdot \left(-6 \cdot \left(k \cdot k\right) + -36\right)}} \]

   10. Simplified 64.3%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{t \cdot \left(-6 \cdot \left(k \cdot k\right) + -36\right)}} \]

   if 9.9999999999999997e-241 < (*.f64 l l)

   1. Initial program 54.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 54.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 54.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 53.7%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 53.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 53.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 53.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 53.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 53.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 53.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 53.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 53.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 63.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k}}} \]
   5. Step-by-step derivation

      1. associate-/l* 63.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}}} \]

      2. unpow2 63.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}} \]

      3. associate-/r* 63.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

   6. Simplified 63.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

   7. Taylor expanded in k around 0 52.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{1}{{k}^{2} \cdot t}}}} \]
   8. Step-by-step derivation

      1. *-commutative 52.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\frac{1}{\color{blue}{t \cdot {k}^{2}}}}} \]

      2. associate-/r* 52.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\frac{1}{t}}{{k}^{2}}}}} \]

      3. unpow2 52.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\frac{\frac{1}{t}}{\color{blue}{k \cdot k}}}} \]

   9. Simplified 52.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\frac{1}{t}}{k \cdot k}}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 52.4%

          \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\frac{k \cdot k}{\frac{\frac{1}{t}}{k \cdot k}}}} \]

       2. associate-/r/ 52.4%

          \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\color{blue}{\frac{k \cdot k}{\frac{1}{t}} \cdot \left(k \cdot k\right)}} \]

   11. Applied egg-rr 52.4%

       \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\frac{k \cdot k}{\frac{1}{t}} \cdot \left(k \cdot k\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-240}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot -6 + -36\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \frac{k \cdot k}{\frac{1}{t}}}\\ \end{array} \]

Alternative 17: 45.4% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.7 \cdot 10^{-146}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot -6 + -36\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{0.16666666666666666 + \frac{-1}{k \cdot k}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 1.7e-146)
   (* (* l l) (/ 2.0 (* t (+ (* (* k k) -6.0) -36.0))))
   (*
    -2.0
    (* (/ (* l l) (* k k)) (/ (+ 0.16666666666666666 (/ -1.0 (* k k))) t)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.7e-146) {
		tmp = (l * l) * (2.0 / (t * (((k * k) * -6.0) + -36.0)));
	} else {
		tmp = -2.0 * (((l * l) / (k * k)) * ((0.16666666666666666 + (-1.0 / (k * k))) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1.7d-146) then
        tmp = (l * l) * (2.0d0 / (t * (((k * k) * (-6.0d0)) + (-36.0d0))))
    else
        tmp = (-2.0d0) * (((l * l) / (k * k)) * ((0.16666666666666666d0 + ((-1.0d0) / (k * k))) / t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.7e-146) {
		tmp = (l * l) * (2.0 / (t * (((k * k) * -6.0) + -36.0)));
	} else {
		tmp = -2.0 * (((l * l) / (k * k)) * ((0.16666666666666666 + (-1.0 / (k * k))) / t));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 1.7e-146:
		tmp = (l * l) * (2.0 / (t * (((k * k) * -6.0) + -36.0)))
	else:
		tmp = -2.0 * (((l * l) / (k * k)) * ((0.16666666666666666 + (-1.0 / (k * k))) / t))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 1.7e-146)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t * Float64(Float64(Float64(k * k) * -6.0) + -36.0))));
	else
		tmp = Float64(-2.0 * Float64(Float64(Float64(l * l) / Float64(k * k)) * Float64(Float64(0.16666666666666666 + Float64(-1.0 / Float64(k * k))) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 1.7e-146)
		tmp = (l * l) * (2.0 / (t * (((k * k) * -6.0) + -36.0)));
	else
		tmp = -2.0 * (((l * l) / (k * k)) * ((0.16666666666666666 + (-1.0 / (k * k))) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 1.7e-146], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t * N[(N[(N[(k * k), $MachinePrecision] * -6.0), $MachinePrecision] + -36.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(0.16666666666666666 + N[(-1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.7e-146

   1. Initial program 56.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 56.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 57.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 55.6%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 55.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 55.6%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 55.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 55.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 55.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 55.6%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 55.6%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 55.6%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 58.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k}}} \]
   5. Step-by-step derivation

      1. associate-/l* 58.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}}} \]

      2. unpow2 58.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}} \]

      3. associate-/r* 58.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

   6. Simplified 58.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

   7. Taylor expanded in k around 0 54.4%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}}}} \]

   8. Taylor expanded in k around inf 46.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{-36 \cdot t + -6 \cdot \left({k}^{2} \cdot t\right)}} \]
   9. Step-by-step derivation

      1. +-commutative 46.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{-6 \cdot \left({k}^{2} \cdot t\right) + -36 \cdot t}} \]

      2. unpow2 46.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{-6 \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) + -36 \cdot t} \]

      3. associate-*r* 46.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(-6 \cdot \left(k \cdot k\right)\right) \cdot t} + -36 \cdot t} \]

      4. distribute-rgt-out 46.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{t \cdot \left(-6 \cdot \left(k \cdot k\right) + -36\right)}} \]

   10. Simplified 46.1%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{t \cdot \left(-6 \cdot \left(k \cdot k\right) + -36\right)}} \]

   if 1.7e-146 < l

   1. Initial program 48.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 48.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 49.5%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 49.2%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 48.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 48.0%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 48.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 48.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 48.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 48.0%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 48.0%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 48.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 61.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k}}} \]
   5. Step-by-step derivation

      1. associate-/l* 61.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}}} \]

      2. unpow2 61.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}} \]

      3. associate-/r* 61.1%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

   6. Simplified 61.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

   7. Taylor expanded in k around 0 50.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}}}} \]

   8. Taylor expanded in t around -inf 53.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{{\ell}^{2} \cdot \left(0.16666666666666666 - \frac{1}{{k}^{2}}\right)}{{k}^{2} \cdot t}} \]
   9. Step-by-step derivation

      1. unpow2 53.4%

         \[\leadsto -2 \cdot \frac{{\ell}^{2} \cdot \left(0.16666666666666666 - \frac{1}{{k}^{2}}\right)}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

      2. times-frac 52.1%

         \[\leadsto -2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{k \cdot k} \cdot \frac{0.16666666666666666 - \frac{1}{{k}^{2}}}{t}\right)} \]

      3. unpow2 52.1%

         \[\leadsto -2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{k \cdot k} \cdot \frac{0.16666666666666666 - \frac{1}{{k}^{2}}}{t}\right) \]

      4. unpow2 52.1%

         \[\leadsto -2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{0.16666666666666666 - \frac{1}{\color{blue}{k \cdot k}}}{t}\right) \]

   10. Simplified 52.1%

       \[\leadsto \color{blue}{-2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{0.16666666666666666 - \frac{1}{k \cdot k}}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.7 \cdot 10^{-146}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot -6 + -36\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{0.16666666666666666 + \frac{-1}{k \cdot k}}{t}\right)\\ \end{array} \]

Alternative 18: 58.0% accurate, 22.2× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right) \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  2.0
  (* (* (/ l (* k k)) (/ l t)) (+ -0.16666666666666666 (/ 1.0 (* k k))))))

C

double code(double t, double l, double k) {
	return 2.0 * (((l / (k * k)) * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (((l / (k * k)) * (l / t)) * ((-0.16666666666666666d0) + (1.0d0 / (k * k))))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (((l / (k * k)) * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))));
}

Python

def code(t, l, k):
	return 2.0 * (((l / (k * k)) * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(Float64(l / Float64(k * k)) * Float64(l / t)) * Float64(-0.16666666666666666 + Float64(1.0 / Float64(k * k)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (((l / (k * k)) * (l / t)) * (-0.16666666666666666 + (1.0 / (k * k))));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 + N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 54.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   2. associate-*l/ 54.7%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

   3. associate-*l/ 53.5%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

   4. associate-/r/ 53.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

   5. *-commutative 53.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

   6. associate-/l/ 53.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   7. associate-*r* 53.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   8. *-commutative 53.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

   9. associate-*r* 53.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

   10. *-commutative 53.2%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 53.2%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

4. Taylor expanded in k around inf 59.1%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k}}} \]
5. Step-by-step derivation

   1. associate-/l* 59.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}}} \]

   2. unpow2 59.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}} \]

   3. associate-/r* 59.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

6. Simplified 59.1%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

7. Taylor expanded in k around 0 53.0%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}}}} \]

8. Taylor expanded in l around 0 55.5%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}{{k}^{2}}} \]
9. Step-by-step derivation

   1. unpow2 55.5%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}{\color{blue}{k \cdot k}} \]

   2. associate-*l/ 55.2%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{k \cdot k} \cdot \left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right)} \]

   3. unpow2 55.2%

      \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{k \cdot k} \cdot \left(\frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

   4. associate-/r* 55.2%

      \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{k \cdot k} \cdot \left(\color{blue}{\frac{\frac{1}{k \cdot k}}{t}} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

   5. associate-*r/ 55.2%

      \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{k \cdot k} \cdot \left(\frac{\frac{1}{k \cdot k}}{t} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

   6. metadata-eval 55.2%

      \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{k \cdot k} \cdot \left(\frac{\frac{1}{k \cdot k}}{t} - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

   7. div-sub 55.2%

      \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{k \cdot k} \cdot \color{blue}{\frac{\frac{1}{k \cdot k} - 0.16666666666666666}{t}}\right) \]

   8. unpow2 55.2%

      \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{k \cdot k} \cdot \frac{\frac{1}{\color{blue}{{k}^{2}}} - 0.16666666666666666}{t}\right) \]

   9. times-frac 55.6%

      \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}{\left(k \cdot k\right) \cdot t}} \]

   10. *-commutative 55.6%

       \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]

   11. associate-/l* 53.5%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{t \cdot \left(k \cdot k\right)}{\frac{1}{{k}^{2}} - 0.16666666666666666}}} \]

   12. associate-/r/ 56.0%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{t \cdot \left(k \cdot k\right)} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)\right)} \]

   13. unpow2 56.0%

       \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{t \cdot \left(k \cdot k\right)} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)\right) \]

   14. times-frac 63.2%

       \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)} \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)\right) \]

10. Simplified 63.2%

    \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)\right)} \]

11. Final simplification 63.2%

    \[\leadsto 2 \cdot \left(\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right) \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)\right) \]

Alternative 19: 35.9% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot -6 + -36\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* l l) (/ 2.0 (* t (+ (* (* k k) -6.0) -36.0)))))

C

double code(double t, double l, double k) {
	return (l * l) * (2.0 / (t * (((k * k) * -6.0) + -36.0)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l * l) * (2.0d0 / (t * (((k * k) * (-6.0d0)) + (-36.0d0))))
end function

Java

public static double code(double t, double l, double k) {
	return (l * l) * (2.0 / (t * (((k * k) * -6.0) + -36.0)));
}

Python

def code(t, l, k):
	return (l * l) * (2.0 / (t * (((k * k) * -6.0) + -36.0)))

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(2.0 / Float64(t * Float64(Float64(Float64(k * k) * -6.0) + -36.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * l) * (2.0 / (t * (((k * k) * -6.0) + -36.0)));
end

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t * N[(N[(N[(k * k), $MachinePrecision] * -6.0), $MachinePrecision] + -36.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 54.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   2. associate-*l/ 54.7%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

   3. associate-*l/ 53.5%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

   4. associate-/r/ 53.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

   5. *-commutative 53.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

   6. associate-/l/ 53.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   7. associate-*r* 53.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   8. *-commutative 53.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

   9. associate-*r* 53.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

   10. *-commutative 53.2%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 53.2%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

4. Taylor expanded in k around inf 59.1%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k}}} \]
5. Step-by-step derivation

   1. associate-/l* 59.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}}} \]

   2. unpow2 59.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}} \]

   3. associate-/r* 59.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

6. Simplified 59.1%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

7. Taylor expanded in k around 0 53.0%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}}}} \]

8. Taylor expanded in k around inf 40.3%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{-36 \cdot t + -6 \cdot \left({k}^{2} \cdot t\right)}} \]
9. Step-by-step derivation

   1. +-commutative 40.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{-6 \cdot \left({k}^{2} \cdot t\right) + -36 \cdot t}} \]

   2. unpow2 40.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{-6 \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) + -36 \cdot t} \]

   3. associate-*r* 40.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(-6 \cdot \left(k \cdot k\right)\right) \cdot t} + -36 \cdot t} \]

   4. distribute-rgt-out 40.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{t \cdot \left(-6 \cdot \left(k \cdot k\right) + -36\right)}} \]

10. Simplified 40.3%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{t \cdot \left(-6 \cdot \left(k \cdot k\right) + -36\right)}} \]

11. Final simplification 40.3%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot -6 + -36\right)} \]

Alternative 20: 32.2% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right) \cdot -0.3333333333333333 \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (* (/ l (* k k)) (/ l t)) -0.3333333333333333))

C

double code(double t, double l, double k) {
	return ((l / (k * k)) * (l / t)) * -0.3333333333333333;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((l / (k * k)) * (l / t)) * (-0.3333333333333333d0)
end function

Java

public static double code(double t, double l, double k) {
	return ((l / (k * k)) * (l / t)) * -0.3333333333333333;
}

Python

def code(t, l, k):
	return ((l / (k * k)) * (l / t)) * -0.3333333333333333

Julia

function code(t, l, k)
	return Float64(Float64(Float64(l / Float64(k * k)) * Float64(l / t)) * -0.3333333333333333)
end

MATLAB

function tmp = code(t, l, k)
	tmp = ((l / (k * k)) * (l / t)) * -0.3333333333333333;
end

Wolfram

code[t_, l_, k_] := N[(N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 54.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   2. associate-*l/ 54.7%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

   3. associate-*l/ 53.5%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

   4. associate-/r/ 53.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

   5. *-commutative 53.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

   6. associate-/l/ 53.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   7. associate-*r* 53.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   8. *-commutative 53.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

   9. associate-*r* 53.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

   10. *-commutative 53.2%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 53.2%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

4. Taylor expanded in k around inf 59.1%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k}}} \]
5. Step-by-step derivation

   1. associate-/l* 59.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}}} \]

   2. unpow2 59.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}} \]

   3. associate-/r* 59.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

6. Simplified 59.1%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

7. Taylor expanded in k around 0 53.0%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}}}} \]

8. Taylor expanded in k around inf 35.3%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
9. Step-by-step derivation

   1. unpow2 35.3%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} \]

   2. unpow2 35.3%

      \[\leadsto -0.3333333333333333 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

   3. *-commutative 35.3%

      \[\leadsto -0.3333333333333333 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]

   4. times-frac 36.9%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)} \]

10. Simplified 36.9%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)} \]

11. Final simplification 36.9%

    \[\leadsto \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right) \cdot -0.3333333333333333 \]

Alternative 21: 32.6% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{t}}{k \cdot k} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* -0.3333333333333333 (/ (* l (/ l t)) (* k k))))

C

double code(double t, double l, double k) {
	return -0.3333333333333333 * ((l * (l / t)) / (k * k));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (-0.3333333333333333d0) * ((l * (l / t)) / (k * k))
end function

Java

public static double code(double t, double l, double k) {
	return -0.3333333333333333 * ((l * (l / t)) / (k * k));
}

Python

def code(t, l, k):
	return -0.3333333333333333 * ((l * (l / t)) / (k * k))

Julia

function code(t, l, k)
	return Float64(-0.3333333333333333 * Float64(Float64(l * Float64(l / t)) / Float64(k * k)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = -0.3333333333333333 * ((l * (l / t)) / (k * k));
end

Wolfram

code[t_, l_, k_] := N[(-0.3333333333333333 * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l/ 54.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

   2. associate-*l/ 54.7%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

   3. associate-*l/ 53.5%

      \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

   4. associate-/r/ 53.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

   5. *-commutative 53.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

   6. associate-/l/ 53.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

   7. associate-*r* 53.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   8. *-commutative 53.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

   9. associate-*r* 53.2%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

   10. *-commutative 53.2%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 53.2%

   \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

4. Taylor expanded in k around inf 59.1%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k}}} \]
5. Step-by-step derivation

   1. associate-/l* 59.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}}} \]

   2. unpow2 59.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\cos k}{{\sin k}^{2} \cdot t}}} \]

   3. associate-/r* 59.1%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

6. Simplified 59.1%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\frac{\cos k}{{\sin k}^{2}}}{t}}}} \]

7. Taylor expanded in k around 0 53.0%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{k \cdot k}{\color{blue}{\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}}}} \]

8. Taylor expanded in k around inf 35.3%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
9. Step-by-step derivation

   1. unpow2 35.3%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} \]

   2. unpow2 35.3%

      \[\leadsto -0.3333333333333333 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

   3. *-commutative 35.3%

      \[\leadsto -0.3333333333333333 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot \left(k \cdot k\right)}} \]

   4. times-frac 36.9%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)} \]

10. Simplified 36.9%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)} \]
11. Step-by-step derivation

    1. associate-*r/ 37.4%

       \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{\ell}{t} \cdot \ell}{k \cdot k}} \]

    2. *-commutative 37.4%

       \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\ell \cdot \frac{\ell}{t}}}{k \cdot k} \]

12. Applied egg-rr 37.4%

    \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\ell \cdot \frac{\ell}{t}}{k \cdot k}} \]

13. Final simplification 37.4%

    \[\leadsto -0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{t}}{k \cdot k} \]

Reproduce

herbie shell --seed 2023216 
(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))))