Toniolo and Linder, Equation (10+)

Percentage Accurate: 52.8% → 74.7%

Time: 32.9s

Alternatives: 23

Speedup: 3.8×

• 26.7% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 6.170674341151165e-249
  2. l = 2.5390842128540495e+300
  3. k = 9.601373265556763e+270

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: 52.8% 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: 74.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{t}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 10^{+26}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{t_2 \cdot {t_2}^{2}}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2}} \cdot \left(\frac{\frac{\cos k}{\frac{k}{\ell}}}{t} \cdot \frac{\ell}{k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)) (t_2 (/ t (cbrt (/ (* l l) (sin k))))))
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ 1.0 (+ 1.0 t_1))))
        1e+26)
     (/ (/ (/ 2.0 (tan k)) (* t_2 (pow t_2 2.0))) (+ 2.0 t_1))
     (* (/ 2.0 (pow (sin k) 2.0)) (* (/ (/ (cos k) (/ k l)) t) (/ l k))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double t_2 = t / cbrt(((l * l) / sin(k)));
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 1e+26) {
		tmp = ((2.0 / tan(k)) / (t_2 * pow(t_2, 2.0))) / (2.0 + t_1);
	} else {
		tmp = (2.0 / pow(sin(k), 2.0)) * (((cos(k) / (k / l)) / t) * (l / k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double t_2 = t / Math.cbrt(((l * l) / Math.sin(k)));
	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+26) {
		tmp = ((2.0 / Math.tan(k)) / (t_2 * Math.pow(t_2, 2.0))) / (2.0 + t_1);
	} else {
		tmp = (2.0 / Math.pow(Math.sin(k), 2.0)) * (((Math.cos(k) / (k / l)) / t) * (l / k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	t_2 = Float64(t / cbrt(Float64(Float64(l * l) / sin(k))))
	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+26)
		tmp = Float64(Float64(Float64(2.0 / tan(k)) / Float64(t_2 * (t_2 ^ 2.0))) / Float64(2.0 + t_1));
	else
		tmp = Float64(Float64(2.0 / (sin(k) ^ 2.0)) * Float64(Float64(Float64(cos(k) / Float64(k / l)) / t) * Float64(l / k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t / N[Power[N[(N[(l * l), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $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+26], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l / k), $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))) < 1.00000000000000005e26

   1. Initial program 72.6%

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

      1. associate-/r* 72.7%

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

      2. associate-*l* 67.2%

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

      3. sqr-neg 67.2%

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

      4. associate-*l* 72.7%

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

      5. *-commutative 72.7%

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

      6. sqr-neg 72.7%

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

      7. associate-/r* 72.7%

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

   3. Simplified 73.7%

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

      1. add-cube-cbrt 73.6%

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

      2. pow2 73.6%

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

      3. cbrt-div 73.6%

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

      4. rem-cbrt-cube 73.7%

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

      5. cbrt-div 73.7%

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

      6. rem-cbrt-cube 79.7%

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

   5. Applied egg-rr 79.7%

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

   if 1.00000000000000005e26 < (/.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 21.8%

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

   2. Taylor expanded in t around 0 53.8%

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

      1. associate-*r* 53.8%

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

      2. times-frac 54.6%

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

      3. unpow2 54.6%

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

      4. associate-*l* 57.6%

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

      5. unpow2 57.6%

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

   4. Simplified 57.6%

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

      1. associate-*r/ 57.6%

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

      2. times-frac 74.5%

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

   6. Applied egg-rr 74.5%

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

   7. Taylor expanded in k around inf 53.8%

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

      1. associate-*r* 53.9%

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

      2. unpow2 53.9%

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

      3. associate-*r* 56.9%

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

      4. associate-*l/ 57.6%

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

      5. unpow2 57.6%

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

      6. *-commutative 57.6%

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

      7. times-frac 74.5%

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

      8. associate-*l/ 76.9%

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

      9. *-commutative 76.9%

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

   9. Simplified 76.9%

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

       1. div-inv 76.9%

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

       2. associate-/l* 76.9%

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

   11. Applied egg-rr 76.9%

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

       1. associate-*r/ 76.9%

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

       2. metadata-eval 76.9%

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

       3. associate-/r/ 76.9%

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

       4. associate-/r* 77.0%

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

       5. associate-*r/ 74.5%

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

       6. associate-/l* 77.0%

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

       7. associate-/r/ 77.0%

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

   13. Simplified 77.0%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 10^{+26}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\frac{t}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}} \cdot {\left(\frac{t}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2}} \cdot \left(\frac{\frac{\cos k}{\frac{k}{\ell}}}{t} \cdot \frac{\ell}{k}\right)\\ \end{array} \]

Alternative 2: 74.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot t_1} \leq 10^{+26}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\tan k \cdot {\left(\frac{t}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2}} \cdot \left(\frac{\frac{\cos k}{\frac{k}{\ell}}}{t} \cdot \frac{\ell}{k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))
   (if (<=
        (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) t_1))
        1e+26)
     (/ 2.0 (* t_1 (* (tan k) (pow (/ t (cbrt (/ (* l l) (sin k)))) 3.0))))
     (* (/ 2.0 (pow (sin k) 2.0)) (* (/ (/ (cos k) (/ k l)) t) (/ l k))))))

C

double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + pow((k / t), 2.0));
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * t_1)) <= 1e+26) {
		tmp = 2.0 / (t_1 * (tan(k) * pow((t / cbrt(((l * l) / sin(k)))), 3.0)));
	} else {
		tmp = (2.0 / pow(sin(k), 2.0)) * (((cos(k) / (k / l)) / t) * (l / k));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + Math.pow((k / t), 2.0));
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * t_1)) <= 1e+26) {
		tmp = 2.0 / (t_1 * (Math.tan(k) * Math.pow((t / Math.cbrt(((l * l) / Math.sin(k)))), 3.0)));
	} else {
		tmp = (2.0 / Math.pow(Math.sin(k), 2.0)) * (((Math.cos(k) / (k / l)) / t) * (l / k));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(1.0 + Float64(1.0 + (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)) * t_1)) <= 1e+26)
		tmp = Float64(2.0 / Float64(t_1 * Float64(tan(k) * (Float64(t / cbrt(Float64(Float64(l * l) / sin(k)))) ^ 3.0))));
	else
		tmp = Float64(Float64(2.0 / (sin(k) ^ 2.0)) * Float64(Float64(Float64(cos(k) / Float64(k / l)) / t) * Float64(l / k)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $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] * t$95$1), $MachinePrecision]), $MachinePrecision], 1e+26], N[(2.0 / N[(t$95$1 * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t / N[Power[N[(N[(l * l), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l / k), $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))) < 1.00000000000000005e26

   1. Initial program 72.6%

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

      1. associate-/r/ 73.7%

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

      2. add-cube-cbrt 73.6%

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

      3. pow3 73.6%

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

      4. cbrt-div 73.5%

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

      5. rem-cbrt-cube 79.6%

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

   3. Applied egg-rr 79.6%

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

   if 1.00000000000000005e26 < (/.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 21.8%

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

   2. Taylor expanded in t around 0 53.8%

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

      1. associate-*r* 53.8%

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

      2. times-frac 54.6%

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

      3. unpow2 54.6%

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

      4. associate-*l* 57.6%

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

      5. unpow2 57.6%

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

   4. Simplified 57.6%

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

      1. associate-*r/ 57.6%

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

      2. times-frac 74.5%

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

   6. Applied egg-rr 74.5%

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

   7. Taylor expanded in k around inf 53.8%

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

      1. associate-*r* 53.9%

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

      2. unpow2 53.9%

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

      3. associate-*r* 56.9%

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

      4. associate-*l/ 57.6%

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

      5. unpow2 57.6%

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

      6. *-commutative 57.6%

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

      7. times-frac 74.5%

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

      8. associate-*l/ 76.9%

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

      9. *-commutative 76.9%

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

   9. Simplified 76.9%

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

       1. div-inv 76.9%

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

       2. associate-/l* 76.9%

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

   11. Applied egg-rr 76.9%

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

       1. associate-*r/ 76.9%

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

       2. metadata-eval 76.9%

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

       3. associate-/r/ 76.9%

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

       4. associate-/r* 77.0%

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

       5. associate-*r/ 74.5%

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

       6. associate-/l* 77.0%

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

       7. associate-/r/ 77.0%

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

   13. Simplified 77.0%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 10^{+26}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot {\left(\frac{t}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2}} \cdot \left(\frac{\frac{\cos k}{\frac{k}{\ell}}}{t} \cdot \frac{\ell}{k}\right)\\ \end{array} \]

Alternative 3: 74.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2}} \cdot \left(\frac{\frac{\cos k}{\frac{k}{\ell}}}{t} \cdot \frac{\ell}{k}\right)\\ \end{array} \end{array} \]

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))))
        5e+307)
     (/
      (* (* l l) (/ 2.0 (* (tan k) (pow (* t (cbrt (sin k))) 3.0))))
      (+ 2.0 t_1))
     (* (/ 2.0 (pow (sin k) 2.0)) (* (/ (/ (cos k) (/ k l)) t) (/ l 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)))) <= 5e+307) {
		tmp = ((l * l) * (2.0 / (tan(k) * pow((t * cbrt(sin(k))), 3.0)))) / (2.0 + t_1);
	} else {
		tmp = (2.0 / pow(sin(k), 2.0)) * (((cos(k) / (k / l)) / t) * (l / k));
	}
	return tmp;
}

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)))) <= 5e+307) {
		tmp = ((l * l) * (2.0 / (Math.tan(k) * Math.pow((t * Math.cbrt(Math.sin(k))), 3.0)))) / (2.0 + t_1);
	} else {
		tmp = (2.0 / Math.pow(Math.sin(k), 2.0)) * (((Math.cos(k) / (k / l)) / t) * (l / 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)))) <= 5e+307)
		tmp = Float64(Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * (Float64(t * cbrt(sin(k))) ^ 3.0)))) / Float64(2.0 + t_1));
	else
		tmp = Float64(Float64(2.0 / (sin(k) ^ 2.0)) * Float64(Float64(Float64(cos(k) / Float64(k / l)) / t) * Float64(l / k)));
	end
	return 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], 5e+307], N[(N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l / k), $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))) < 5e307

   1. Initial program 72.8%

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

      1. associate-/r* 72.9%

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

      2. associate-*l* 67.4%

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

      3. sqr-neg 67.4%

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

      4. associate-*l* 72.9%

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

      5. *-commutative 72.9%

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

      6. sqr-neg 72.9%

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

      7. associate-*l/ 74.7%

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

      8. associate-*r/ 74.0%

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

      9. associate-/r/ 73.2%

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

   3. Simplified 73.2%

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

      1. add-cube-cbrt 73.1%

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

      2. pow3 73.1%

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

      3. cbrt-prod 73.0%

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

      4. rem-cbrt-cube 77.6%

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

   5. Applied egg-rr 77.6%

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

   if 5e307 < (/.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 21.2%

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

   2. Taylor expanded in t around 0 54.1%

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

      1. associate-*r* 54.1%

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

      2. times-frac 54.9%

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

      3. unpow2 54.9%

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

      4. associate-*l* 57.9%

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

      5. unpow2 57.9%

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

   4. Simplified 57.9%

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

      1. associate-*r/ 57.9%

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

      2. times-frac 74.9%

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

   6. Applied egg-rr 74.9%

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

   7. Taylor expanded in k around inf 54.1%

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

      1. associate-*r* 54.2%

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

      2. unpow2 54.2%

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

      3. associate-*r* 57.2%

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

      4. associate-*l/ 57.9%

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

      5. unpow2 57.9%

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

      6. *-commutative 57.9%

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

      7. times-frac 74.9%

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

      8. associate-*l/ 77.3%

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

      9. *-commutative 77.3%

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

   9. Simplified 77.3%

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

       1. div-inv 77.3%

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

       2. associate-/l* 77.4%

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

   11. Applied egg-rr 77.4%

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

       1. associate-*r/ 77.4%

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

       2. metadata-eval 77.4%

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

       3. associate-/r/ 77.4%

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

       4. associate-/r* 77.4%

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

       5. associate-*r/ 75.0%

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

       6. associate-/l* 77.4%

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

       7. associate-/r/ 77.5%

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

   13. Simplified 77.5%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2}} \cdot \left(\frac{\frac{\cos k}{\frac{k}{\ell}}}{t} \cdot \frac{\ell}{k}\right)\\ \end{array} \]

Alternative 4: 74.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 10^{+26}:\\ \;\;\;\;\frac{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\tan k}}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2}} \cdot \left(\frac{\frac{\cos k}{\frac{k}{\ell}}}{t} \cdot \frac{\ell}{k}\right)\\ \end{array} \end{array} \]

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+26)
     (/
      (/ (/ (* 2.0 (* l l)) (tan k)) (pow (* t (cbrt (sin k))) 3.0))
      (+ 2.0 t_1))
     (* (/ 2.0 (pow (sin k) 2.0)) (* (/ (/ (cos k) (/ k l)) t) (/ l 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+26) {
		tmp = (((2.0 * (l * l)) / tan(k)) / pow((t * cbrt(sin(k))), 3.0)) / (2.0 + t_1);
	} else {
		tmp = (2.0 / pow(sin(k), 2.0)) * (((cos(k) / (k / l)) / t) * (l / k));
	}
	return tmp;
}

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+26) {
		tmp = (((2.0 * (l * l)) / Math.tan(k)) / Math.pow((t * Math.cbrt(Math.sin(k))), 3.0)) / (2.0 + t_1);
	} else {
		tmp = (2.0 / Math.pow(Math.sin(k), 2.0)) * (((Math.cos(k) / (k / l)) / t) * (l / 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+26)
		tmp = Float64(Float64(Float64(Float64(2.0 * Float64(l * l)) / tan(k)) / (Float64(t * cbrt(sin(k))) ^ 3.0)) / Float64(2.0 + t_1));
	else
		tmp = Float64(Float64(2.0 / (sin(k) ^ 2.0)) * Float64(Float64(Float64(cos(k) / Float64(k / l)) / t) * Float64(l / k)));
	end
	return 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+26], N[(N[(N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l / k), $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))) < 1.00000000000000005e26

   1. Initial program 72.6%

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

      1. associate-/r* 72.7%

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

      2. associate-*l* 67.2%

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

      3. sqr-neg 67.2%

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

      4. associate-*l* 72.7%

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

      5. *-commutative 72.7%

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

      6. sqr-neg 72.7%

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

      7. associate-*l/ 74.6%

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

      8. associate-*r/ 73.8%

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

      9. associate-/r/ 73.0%

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

   3. Simplified 73.0%

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

      1. add-cube-cbrt 72.9%

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

      2. pow3 72.9%

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

      3. cbrt-prod 72.9%

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

      4. rem-cbrt-cube 77.5%

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

   5. Applied egg-rr 77.5%

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

      1. associate-*l/ 78.3%

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

   7. Applied egg-rr 78.3%

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

      1. associate-/r* 78.1%

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

   9. Simplified 78.1%

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

   if 1.00000000000000005e26 < (/.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 21.8%

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

   2. Taylor expanded in t around 0 53.8%

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

      1. associate-*r* 53.8%

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

      2. times-frac 54.6%

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

      3. unpow2 54.6%

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

      4. associate-*l* 57.6%

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

      5. unpow2 57.6%

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

   4. Simplified 57.6%

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

      1. associate-*r/ 57.6%

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

      2. times-frac 74.5%

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

   6. Applied egg-rr 74.5%

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

   7. Taylor expanded in k around inf 53.8%

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

      1. associate-*r* 53.9%

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

      2. unpow2 53.9%

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

      3. associate-*r* 56.9%

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

      4. associate-*l/ 57.6%

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

      5. unpow2 57.6%

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

      6. *-commutative 57.6%

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

      7. times-frac 74.5%

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

      8. associate-*l/ 76.9%

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

      9. *-commutative 76.9%

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

   9. Simplified 76.9%

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

       1. div-inv 76.9%

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

       2. associate-/l* 76.9%

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

   11. Applied egg-rr 76.9%

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

       1. associate-*r/ 76.9%

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

       2. metadata-eval 76.9%

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

       3. associate-/r/ 76.9%

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

       4. associate-/r* 77.0%

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

       5. associate-*r/ 74.5%

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

       6. associate-/l* 77.0%

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

       7. associate-/r/ 77.0%

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

   13. Simplified 77.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 10^{+26}:\\ \;\;\;\;\frac{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\tan k}}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2}} \cdot \left(\frac{\frac{\cos k}{\frac{k}{\ell}}}{t} \cdot \frac{\ell}{k}\right)\\ \end{array} \]

Alternative 5: 74.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\ell \cdot \frac{2 \cdot \ell}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2}} \cdot \left(\frac{\frac{\cos k}{\frac{k}{\ell}}}{t} \cdot \frac{\ell}{k}\right)\\ \end{array} \end{array} \]

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))))
        5e+307)
     (/ (* l (/ (* 2.0 l) (* (tan k) (* (pow t 3.0) (sin k))))) (+ 2.0 t_1))
     (* (/ 2.0 (pow (sin k) 2.0)) (* (/ (/ (cos k) (/ k l)) t) (/ l 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)))) <= 5e+307) {
		tmp = (l * ((2.0 * l) / (tan(k) * (pow(t, 3.0) * sin(k))))) / (2.0 + t_1);
	} else {
		tmp = (2.0 / pow(sin(k), 2.0)) * (((cos(k) / (k / l)) / t) * (l / 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)))) <= 5d+307) then
        tmp = (l * ((2.0d0 * l) / (tan(k) * ((t ** 3.0d0) * sin(k))))) / (2.0d0 + t_1)
    else
        tmp = (2.0d0 / (sin(k) ** 2.0d0)) * (((cos(k) / (k / l)) / t) * (l / 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)))) <= 5e+307) {
		tmp = (l * ((2.0 * l) / (Math.tan(k) * (Math.pow(t, 3.0) * Math.sin(k))))) / (2.0 + t_1);
	} else {
		tmp = (2.0 / Math.pow(Math.sin(k), 2.0)) * (((Math.cos(k) / (k / l)) / t) * (l / 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)))) <= 5e+307:
		tmp = (l * ((2.0 * l) / (math.tan(k) * (math.pow(t, 3.0) * math.sin(k))))) / (2.0 + t_1)
	else:
		tmp = (2.0 / math.pow(math.sin(k), 2.0)) * (((math.cos(k) / (k / l)) / t) * (l / 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)))) <= 5e+307)
		tmp = Float64(Float64(l * Float64(Float64(2.0 * l) / Float64(tan(k) * Float64((t ^ 3.0) * sin(k))))) / Float64(2.0 + t_1));
	else
		tmp = Float64(Float64(2.0 / (sin(k) ^ 2.0)) * Float64(Float64(Float64(cos(k) / Float64(k / l)) / t) * Float64(l / 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)))) <= 5e+307)
		tmp = (l * ((2.0 * l) / (tan(k) * ((t ^ 3.0) * sin(k))))) / (2.0 + t_1);
	else
		tmp = (2.0 / (sin(k) ^ 2.0)) * (((cos(k) / (k / l)) / t) * (l / 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], 5e+307], N[(N[(l * N[(N[(2.0 * l), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l / k), $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))) < 5e307

   1. Initial program 72.8%

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

      1. associate-/r* 72.9%

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

      2. associate-*l* 67.4%

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

      3. sqr-neg 67.4%

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

      4. associate-*l* 72.9%

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

      5. *-commutative 72.9%

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

      6. sqr-neg 72.9%

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

      7. associate-*l/ 74.7%

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

      8. associate-*r/ 74.0%

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

      9. associate-/r/ 73.2%

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

   3. Simplified 73.2%

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

      1. add-cube-cbrt 73.1%

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

      2. pow3 73.1%

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

      3. cbrt-prod 73.0%

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

      4. rem-cbrt-cube 77.6%

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

   5. Applied egg-rr 77.6%

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

      1. associate-*l/ 78.4%

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

   7. Applied egg-rr 78.4%

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

      1. unpow2 78.4%

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

      2. *-commutative 78.4%

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

      3. associate-*r/ 77.6%

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

      4. unpow2 77.6%

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

      5. associate-*l* 81.6%

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

      6. associate-*r/ 82.4%

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

      7. *-commutative 82.4%

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

      8. cube-prod 77.3%

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

      9. rem-cube-cbrt 77.4%

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

   9. Simplified 77.4%

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

   if 5e307 < (/.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 21.2%

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

   2. Taylor expanded in t around 0 54.1%

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

      1. associate-*r* 54.1%

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

      2. times-frac 54.9%

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

      3. unpow2 54.9%

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

      4. associate-*l* 57.9%

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

      5. unpow2 57.9%

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

   4. Simplified 57.9%

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

      1. associate-*r/ 57.9%

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

      2. times-frac 74.9%

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

   6. Applied egg-rr 74.9%

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

   7. Taylor expanded in k around inf 54.1%

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

      1. associate-*r* 54.2%

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

      2. unpow2 54.2%

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

      3. associate-*r* 57.2%

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

      4. associate-*l/ 57.9%

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

      5. unpow2 57.9%

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

      6. *-commutative 57.9%

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

      7. times-frac 74.9%

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

      8. associate-*l/ 77.3%

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

      9. *-commutative 77.3%

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

   9. Simplified 77.3%

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

       1. div-inv 77.3%

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

       2. associate-/l* 77.4%

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

   11. Applied egg-rr 77.4%

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

       1. associate-*r/ 77.4%

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

       2. metadata-eval 77.4%

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

       3. associate-/r/ 77.4%

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

       4. associate-/r* 77.4%

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

       5. associate-*r/ 75.0%

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

       6. associate-/l* 77.4%

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

       7. associate-/r/ 77.5%

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

   13. Simplified 77.5%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\ell \cdot \frac{2 \cdot \ell}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2}} \cdot \left(\frac{\frac{\cos k}{\frac{k}{\ell}}}{t} \cdot \frac{\ell}{k}\right)\\ \end{array} \]

Alternative 6: 71.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+152} \lor \neg \left(t \leq 3800000000000\right):\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2}} \cdot \left(\frac{\frac{\cos k}{\frac{k}{\ell}}}{t} \cdot \frac{\ell}{k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.85e+152) (not (<= t 3800000000000.0)))
   (/
    2.0
    (*
     (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))
     (* (tan k) (* (/ k l) (/ (pow t 3.0) l)))))
   (* (/ 2.0 (pow (sin k) 2.0)) (* (/ (/ (cos k) (/ k l)) t) (/ l k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.85e+152) || !(t <= 3800000000000.0)) {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t), 2.0))) * (tan(k) * ((k / l) * (pow(t, 3.0) / l))));
	} else {
		tmp = (2.0 / pow(sin(k), 2.0)) * (((cos(k) / (k / l)) / t) * (l / 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 <= (-1.85d+152)) .or. (.not. (t <= 3800000000000.0d0))) then
        tmp = 2.0d0 / ((1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))) * (tan(k) * ((k / l) * ((t ** 3.0d0) / l))))
    else
        tmp = (2.0d0 / (sin(k) ** 2.0d0)) * (((cos(k) / (k / l)) / t) * (l / k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.85e+152) || !(t <= 3800000000000.0)) {
		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 {
		tmp = (2.0 / Math.pow(Math.sin(k), 2.0)) * (((Math.cos(k) / (k / l)) / t) * (l / k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -1.85e+152) or not (t <= 3800000000000.0):
		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:
		tmp = (2.0 / math.pow(math.sin(k), 2.0)) * (((math.cos(k) / (k / l)) / t) * (l / k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.85e+152) || !(t <= 3800000000000.0))
		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)))));
	else
		tmp = Float64(Float64(2.0 / (sin(k) ^ 2.0)) * Float64(Float64(Float64(cos(k) / Float64(k / l)) / t) * Float64(l / k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.85e+152) || ~((t <= 3800000000000.0)))
		tmp = 2.0 / ((1.0 + (1.0 + ((k / t) ^ 2.0))) * (tan(k) * ((k / l) * ((t ^ 3.0) / l))));
	else
		tmp = (2.0 / (sin(k) ^ 2.0)) * (((cos(k) / (k / l)) / t) * (l / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -1.85e+152], N[Not[LessEqual[t, 3800000000000.0]], $MachinePrecision]], 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], N[(N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.84999999999999998e152 or 3.8e12 < t

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0 56.7%

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

      1. *-commutative 56.7%

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

      2. unpow2 56.7%

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

      3. times-frac 60.0%

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

   4. Simplified 60.0%

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

   if -1.84999999999999998e152 < t < 3.8e12

   1. Initial program 43.0%

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

   2. Taylor expanded in t around 0 66.0%

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

      1. associate-*r* 66.0%

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

      2. times-frac 66.5%

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

      3. unpow2 66.5%

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

      4. associate-*l* 68.8%

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

      5. unpow2 68.8%

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

   4. Simplified 68.8%

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

      1. associate-*r/ 68.8%

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

      2. times-frac 81.6%

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

   6. Applied egg-rr 81.6%

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

   7. Taylor expanded in k around inf 66.0%

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

      1. associate-*r* 66.0%

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

      2. unpow2 66.0%

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

      3. associate-*r* 68.3%

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

      4. associate-*l/ 68.8%

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

      5. unpow2 68.8%

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

      6. *-commutative 68.8%

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

      7. times-frac 81.6%

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

      8. associate-*l/ 84.0%

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

      9. *-commutative 84.0%

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

   9. Simplified 84.0%

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

       1. div-inv 84.0%

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

       2. associate-/l* 84.0%

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

   11. Applied egg-rr 84.0%

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

       1. associate-*r/ 84.0%

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

       2. metadata-eval 84.0%

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

       3. associate-/r/ 84.0%

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

       4. associate-/r* 84.2%

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

       5. associate-*r/ 81.7%

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

       6. associate-/l* 84.2%

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

       7. associate-/r/ 84.2%

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

   13. Simplified 84.2%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+152} \lor \neg \left(t \leq 3800000000000\right):\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2}} \cdot \left(\frac{\frac{\cos k}{\frac{k}{\ell}}}{t} \cdot \frac{\ell}{k}\right)\\ \end{array} \]

Alternative 7: 71.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{{t}^{3}}{\ell}\\ t_2 := 1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\\ \mathbf{if}\;t \leq -1.25 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\tan k \cdot \frac{k}{\frac{\ell}{t_1}}\right)}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{{\sin k}^{2}} \cdot \left(\frac{\frac{\cos k}{\frac{k}{\ell}}}{t} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\tan k \cdot \left(\frac{k}{\ell} \cdot t_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (pow t 3.0) l)) (t_2 (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))
   (if (<= t -1.25e+148)
     (/ 2.0 (* t_2 (* (tan k) (/ k (/ l t_1)))))
     (if (<= t 2.1e+14)
       (* (/ 2.0 (pow (sin k) 2.0)) (* (/ (/ (cos k) (/ k l)) t) (/ l k)))
       (/ 2.0 (* t_2 (* (tan k) (* (/ k l) t_1))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(t, 3.0) / l;
	double t_2 = 1.0 + (1.0 + pow((k / t), 2.0));
	double tmp;
	if (t <= -1.25e+148) {
		tmp = 2.0 / (t_2 * (tan(k) * (k / (l / t_1))));
	} else if (t <= 2.1e+14) {
		tmp = (2.0 / pow(sin(k), 2.0)) * (((cos(k) / (k / l)) / t) * (l / k));
	} else {
		tmp = 2.0 / (t_2 * (tan(k) * ((k / l) * t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t ** 3.0d0) / l
    t_2 = 1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))
    if (t <= (-1.25d+148)) then
        tmp = 2.0d0 / (t_2 * (tan(k) * (k / (l / t_1))))
    else if (t <= 2.1d+14) then
        tmp = (2.0d0 / (sin(k) ** 2.0d0)) * (((cos(k) / (k / l)) / t) * (l / k))
    else
        tmp = 2.0d0 / (t_2 * (tan(k) * ((k / l) * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(t, 3.0) / l;
	double t_2 = 1.0 + (1.0 + Math.pow((k / t), 2.0));
	double tmp;
	if (t <= -1.25e+148) {
		tmp = 2.0 / (t_2 * (Math.tan(k) * (k / (l / t_1))));
	} else if (t <= 2.1e+14) {
		tmp = (2.0 / Math.pow(Math.sin(k), 2.0)) * (((Math.cos(k) / (k / l)) / t) * (l / k));
	} else {
		tmp = 2.0 / (t_2 * (Math.tan(k) * ((k / l) * t_1)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(t, 3.0) / l
	t_2 = 1.0 + (1.0 + math.pow((k / t), 2.0))
	tmp = 0
	if t <= -1.25e+148:
		tmp = 2.0 / (t_2 * (math.tan(k) * (k / (l / t_1))))
	elif t <= 2.1e+14:
		tmp = (2.0 / math.pow(math.sin(k), 2.0)) * (((math.cos(k) / (k / l)) / t) * (l / k))
	else:
		tmp = 2.0 / (t_2 * (math.tan(k) * ((k / l) * t_1)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64((t ^ 3.0) / l)
	t_2 = Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))
	tmp = 0.0
	if (t <= -1.25e+148)
		tmp = Float64(2.0 / Float64(t_2 * Float64(tan(k) * Float64(k / Float64(l / t_1)))));
	elseif (t <= 2.1e+14)
		tmp = Float64(Float64(2.0 / (sin(k) ^ 2.0)) * Float64(Float64(Float64(cos(k) / Float64(k / l)) / t) * Float64(l / k)));
	else
		tmp = Float64(2.0 / Float64(t_2 * Float64(tan(k) * Float64(Float64(k / l) * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (t ^ 3.0) / l;
	t_2 = 1.0 + (1.0 + ((k / t) ^ 2.0));
	tmp = 0.0;
	if (t <= -1.25e+148)
		tmp = 2.0 / (t_2 * (tan(k) * (k / (l / t_1))));
	elseif (t <= 2.1e+14)
		tmp = (2.0 / (sin(k) ^ 2.0)) * (((cos(k) / (k / l)) / t) * (l / k));
	else
		tmp = 2.0 / (t_2 * (tan(k) * ((k / l) * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.25e+148], N[(2.0 / N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[(k / N[(l / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.1e+14], N[(N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.25000000000000006e148

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0 54.8%

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

      1. associate-/l* 54.8%

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

      2. unpow2 54.8%

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

      3. associate-/l* 59.9%

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

   4. Simplified 59.9%

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

   if -1.25000000000000006e148 < t < 2.1e14

   1. Initial program 43.0%

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

   2. Taylor expanded in t around 0 66.0%

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

      1. associate-*r* 66.0%

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

      2. times-frac 66.5%

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

      3. unpow2 66.5%

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

      4. associate-*l* 68.8%

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

      5. unpow2 68.8%

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

   4. Simplified 68.8%

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

      1. associate-*r/ 68.8%

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

      2. times-frac 81.6%

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

   6. Applied egg-rr 81.6%

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

   7. Taylor expanded in k around inf 66.0%

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

      1. associate-*r* 66.0%

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

      2. unpow2 66.0%

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

      3. associate-*r* 68.3%

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

      4. associate-*l/ 68.8%

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

      5. unpow2 68.8%

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

      6. *-commutative 68.8%

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

      7. times-frac 81.6%

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

      8. associate-*l/ 84.0%

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

      9. *-commutative 84.0%

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

   9. Simplified 84.0%

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

       1. div-inv 84.0%

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

       2. associate-/l* 84.0%

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

   11. Applied egg-rr 84.0%

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

       1. associate-*r/ 84.0%

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

       2. metadata-eval 84.0%

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

       3. associate-/r/ 84.0%

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

       4. associate-/r* 84.2%

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

       5. associate-*r/ 81.7%

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

       6. associate-/l* 84.2%

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

       7. associate-/r/ 84.2%

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

   13. Simplified 84.2%

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

   if 2.1e14 < t

   1. Initial program 54.8%

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

   2. Taylor expanded in k around 0 57.5%

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

      1. *-commutative 57.5%

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

      2. unpow2 57.5%

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

      3. times-frac 60.2%

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

   4. Simplified 60.2%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \frac{k}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}\right)}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{{\sin k}^{2}} \cdot \left(\frac{\frac{\cos k}{\frac{k}{\ell}}}{t} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \end{array} \]

Alternative 8: 67.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.3 \cdot 10^{-11} \lor \neg \left(t \leq 4500\right):\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3}}\right) \cdot \frac{\cos k}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{\frac{t}{\ell}} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \cdot \frac{2}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -5.3e-11) (not (<= t 4500.0)))
   (* (* (/ l k) (/ l (pow t 3.0))) (/ (cos k) (sin k)))
   (* (* (/ l (/ t l)) (/ (cos k) (pow (sin k) 2.0))) (/ 2.0 (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -5.3e-11) || !(t <= 4500.0)) {
		tmp = ((l / k) * (l / pow(t, 3.0))) * (cos(k) / sin(k));
	} else {
		tmp = ((l / (t / l)) * (cos(k) / pow(sin(k), 2.0))) * (2.0 / (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 <= (-5.3d-11)) .or. (.not. (t <= 4500.0d0))) then
        tmp = ((l / k) * (l / (t ** 3.0d0))) * (cos(k) / sin(k))
    else
        tmp = ((l / (t / l)) * (cos(k) / (sin(k) ** 2.0d0))) * (2.0d0 / (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -5.3e-11) || !(t <= 4500.0)) {
		tmp = ((l / k) * (l / Math.pow(t, 3.0))) * (Math.cos(k) / Math.sin(k));
	} else {
		tmp = ((l / (t / l)) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0))) * (2.0 / (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -5.3e-11) or not (t <= 4500.0):
		tmp = ((l / k) * (l / math.pow(t, 3.0))) * (math.cos(k) / math.sin(k))
	else:
		tmp = ((l / (t / l)) * (math.cos(k) / math.pow(math.sin(k), 2.0))) * (2.0 / (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -5.3e-11) || !(t <= 4500.0))
		tmp = Float64(Float64(Float64(l / k) * Float64(l / (t ^ 3.0))) * Float64(cos(k) / sin(k)));
	else
		tmp = Float64(Float64(Float64(l / Float64(t / l)) * Float64(cos(k) / (sin(k) ^ 2.0))) * Float64(2.0 / Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -5.3e-11) || ~((t <= 4500.0)))
		tmp = ((l / k) * (l / (t ^ 3.0))) * (cos(k) / sin(k));
	else
		tmp = ((l / (t / l)) * (cos(k) / (sin(k) ^ 2.0))) * (2.0 / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -5.3e-11], N[Not[LessEqual[t, 4500.0]], $MachinePrecision]], N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.2999999999999998e-11 or 4500 < t

   1. Initial program 55.1%

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

      1. associate-/r* 55.2%

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

      2. associate-*l* 49.4%

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

      3. sqr-neg 49.4%

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

      4. associate-*l* 55.2%

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

      5. *-commutative 55.2%

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

      6. sqr-neg 55.2%

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

      7. associate-*l/ 57.2%

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

      8. associate-*r/ 56.4%

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

      9. associate-/r/ 56.5%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in k around 0 54.9%

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

   5. Taylor expanded in t around inf 55.0%

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

      1. associate-*r* 55.0%

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

      2. times-frac 55.7%

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

      3. unpow2 55.7%

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

      4. *-commutative 55.7%

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

      5. times-frac 57.4%

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

   7. Simplified 57.4%

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

   if -5.2999999999999998e-11 < t < 4500

   1. Initial program 39.9%

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

   2. Taylor expanded in t around 0 71.1%

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

      1. associate-*r* 71.1%

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

      2. times-frac 71.8%

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

      3. unpow2 71.8%

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

      4. associate-*l* 74.6%

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

      5. unpow2 74.6%

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

   4. Simplified 74.6%

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

      1. associate-*r/ 74.6%

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

      2. times-frac 87.8%

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

   6. Applied egg-rr 87.8%

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

   7. Taylor expanded in k around inf 71.1%

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

      1. *-commutative 71.1%

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

      2. associate-*l/ 71.1%

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

      3. *-commutative 71.1%

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

      4. times-frac 70.1%

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

      5. times-frac 70.9%

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

      6. unpow2 70.9%

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

      7. associate-/l* 77.1%

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

      8. unpow2 77.1%

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

   9. Simplified 77.1%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.3 \cdot 10^{-11} \lor \neg \left(t \leq 4500\right):\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3}}\right) \cdot \frac{\cos k}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{\frac{t}{\ell}} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \cdot \frac{2}{k \cdot k}\\ \end{array} \]

Alternative 9: 71.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+149} \lor \neg \left(t \leq 34000000000000\right):\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3}}\right) \cdot \frac{\cos k}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2}} \cdot \left(\frac{\frac{\cos k}{\frac{k}{\ell}}}{t} \cdot \frac{\ell}{k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.5e+149) (not (<= t 34000000000000.0)))
   (* (* (/ l k) (/ l (pow t 3.0))) (/ (cos k) (sin k)))
   (* (/ 2.0 (pow (sin k) 2.0)) (* (/ (/ (cos k) (/ k l)) t) (/ l k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.5e+149) || !(t <= 34000000000000.0)) {
		tmp = ((l / k) * (l / pow(t, 3.0))) * (cos(k) / sin(k));
	} else {
		tmp = (2.0 / pow(sin(k), 2.0)) * (((cos(k) / (k / l)) / t) * (l / 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 <= (-1.5d+149)) .or. (.not. (t <= 34000000000000.0d0))) then
        tmp = ((l / k) * (l / (t ** 3.0d0))) * (cos(k) / sin(k))
    else
        tmp = (2.0d0 / (sin(k) ** 2.0d0)) * (((cos(k) / (k / l)) / t) * (l / k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.5e+149) || !(t <= 34000000000000.0)) {
		tmp = ((l / k) * (l / Math.pow(t, 3.0))) * (Math.cos(k) / Math.sin(k));
	} else {
		tmp = (2.0 / Math.pow(Math.sin(k), 2.0)) * (((Math.cos(k) / (k / l)) / t) * (l / k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -1.5e+149) or not (t <= 34000000000000.0):
		tmp = ((l / k) * (l / math.pow(t, 3.0))) * (math.cos(k) / math.sin(k))
	else:
		tmp = (2.0 / math.pow(math.sin(k), 2.0)) * (((math.cos(k) / (k / l)) / t) * (l / k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.5e+149) || !(t <= 34000000000000.0))
		tmp = Float64(Float64(Float64(l / k) * Float64(l / (t ^ 3.0))) * Float64(cos(k) / sin(k)));
	else
		tmp = Float64(Float64(2.0 / (sin(k) ^ 2.0)) * Float64(Float64(Float64(cos(k) / Float64(k / l)) / t) * Float64(l / k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.5e+149) || ~((t <= 34000000000000.0)))
		tmp = ((l / k) * (l / (t ^ 3.0))) * (cos(k) / sin(k));
	else
		tmp = (2.0 / (sin(k) ^ 2.0)) * (((cos(k) / (k / l)) / t) * (l / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -1.5e+149], N[Not[LessEqual[t, 34000000000000.0]], $MachinePrecision]], N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.50000000000000002e149 or 3.4e13 < t

   1. Initial program 54.8%

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

      1. associate-/r* 55.0%

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

      2. associate-*l* 48.8%

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

      3. sqr-neg 48.8%

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

      4. associate-*l* 55.0%

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

      5. *-commutative 55.0%

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

      6. sqr-neg 55.0%

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

      7. associate-*l/ 57.9%

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

      8. associate-*r/ 56.7%

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

      9. associate-/r/ 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in k around 0 55.6%

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

   5. Taylor expanded in t around inf 55.6%

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

      1. associate-*r* 55.6%

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

      2. times-frac 56.7%

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

      3. unpow2 56.7%

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

      4. *-commutative 56.7%

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

      5. times-frac 59.7%

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

   7. Simplified 59.7%

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

   if -1.50000000000000002e149 < t < 3.4e13

   1. Initial program 43.0%

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

   2. Taylor expanded in t around 0 66.0%

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

      1. associate-*r* 66.0%

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

      2. times-frac 66.5%

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

      3. unpow2 66.5%

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

      4. associate-*l* 68.8%

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

      5. unpow2 68.8%

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

   4. Simplified 68.8%

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

      1. associate-*r/ 68.8%

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

      2. times-frac 81.6%

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

   6. Applied egg-rr 81.6%

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

   7. Taylor expanded in k around inf 66.0%

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

      1. associate-*r* 66.0%

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

      2. unpow2 66.0%

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

      3. associate-*r* 68.3%

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

      4. associate-*l/ 68.8%

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

      5. unpow2 68.8%

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

      6. *-commutative 68.8%

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

      7. times-frac 81.6%

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

      8. associate-*l/ 84.0%

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

      9. *-commutative 84.0%

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

   9. Simplified 84.0%

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

       1. div-inv 84.0%

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

       2. associate-/l* 84.0%

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

   11. Applied egg-rr 84.0%

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

       1. associate-*r/ 84.0%

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

       2. metadata-eval 84.0%

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

       3. associate-/r/ 84.0%

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

       4. associate-/r* 84.2%

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

       5. associate-*r/ 81.7%

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

       6. associate-/l* 84.2%

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

       7. associate-/r/ 84.2%

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

   13. Simplified 84.2%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+149} \lor \neg \left(t \leq 34000000000000\right):\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3}}\right) \cdot \frac{\cos k}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\sin k}^{2}} \cdot \left(\frac{\frac{\cos k}{\frac{k}{\ell}}}{t} \cdot \frac{\ell}{k}\right)\\ \end{array} \]

Alternative 10: 62.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6.6 \cdot 10^{-170}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t \cdot \frac{k}{\ell}\right)}{\ell} \cdot \frac{{k}^{2}}{\cos k}}\\ \mathbf{elif}\;t \leq 7800:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3}}\right) \cdot \frac{\cos k}{\sin k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 6.6e-170)
   (/ 2.0 (* (/ (* k (* t (/ k l))) l) (/ (pow k 2.0) (cos k))))
   (if (<= t 7800.0)
     (* 2.0 (* (/ (* l l) (* k k)) (/ (cos k) (* t (pow (sin k) 2.0)))))
     (* (* (/ l k) (/ l (pow t 3.0))) (/ (cos k) (sin k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 6.6e-170) {
		tmp = 2.0 / (((k * (t * (k / l))) / l) * (pow(k, 2.0) / cos(k)));
	} else if (t <= 7800.0) {
		tmp = 2.0 * (((l * l) / (k * k)) * (cos(k) / (t * pow(sin(k), 2.0))));
	} else {
		tmp = ((l / k) * (l / pow(t, 3.0))) * (cos(k) / sin(k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 6.6d-170) then
        tmp = 2.0d0 / (((k * (t * (k / l))) / l) * ((k ** 2.0d0) / cos(k)))
    else if (t <= 7800.0d0) then
        tmp = 2.0d0 * (((l * l) / (k * k)) * (cos(k) / (t * (sin(k) ** 2.0d0))))
    else
        tmp = ((l / k) * (l / (t ** 3.0d0))) * (cos(k) / sin(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 6.6e-170) {
		tmp = 2.0 / (((k * (t * (k / l))) / l) * (Math.pow(k, 2.0) / Math.cos(k)));
	} else if (t <= 7800.0) {
		tmp = 2.0 * (((l * l) / (k * k)) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	} else {
		tmp = ((l / k) * (l / Math.pow(t, 3.0))) * (Math.cos(k) / Math.sin(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 6.6e-170:
		tmp = 2.0 / (((k * (t * (k / l))) / l) * (math.pow(k, 2.0) / math.cos(k)))
	elif t <= 7800.0:
		tmp = 2.0 * (((l * l) / (k * k)) * (math.cos(k) / (t * math.pow(math.sin(k), 2.0))))
	else:
		tmp = ((l / k) * (l / math.pow(t, 3.0))) * (math.cos(k) / math.sin(k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 6.6e-170)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(t * Float64(k / l))) / l) * Float64((k ^ 2.0) / cos(k))));
	elseif (t <= 7800.0)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * l) / Float64(k * k)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(l / (t ^ 3.0))) * Float64(cos(k) / sin(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 6.6e-170)
		tmp = 2.0 / (((k * (t * (k / l))) / l) * ((k ^ 2.0) / cos(k)));
	elseif (t <= 7800.0)
		tmp = 2.0 * (((l * l) / (k * k)) * (cos(k) / (t * (sin(k) ^ 2.0))));
	else
		tmp = ((l / k) * (l / (t ^ 3.0))) * (cos(k) / sin(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 6.6e-170], N[(2.0 / N[(N[(N[(k * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7800.0], N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 6.60000000000000007e-170

   1. Initial program 44.3%

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

   2. Taylor expanded in t around 0 59.4%

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

      1. associate-*r* 59.4%

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

      2. times-frac 60.0%

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

      3. unpow2 60.0%

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

      4. associate-*l* 61.9%

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

      5. unpow2 61.9%

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

   4. Simplified 61.9%

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

   5. Taylor expanded in k around 0 56.9%

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

      1. expm1-log1p-u 36.0%

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

      2. times-frac 40.7%

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

   7. Applied egg-rr 40.7%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right)\right)} \cdot \frac{{k}^{2}}{\cos k}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 65.0%

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

      2. associate-*l/ 65.0%

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

      3. *-commutative 65.0%

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

      4. associate-*r/ 65.8%

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

   9. Applied egg-rr 65.8%

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

   if 6.60000000000000007e-170 < t < 7800

   1. Initial program 43.5%

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

      1. associate-/r* 43.3%

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

      2. associate-*l* 43.2%

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

      3. sqr-neg 43.2%

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

      4. associate-*l* 43.3%

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

      5. *-commutative 43.3%

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

      6. sqr-neg 43.3%

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

      7. associate-*l/ 43.2%

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

      8. associate-*r/ 43.2%

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

      9. associate-/r/ 40.5%

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

   3. Simplified 40.5%

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

   4. Taylor expanded in k around inf 80.3%

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

      1. times-frac 80.3%

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

      2. unpow2 80.3%

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

      3. unpow2 80.3%

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

   6. Simplified 80.3%

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

   if 7800 < t

   1. Initial program 55.4%

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

      1. associate-/r* 55.6%

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

      2. associate-*l* 49.2%

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

      3. sqr-neg 49.2%

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

      4. associate-*l* 55.6%

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

      5. *-commutative 55.6%

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

      6. sqr-neg 55.6%

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

      7. associate-*l/ 59.3%

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

      8. associate-*r/ 57.9%

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

      9. associate-/r/ 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in k around 0 55.0%

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

   5. Taylor expanded in t around inf 55.1%

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

      1. associate-*r* 55.1%

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

      2. times-frac 56.5%

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

      3. unpow2 56.5%

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

      4. *-commutative 56.5%

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

      5. times-frac 58.9%

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

   7. Simplified 58.9%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.6 \cdot 10^{-170}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t \cdot \frac{k}{\ell}\right)}{\ell} \cdot \frac{{k}^{2}}{\cos k}}\\ \mathbf{elif}\;t \leq 7800:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3}}\right) \cdot \frac{\cos k}{\sin k}\\ \end{array} \]

Alternative 11: 63.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{-133}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3}}\right) \cdot \frac{\cos k}{\sin k}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t \cdot \frac{k}{\ell}\right)}{\ell} \cdot \frac{{k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.15e-133)
   (* (* (/ l k) (/ l (pow t 3.0))) (/ (cos k) (sin k)))
   (if (<= k 3.8e+16)
     (/ 2.0 (* (/ (* k (* t (/ k l))) l) (/ (pow k 2.0) (cos k))))
     (* (/ 2.0 (* k (* t k))) (/ (* (* l l) (cos k)) (pow (sin k) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.15e-133) {
		tmp = ((l / k) * (l / pow(t, 3.0))) * (cos(k) / sin(k));
	} else if (k <= 3.8e+16) {
		tmp = 2.0 / (((k * (t * (k / l))) / l) * (pow(k, 2.0) / cos(k)));
	} else {
		tmp = (2.0 / (k * (t * k))) * (((l * l) * cos(k)) / pow(sin(k), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.15d-133) then
        tmp = ((l / k) * (l / (t ** 3.0d0))) * (cos(k) / sin(k))
    else if (k <= 3.8d+16) then
        tmp = 2.0d0 / (((k * (t * (k / l))) / l) * ((k ** 2.0d0) / cos(k)))
    else
        tmp = (2.0d0 / (k * (t * k))) * (((l * l) * cos(k)) / (sin(k) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.15e-133) {
		tmp = ((l / k) * (l / Math.pow(t, 3.0))) * (Math.cos(k) / Math.sin(k));
	} else if (k <= 3.8e+16) {
		tmp = 2.0 / (((k * (t * (k / l))) / l) * (Math.pow(k, 2.0) / Math.cos(k)));
	} else {
		tmp = (2.0 / (k * (t * k))) * (((l * l) * Math.cos(k)) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.15e-133:
		tmp = ((l / k) * (l / math.pow(t, 3.0))) * (math.cos(k) / math.sin(k))
	elif k <= 3.8e+16:
		tmp = 2.0 / (((k * (t * (k / l))) / l) * (math.pow(k, 2.0) / math.cos(k)))
	else:
		tmp = (2.0 / (k * (t * k))) * (((l * l) * math.cos(k)) / math.pow(math.sin(k), 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.15e-133)
		tmp = Float64(Float64(Float64(l / k) * Float64(l / (t ^ 3.0))) * Float64(cos(k) / sin(k)));
	elseif (k <= 3.8e+16)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(t * Float64(k / l))) / l) * Float64((k ^ 2.0) / cos(k))));
	else
		tmp = Float64(Float64(2.0 / Float64(k * Float64(t * k))) * Float64(Float64(Float64(l * l) * cos(k)) / (sin(k) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.15e-133)
		tmp = ((l / k) * (l / (t ^ 3.0))) * (cos(k) / sin(k));
	elseif (k <= 3.8e+16)
		tmp = 2.0 / (((k * (t * (k / l))) / l) * ((k ^ 2.0) / cos(k)));
	else
		tmp = (2.0 / (k * (t * k))) * (((l * l) * cos(k)) / (sin(k) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.15e-133], N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.8e+16], N[(2.0 / N[(N[(N[(k * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.15000000000000008e-133

   1. Initial program 48.9%

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

      1. associate-/r* 49.0%

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

      2. associate-*l* 44.9%

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

      3. sqr-neg 44.9%

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

      4. associate-*l* 49.0%

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

      5. *-commutative 49.0%

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

      6. sqr-neg 49.0%

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

      7. associate-*l/ 50.4%

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

      8. associate-*r/ 49.8%

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

      9. associate-/r/ 49.2%

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

   3. Simplified 49.2%

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

   4. Taylor expanded in k around 0 49.3%

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

   5. Taylor expanded in t around inf 50.4%

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

      1. associate-*r* 50.4%

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

      2. times-frac 51.0%

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

      3. unpow2 51.0%

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

      4. *-commutative 51.0%

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

      5. times-frac 55.5%

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

   7. Simplified 55.5%

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

   if 2.15000000000000008e-133 < k < 3.8e16

   1. Initial program 50.5%

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

   2. Taylor expanded in t around 0 58.2%

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

      1. associate-*r* 58.2%

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

      2. times-frac 58.4%

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

      3. unpow2 58.4%

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

      4. associate-*l* 58.4%

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

      5. unpow2 58.4%

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

   4. Simplified 58.4%

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

   5. Taylor expanded in k around 0 57.7%

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

      1. expm1-log1p-u 48.4%

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

      2. times-frac 53.6%

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

   7. Applied egg-rr 53.6%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right)\right)} \cdot \frac{{k}^{2}}{\cos k}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 68.9%

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

      2. associate-*l/ 68.8%

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

      3. *-commutative 68.8%

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

      4. associate-*r/ 72.8%

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

   9. Applied egg-rr 72.8%

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

   if 3.8e16 < k

   1. Initial program 38.2%

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

      1. associate-/r* 38.2%

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

      2. associate-*l* 38.2%

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

      3. sqr-neg 38.2%

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

      4. associate-*l* 38.2%

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

      5. *-commutative 38.2%

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

      6. sqr-neg 38.2%

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

      7. associate-*l/ 38.2%

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

      8. associate-*r/ 38.2%

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

      9. associate-/r/ 38.2%

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

   3. Simplified 38.2%

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

   4. Taylor expanded in k around inf 67.9%

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

      1. associate-*r/ 67.9%

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

      2. associate-*r* 68.0%

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

      3. times-frac 68.0%

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

      4. unpow2 68.0%

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

      5. associate-*l* 71.8%

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

      6. unpow2 71.8%

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

   6. Simplified 71.8%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{-133}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3}}\right) \cdot \frac{\cos k}{\sin k}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t \cdot \frac{k}{\ell}\right)}{\ell} \cdot \frac{{k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\sin k}^{2}}\\ \end{array} \]

Alternative 12: 68.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 15000000000000:\\ \;\;\;\;\cos k \cdot \frac{2}{{\sin k}^{2} \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3}}\right) \cdot \frac{\cos k}{\sin k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 15000000000000.0)
   (* (cos k) (/ 2.0 (* (pow (sin k) 2.0) (* (/ k l) (* t (/ k l))))))
   (* (* (/ l k) (/ l (pow t 3.0))) (/ (cos k) (sin k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 15000000000000.0) {
		tmp = cos(k) * (2.0 / (pow(sin(k), 2.0) * ((k / l) * (t * (k / l)))));
	} else {
		tmp = ((l / k) * (l / pow(t, 3.0))) * (cos(k) / sin(k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 15000000000000.0d0) then
        tmp = cos(k) * (2.0d0 / ((sin(k) ** 2.0d0) * ((k / l) * (t * (k / l)))))
    else
        tmp = ((l / k) * (l / (t ** 3.0d0))) * (cos(k) / sin(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 15000000000000.0) {
		tmp = Math.cos(k) * (2.0 / (Math.pow(Math.sin(k), 2.0) * ((k / l) * (t * (k / l)))));
	} else {
		tmp = ((l / k) * (l / Math.pow(t, 3.0))) * (Math.cos(k) / Math.sin(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 15000000000000.0:
		tmp = math.cos(k) * (2.0 / (math.pow(math.sin(k), 2.0) * ((k / l) * (t * (k / l)))))
	else:
		tmp = ((l / k) * (l / math.pow(t, 3.0))) * (math.cos(k) / math.sin(k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 15000000000000.0)
		tmp = Float64(cos(k) * Float64(2.0 / Float64((sin(k) ^ 2.0) * Float64(Float64(k / l) * Float64(t * Float64(k / l))))));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(l / (t ^ 3.0))) * Float64(cos(k) / sin(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 15000000000000.0)
		tmp = cos(k) * (2.0 / ((sin(k) ^ 2.0) * ((k / l) * (t * (k / l)))));
	else
		tmp = ((l / k) * (l / (t ^ 3.0))) * (cos(k) / sin(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 15000000000000.0], N[(N[Cos[k], $MachinePrecision] * N[(2.0 / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.5e13

   1. Initial program 44.6%

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

   2. Taylor expanded in t around 0 63.0%

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

      1. associate-*r* 63.0%

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

      2. times-frac 63.4%

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

      3. unpow2 63.4%

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

      4. associate-*l* 65.4%

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

      5. unpow2 65.4%

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

   4. Simplified 65.4%

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

      1. associate-*r/ 65.4%

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

      2. times-frac 76.6%

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

   6. Applied egg-rr 76.6%

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

   7. Taylor expanded in k around inf 63.0%

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

      1. associate-*r* 63.0%

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

      2. unpow2 63.0%

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

      3. associate-*r* 65.0%

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

      4. associate-*l/ 65.4%

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

      5. unpow2 65.4%

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

      6. *-commutative 65.4%

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

      7. times-frac 76.6%

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

      8. associate-*l/ 78.7%

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

      9. *-commutative 78.7%

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

   9. Simplified 78.7%

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

       1. associate-/r/ 78.7%

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

   11. Applied egg-rr 78.7%

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

   if 1.5e13 < t

   1. Initial program 54.8%

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

      1. associate-/r* 55.1%

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

      2. associate-*l* 48.1%

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

      3. sqr-neg 48.1%

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

      4. associate-*l* 55.1%

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

      5. *-commutative 55.1%

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

      6. sqr-neg 55.1%

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

      7. associate-*l/ 59.1%

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

      8. associate-*r/ 57.6%

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

      9. associate-/r/ 57.6%

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

   3. Simplified 57.6%

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

   4. Taylor expanded in k around 0 56.0%

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

   5. Taylor expanded in t around inf 55.9%

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

      1. associate-*r* 55.9%

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

      2. times-frac 57.5%

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

      3. unpow2 57.5%

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

      4. *-commutative 57.5%

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

      5. times-frac 60.0%

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

   7. Simplified 60.0%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 15000000000000:\\ \;\;\;\;\cos k \cdot \frac{2}{{\sin k}^{2} \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3}}\right) \cdot \frac{\cos k}{\sin k}\\ \end{array} \]

Alternative 13: 63.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t \cdot \frac{k}{\ell}\right)}{\ell} \cdot \frac{{k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3}}\right) \cdot \frac{\cos k}{\sin k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1.6e-70)
   (/ 2.0 (* (/ (* k (* t (/ k l))) l) (/ (pow k 2.0) (cos k))))
   (* (* (/ l k) (/ l (pow t 3.0))) (/ (cos k) (sin k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.6e-70) {
		tmp = 2.0 / (((k * (t * (k / l))) / l) * (pow(k, 2.0) / cos(k)));
	} else {
		tmp = ((l / k) * (l / pow(t, 3.0))) * (cos(k) / sin(k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 1.6d-70) then
        tmp = 2.0d0 / (((k * (t * (k / l))) / l) * ((k ** 2.0d0) / cos(k)))
    else
        tmp = ((l / k) * (l / (t ** 3.0d0))) * (cos(k) / sin(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.6e-70) {
		tmp = 2.0 / (((k * (t * (k / l))) / l) * (Math.pow(k, 2.0) / Math.cos(k)));
	} else {
		tmp = ((l / k) * (l / Math.pow(t, 3.0))) * (Math.cos(k) / Math.sin(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 1.6e-70:
		tmp = 2.0 / (((k * (t * (k / l))) / l) * (math.pow(k, 2.0) / math.cos(k)))
	else:
		tmp = ((l / k) * (l / math.pow(t, 3.0))) * (math.cos(k) / math.sin(k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 1.6e-70)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(t * Float64(k / l))) / l) * Float64((k ^ 2.0) / cos(k))));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(l / (t ^ 3.0))) * Float64(cos(k) / sin(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1.6e-70)
		tmp = 2.0 / (((k * (t * (k / l))) / l) * ((k ^ 2.0) / cos(k)));
	else
		tmp = ((l / k) * (l / (t ^ 3.0))) * (cos(k) / sin(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1.6e-70], N[(2.0 / N[(N[(N[(k * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.5999999999999999e-70

   1. Initial program 42.4%

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

   2. Taylor expanded in t around 0 62.4%

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

      1. associate-*r* 62.4%

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

      2. times-frac 62.9%

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

      3. unpow2 62.9%

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

      4. associate-*l* 65.1%

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

      5. unpow2 65.1%

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

   4. Simplified 65.1%

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

   5. Taylor expanded in k around 0 58.2%

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

      1. expm1-log1p-u 40.3%

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

      2. times-frac 45.0%

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

   7. Applied egg-rr 45.0%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right)\right)} \cdot \frac{{k}^{2}}{\cos k}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 65.7%

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

      2. associate-*l/ 65.7%

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

      3. *-commutative 65.7%

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

      4. associate-*r/ 66.4%

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

   9. Applied egg-rr 66.4%

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

   if 1.5999999999999999e-70 < t

   1. Initial program 57.8%

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

      1. associate-/r* 58.0%

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

      2. associate-*l* 52.5%

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

      3. sqr-neg 52.5%

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

      4. associate-*l* 58.0%

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

      5. *-commutative 58.0%

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

      6. sqr-neg 58.0%

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

      7. associate-*l/ 61.2%

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

      8. associate-*r/ 59.9%

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

      9. associate-/r/ 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in k around 0 56.3%

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

   5. Taylor expanded in t around inf 54.1%

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

      1. associate-*r* 54.1%

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

      2. times-frac 55.3%

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

      3. unpow2 55.3%

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

      4. *-commutative 55.3%

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

      5. times-frac 59.8%

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

   7. Simplified 59.8%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t \cdot \frac{k}{\ell}\right)}{\ell} \cdot \frac{{k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3}}\right) \cdot \frac{\cos k}{\sin k}\\ \end{array} \]

Alternative 14: 62.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t \cdot \frac{k}{\ell}\right)}{\ell} \cdot \frac{{k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\frac{\ell}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1.1e-80)
   (/ 2.0 (* (/ (* k (* t (/ k l))) l) (/ (pow k 2.0) (cos k))))
   (* (/ l (pow t 3.0)) (/ (/ l k) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.1e-80) {
		tmp = 2.0 / (((k * (t * (k / l))) / l) * (pow(k, 2.0) / cos(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 <= 1.1d-80) then
        tmp = 2.0d0 / (((k * (t * (k / l))) / l) * ((k ** 2.0d0) / cos(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 <= 1.1e-80) {
		tmp = 2.0 / (((k * (t * (k / l))) / l) * (Math.pow(k, 2.0) / Math.cos(k)));
	} else {
		tmp = (l / Math.pow(t, 3.0)) * ((l / k) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 1.1e-80:
		tmp = 2.0 / (((k * (t * (k / l))) / l) * (math.pow(k, 2.0) / math.cos(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 <= 1.1e-80)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(t * Float64(k / l))) / l) * Float64((k ^ 2.0) / cos(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 <= 1.1e-80)
		tmp = 2.0 / (((k * (t * (k / l))) / l) * ((k ^ 2.0) / cos(k)));
	else
		tmp = (l / (t ^ 3.0)) * ((l / k) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1.1e-80], N[(2.0 / N[(N[(N[(k * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] / N[Cos[k], $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 2 regimes

2. if t < 1.10000000000000005e-80

   1. Initial program 41.7%

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

   2. Taylor expanded in t around 0 62.0%

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

      1. associate-*r* 62.0%

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

      2. times-frac 62.5%

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

      3. unpow2 62.5%

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

      4. associate-*l* 64.7%

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

      5. unpow2 64.7%

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

   4. Simplified 64.7%

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

   5. Taylor expanded in k around 0 57.7%

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

      1. expm1-log1p-u 39.7%

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

      2. times-frac 44.4%

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

   7. Applied egg-rr 44.4%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right)\right)} \cdot \frac{{k}^{2}}{\cos k}} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 65.3%

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

      2. associate-*l/ 65.3%

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

      3. *-commutative 65.3%

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

      4. associate-*r/ 66.1%

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

   9. Applied egg-rr 66.1%

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

   if 1.10000000000000005e-80 < t

   1. Initial program 58.9%

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

      1. associate-/r* 59.1%

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

      2. associate-*l* 53.7%

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

      3. sqr-neg 53.7%

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

      4. associate-*l* 59.1%

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

      5. *-commutative 59.1%

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

      6. sqr-neg 59.1%

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

      7. associate-*l/ 62.2%

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

      8. associate-*r/ 60.9%

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

      9. associate-/r/ 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in k around 0 51.2%

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

      1. unpow2 51.2%

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

      2. times-frac 55.4%

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

      3. unpow2 55.4%

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

   6. Simplified 55.4%

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

   7. Taylor expanded in l around 0 55.4%

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

      1. unpow2 55.4%

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

      2. associate-/r* 59.1%

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

   9. Simplified 59.1%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t \cdot \frac{k}{\ell}\right)}{\ell} \cdot \frac{{k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\frac{\ell}{k}}{k}\\ \end{array} \]

Alternative 15: 62.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \left(k \cdot k\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\frac{\ell}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1.05e-79)
   (/ 2.0 (/ (* (* (/ k l) (* t (/ k l))) (* k k)) (cos k)))
   (* (/ l (pow t 3.0)) (/ (/ l k) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.05e-79) {
		tmp = 2.0 / ((((k / l) * (t * (k / l))) * (k * k)) / cos(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 <= 1.05d-79) then
        tmp = 2.0d0 / ((((k / l) * (t * (k / l))) * (k * k)) / cos(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 <= 1.05e-79) {
		tmp = 2.0 / ((((k / l) * (t * (k / l))) * (k * k)) / Math.cos(k));
	} else {
		tmp = (l / Math.pow(t, 3.0)) * ((l / k) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 1.05e-79:
		tmp = 2.0 / ((((k / l) * (t * (k / l))) * (k * k)) / math.cos(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 <= 1.05e-79)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k / l) * Float64(t * Float64(k / l))) * Float64(k * k)) / cos(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 <= 1.05e-79)
		tmp = 2.0 / ((((k / l) * (t * (k / l))) * (k * k)) / cos(k));
	else
		tmp = (l / (t ^ 3.0)) * ((l / k) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1.05e-79], N[(2.0 / N[(N[(N[(N[(k / l), $MachinePrecision] * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $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 2 regimes

2. if t < 1.05e-79

   1. Initial program 41.7%

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

   2. Taylor expanded in t around 0 62.0%

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

      1. associate-*r* 62.0%

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

      2. times-frac 62.5%

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

      3. unpow2 62.5%

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

      4. associate-*l* 64.7%

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

      5. unpow2 64.7%

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

   4. Simplified 64.7%

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

   5. Taylor expanded in k around 0 57.7%

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

      1. expm1-log1p-u 39.7%

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

      2. times-frac 44.4%

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

   7. Applied egg-rr 44.4%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right)\right)} \cdot \frac{{k}^{2}}{\cos k}} \]
   8. Step-by-step derivation

      1. associate-*r/ 44.4%

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

      2. expm1-log1p-u 65.3%

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

      3. *-commutative 65.3%

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

      4. associate-*r/ 66.0%

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

      5. unpow2 66.0%

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

   9. Applied egg-rr 66.0%

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

   if 1.05e-79 < t

   1. Initial program 58.9%

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

      1. associate-/r* 59.1%

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

      2. associate-*l* 53.7%

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

      3. sqr-neg 53.7%

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

      4. associate-*l* 59.1%

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

      5. *-commutative 59.1%

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

      6. sqr-neg 59.1%

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

      7. associate-*l/ 62.2%

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

      8. associate-*r/ 60.9%

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

      9. associate-/r/ 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in k around 0 51.2%

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

      1. unpow2 51.2%

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

      2. times-frac 55.4%

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

      3. unpow2 55.4%

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

   6. Simplified 55.4%

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

   7. Taylor expanded in l around 0 55.4%

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

      1. unpow2 55.4%

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

      2. associate-/r* 59.1%

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

   9. Simplified 59.1%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \left(k \cdot k\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\frac{\ell}{k}}{k}\\ \end{array} \]

Alternative 16: 63.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{{t}^{3}}\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\ell \cdot t_1}{k \cdot k}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\ell}}{\frac{\ell}{{k}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{\frac{\ell}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (pow t 3.0))))
   (if (<= t -3.5e-13)
     (/ (* l t_1) (* k k))
     (if (<= t 1.4e-80)
       (/ 2.0 (/ (* t (/ k l)) (/ l (pow k 3.0))))
       (* t_1 (/ (/ l k) k))))))

C

double code(double t, double l, double k) {
	double t_1 = l / pow(t, 3.0);
	double tmp;
	if (t <= -3.5e-13) {
		tmp = (l * t_1) / (k * k);
	} else if (t <= 1.4e-80) {
		tmp = 2.0 / ((t * (k / l)) / (l / pow(k, 3.0)));
	} else {
		tmp = t_1 * ((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) :: t_1
    real(8) :: tmp
    t_1 = l / (t ** 3.0d0)
    if (t <= (-3.5d-13)) then
        tmp = (l * t_1) / (k * k)
    else if (t <= 1.4d-80) then
        tmp = 2.0d0 / ((t * (k / l)) / (l / (k ** 3.0d0)))
    else
        tmp = t_1 * ((l / k) / k)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l / Math.pow(t, 3.0);
	double tmp;
	if (t <= -3.5e-13) {
		tmp = (l * t_1) / (k * k);
	} else if (t <= 1.4e-80) {
		tmp = 2.0 / ((t * (k / l)) / (l / Math.pow(k, 3.0)));
	} else {
		tmp = t_1 * ((l / k) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l / math.pow(t, 3.0)
	tmp = 0
	if t <= -3.5e-13:
		tmp = (l * t_1) / (k * k)
	elif t <= 1.4e-80:
		tmp = 2.0 / ((t * (k / l)) / (l / math.pow(k, 3.0)))
	else:
		tmp = t_1 * ((l / k) / k)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / (t ^ 3.0))
	tmp = 0.0
	if (t <= -3.5e-13)
		tmp = Float64(Float64(l * t_1) / Float64(k * k));
	elseif (t <= 1.4e-80)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(k / l)) / Float64(l / (k ^ 3.0))));
	else
		tmp = Float64(t_1 * Float64(Float64(l / k) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l / (t ^ 3.0);
	tmp = 0.0;
	if (t <= -3.5e-13)
		tmp = (l * t_1) / (k * k);
	elseif (t <= 1.4e-80)
		tmp = 2.0 / ((t * (k / l)) / (l / (k ^ 3.0)));
	else
		tmp = t_1 * ((l / k) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.5e-13], N[(N[(l * t$95$1), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.4e-80], N[(2.0 / N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(l / N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.5000000000000002e-13

   1. Initial program 54.7%

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

      1. associate-/r* 54.7%

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

      2. associate-*l* 49.7%

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

      3. sqr-neg 49.7%

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

      4. associate-*l* 54.7%

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

      5. *-commutative 54.7%

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

      6. sqr-neg 54.7%

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

      7. associate-*l/ 54.7%

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

      8. associate-*r/ 54.7%

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

      9. associate-/r/ 54.7%

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

   3. Simplified 54.7%

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

   4. Taylor expanded in k around 0 49.9%

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

      1. unpow2 49.9%

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

      2. times-frac 51.7%

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

      3. unpow2 51.7%

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

   6. Simplified 51.7%

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

      1. associate-*l/ 54.6%

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

   8. Applied egg-rr 54.6%

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

   if -3.5000000000000002e-13 < t < 1.39999999999999995e-80

   1. Initial program 36.1%

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

   2. Taylor expanded in t around 0 69.7%

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

      1. associate-*r* 69.8%

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

      2. times-frac 70.5%

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

      3. unpow2 70.5%

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

      4. associate-*l* 73.7%

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

      5. unpow2 73.7%

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

   4. Simplified 73.7%

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

      1. add-cbrt-cube 63.8%

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

      2. times-frac 63.8%

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

      3. times-frac 63.8%

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

      4. times-frac 71.0%

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

   6. Applied egg-rr 71.0%

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

      1. associate-*l* 71.0%

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

      2. cube-unmult 71.1%

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

      3. associate-*l* 71.1%

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

      4. associate-*l/ 69.5%

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

      5. associate-*r/ 69.5%

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

   8. Simplified 69.5%

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

   9. Taylor expanded in k around 0 64.9%

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

       1. *-commutative 64.9%

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

       2. associate-/l* 66.5%

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

   11. Simplified 66.5%

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

       1. rem-cbrt-cube 71.4%

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

       2. associate-*r/ 71.5%

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

       3. *-commutative 71.5%

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

   13. Applied egg-rr 71.5%

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

   if 1.39999999999999995e-80 < t

   1. Initial program 58.9%

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

      1. associate-/r* 59.1%

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

      2. associate-*l* 53.7%

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

      3. sqr-neg 53.7%

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

      4. associate-*l* 59.1%

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

      5. *-commutative 59.1%

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

      6. sqr-neg 59.1%

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

      7. associate-*l/ 62.2%

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

      8. associate-*r/ 60.9%

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

      9. associate-/r/ 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in k around 0 51.2%

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

      1. unpow2 51.2%

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

      2. times-frac 55.4%

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

      3. unpow2 55.4%

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

   6. Simplified 55.4%

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

   7. Taylor expanded in l around 0 55.4%

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

      1. unpow2 55.4%

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

      2. associate-/r* 59.1%

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

   9. Simplified 59.1%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\ell}}{\frac{\ell}{{k}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\frac{\ell}{k}}{k}\\ \end{array} \]

Alternative 17: 59.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-29}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 5.5e-29)
   (* 2.0 (* (/ l t) (/ l (pow k 4.0))))
   (* (/ l (pow t 3.0)) (/ l (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 5.5e-29) {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	} 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 <= 5.5d-29) then
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    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 <= 5.5e-29) {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	} else {
		tmp = (l / Math.pow(t, 3.0)) * (l / (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 5.5e-29:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	else:
		tmp = (l / math.pow(t, 3.0)) * (l / (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 5.5e-29)
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	else
		tmp = Float64(Float64(l / (t ^ 3.0)) * Float64(l / Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 5.5e-29)
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	else
		tmp = (l / (t ^ 3.0)) * (l / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 5.5e-29], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.4999999999999999e-29

   1. Initial program 42.8%

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

   2. Taylor expanded in t around 0 62.3%

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

      1. associate-*r* 62.3%

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

      2. times-frac 62.8%

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

      3. unpow2 62.8%

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

      4. associate-*l* 64.9%

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

      5. unpow2 64.9%

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

   4. Simplified 64.9%

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

   5. Taylor expanded in k around 0 57.3%

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

      1. unpow2 57.3%

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

      2. *-commutative 57.3%

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

      3. times-frac 63.2%

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

   7. Simplified 63.2%

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

   if 5.4999999999999999e-29 < t

   1. Initial program 57.8%

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

      1. associate-/r* 57.9%

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

      2. associate-*l* 52.1%

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

      3. sqr-neg 52.1%

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

      4. associate-*l* 57.9%

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

      5. *-commutative 57.9%

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

      6. sqr-neg 57.9%

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

      7. associate-*l/ 61.4%

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

      8. associate-*r/ 60.1%

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

      9. associate-/r/ 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in k around 0 49.4%

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

      1. unpow2 49.4%

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

      2. times-frac 53.9%

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

      3. unpow2 53.9%

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

   6. Simplified 53.9%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-29}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \end{array} \]

Alternative 18: 60.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{-82}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\frac{\ell}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 6e-82)
   (* 2.0 (* (/ l t) (/ l (pow k 4.0))))
   (* (/ l (pow t 3.0)) (/ (/ l k) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 6e-82) {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	} 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 <= 6d-82) then
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    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 <= 6e-82) {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	} else {
		tmp = (l / Math.pow(t, 3.0)) * ((l / k) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 6e-82:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	else:
		tmp = (l / math.pow(t, 3.0)) * ((l / k) / k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 6e-82)
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	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 <= 6e-82)
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	else
		tmp = (l / (t ^ 3.0)) * ((l / k) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 6e-82], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $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 2 regimes

2. if t < 5.9999999999999998e-82

   1. Initial program 41.7%

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

   2. Taylor expanded in t around 0 62.0%

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

      1. associate-*r* 62.0%

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

      2. times-frac 62.5%

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

      3. unpow2 62.5%

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

      4. associate-*l* 64.7%

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

      5. unpow2 64.7%

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

   4. Simplified 64.7%

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

   5. Taylor expanded in k around 0 56.8%

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

      1. unpow2 56.8%

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

      2. *-commutative 56.8%

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

      3. times-frac 62.9%

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

   7. Simplified 62.9%

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

   if 5.9999999999999998e-82 < t

   1. Initial program 58.9%

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

      1. associate-/r* 59.1%

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

      2. associate-*l* 53.7%

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

      3. sqr-neg 53.7%

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

      4. associate-*l* 59.1%

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

      5. *-commutative 59.1%

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

      6. sqr-neg 59.1%

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

      7. associate-*l/ 62.2%

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

      8. associate-*r/ 60.9%

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

      9. associate-/r/ 61.0%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 61.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 51.2%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. times-frac 55.4%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

      3. unpow2 55.4%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 55.4%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   7. Taylor expanded in l around 0 55.4%

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}}} \cdot \frac{\ell}{{t}^{3}} \]
   8. Step-by-step derivation

      1. unpow2 55.4%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

      2. associate-/r* 59.1%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{k}} \cdot \frac{\ell}{{t}^{3}} \]

   9. Simplified 59.1%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{k}} \cdot \frac{\ell}{{t}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{-82}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\frac{\ell}{k}}{k}\\ \end{array} \]

Alternative 19: 60.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.7 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\frac{\ell}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 4.7e-80)
   (/ 2.0 (* (/ t l) (/ (pow k 4.0) l)))
   (* (/ l (pow t 3.0)) (/ (/ l k) k))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 4.7e-80) {
		tmp = 2.0 / ((t / l) * (pow(k, 4.0) / l));
	} 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 <= 4.7d-80) then
        tmp = 2.0d0 / ((t / l) * ((k ** 4.0d0) / l))
    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 <= 4.7e-80) {
		tmp = 2.0 / ((t / l) * (Math.pow(k, 4.0) / l));
	} else {
		tmp = (l / Math.pow(t, 3.0)) * ((l / k) / k);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 4.7e-80:
		tmp = 2.0 / ((t / l) * (math.pow(k, 4.0) / l))
	else:
		tmp = (l / math.pow(t, 3.0)) * ((l / k) / k)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 4.7e-80)
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64((k ^ 4.0) / l)));
	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 <= 4.7e-80)
		tmp = 2.0 / ((t / l) * ((k ^ 4.0) / l));
	else
		tmp = (l / (t ^ 3.0)) * ((l / k) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 4.7e-80], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / l), $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 2 regimes

2. if t < 4.69999999999999973e-80

   1. Initial program 41.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0 62.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. associate-*r* 62.0%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      2. times-frac 62.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

      3. unpow2 62.5%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      4. associate-*l* 64.7%

         \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

      5. unpow2 64.7%

         \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

   4. Simplified 64.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

   5. Taylor expanded in k around 0 57.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   6. Step-by-step derivation

      1. *-commutative 57.1%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{4}}}{{\ell}^{2}}} \]

      2. unpow2 57.1%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{4}}{\color{blue}{\ell \cdot \ell}}} \]

      3. times-frac 63.2%

         \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}} \]

   7. Simplified 63.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}} \]

   if 4.69999999999999973e-80 < t

   1. Initial program 58.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 59.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 53.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 53.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 59.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 59.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 59.1%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 62.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 60.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 61.0%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 61.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 51.2%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 51.2%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. times-frac 55.4%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

      3. unpow2 55.4%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 55.4%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   7. Taylor expanded in l around 0 55.4%

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}}} \cdot \frac{\ell}{{t}^{3}} \]
   8. Step-by-step derivation

      1. unpow2 55.4%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

      2. associate-/r* 59.1%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{k}} \cdot \frac{\ell}{{t}^{3}} \]

   9. Simplified 59.1%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{k}} \cdot \frac{\ell}{{t}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.7 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\frac{\ell}{k}}{k}\\ \end{array} \]

Alternative 20: 56.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right) \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (* (/ l t) (/ l (pow k 4.0)))))

C

double code(double t, double l, double k) {
	return 2.0 * ((l / t) * (l / pow(k, 4.0)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
}

Python

def code(t, l, k):
	return 2.0 * ((l / t) * (l / math.pow(k, 4.0)))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in t around 0 57.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
3. Step-by-step derivation

   1. associate-*r* 57.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

   2. times-frac 58.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

   3. unpow2 58.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

   4. associate-*l* 60.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

   5. unpow2 60.0%

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

4. Simplified 60.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

5. Taylor expanded in k around 0 53.4%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
6. Step-by-step derivation

   1. unpow2 53.4%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. *-commutative 53.4%

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   3. times-frac 57.4%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

7. Simplified 57.4%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

8. Final simplification 57.4%

   \[\leadsto 2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right) \]

Alternative 21: 55.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{\ell}{\frac{t \cdot {k}^{4}}{\ell}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* 2.0 (/ l (/ (* t (pow k 4.0)) l))))

C

double code(double t, double l, double k) {
	return 2.0 * (l / ((t * pow(k, 4.0)) / l));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (l / ((t * (k ** 4.0d0)) / l))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (l / ((t * Math.pow(k, 4.0)) / l));
}

Python

def code(t, l, k):
	return 2.0 * (l / ((t * math.pow(k, 4.0)) / l))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(l / Float64(Float64(t * (k ^ 4.0)) / l)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (l / ((t * (k ^ 4.0)) / l));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(l / N[(N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in t around 0 57.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
3. Step-by-step derivation

   1. associate-*r* 57.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

   2. times-frac 58.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

   3. unpow2 58.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

   4. associate-*l* 60.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

   5. unpow2 60.0%

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

4. Simplified 60.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]
5. Step-by-step derivation

   1. associate-*r/ 60.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot {\sin k}^{2}}{\cos k}}} \]

   2. times-frac 69.4%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right)} \cdot {\sin k}^{2}}{\cos k}} \]

6. Applied egg-rr 69.4%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot {\sin k}^{2}}{\cos k}}} \]

7. Taylor expanded in k around 0 53.4%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. unpow2 53.4%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. associate-/l* 57.6%

      \[\leadsto 2 \cdot \color{blue}{\frac{\ell}{\frac{{k}^{4} \cdot t}{\ell}}} \]

9. Simplified 57.6%

   \[\leadsto \color{blue}{2 \cdot \frac{\ell}{\frac{{k}^{4} \cdot t}{\ell}}} \]

10. Final simplification 57.6%

    \[\leadsto 2 \cdot \frac{\ell}{\frac{t \cdot {k}^{4}}{\ell}} \]

Alternative 22: 31.8% accurate, 32.4× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(-0.16666666666666666 \cdot \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* -0.16666666666666666 (/ (/ (* l l) (* k k)) t))))

C

double code(double t, double l, double k) {
	return 2.0 * (-0.16666666666666666 * (((l * l) / (k * k)) / t));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((-0.16666666666666666d0) * (((l * l) / (k * k)) / t))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (-0.16666666666666666 * (((l * l) / (k * k)) / t));
}

Python

def code(t, l, k):
	return 2.0 * (-0.16666666666666666 * (((l * l) / (k * k)) / t))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(-0.16666666666666666 * Float64(Float64(Float64(l * l) / Float64(k * k)) / t)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (-0.16666666666666666 * (((l * l) / (k * k)) / t));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(-0.16666666666666666 * N[(N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in t around 0 57.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
3. Step-by-step derivation

   1. associate-*r* 57.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

   2. times-frac 58.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

   3. unpow2 58.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

   4. associate-*l* 60.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

   5. unpow2 60.0%

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

4. Simplified 60.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]
5. Step-by-step derivation

   1. associate-*r/ 60.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot {\sin k}^{2}}{\cos k}}} \]

   2. times-frac 69.4%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right)} \cdot {\sin k}^{2}}{\cos k}} \]

6. Applied egg-rr 69.4%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot {\sin k}^{2}}{\cos k}}} \]

7. Taylor expanded in k around 0 26.9%

   \[\leadsto \color{blue}{2 \cdot \frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. distribute-lft-out 26.9%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

   2. distribute-rgt-out-- 26.9%

      \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   3. unpow2 26.9%

      \[\leadsto 2 \cdot \left(\frac{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}{\color{blue}{k \cdot k}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   4. times-frac 27.8%

      \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{{\ell}^{2}}{t}}{k} \cdot \frac{-0.5 - -0.3333333333333333}{k}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   5. associate-/r* 27.9%

      \[\leadsto 2 \cdot \left(\color{blue}{\frac{{\ell}^{2}}{t \cdot k}} \cdot \frac{-0.5 - -0.3333333333333333}{k} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   6. unpow2 27.9%

      \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{t \cdot k} \cdot \frac{-0.5 - -0.3333333333333333}{k} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   7. times-frac 28.0%

      \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right)} \cdot \frac{-0.5 - -0.3333333333333333}{k} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   8. metadata-eval 28.0%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right) \cdot \frac{\color{blue}{-0.16666666666666666}}{k} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   9. unpow2 28.0%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right) \cdot \frac{-0.16666666666666666}{k} + \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}\right) \]

   10. *-commutative 28.0%

       \[\leadsto 2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right) \cdot \frac{-0.16666666666666666}{k} + \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}\right) \]

   11. times-frac 33.1%

       \[\leadsto 2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right) \cdot \frac{-0.16666666666666666}{k} + \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}}\right) \]

9. Simplified 33.1%

   \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right) \cdot \frac{-0.16666666666666666}{k} + \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

10. Taylor expanded in k around inf 27.1%

    \[\leadsto 2 \cdot \color{blue}{\left(-0.16666666666666666 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
11. Step-by-step derivation

    1. associate-/r* 26.4%

       \[\leadsto 2 \cdot \left(-0.16666666666666666 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}}\right) \]

    2. unpow2 26.4%

       \[\leadsto 2 \cdot \left(-0.16666666666666666 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t}\right) \]

    3. unpow2 26.4%

       \[\leadsto 2 \cdot \left(-0.16666666666666666 \cdot \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t}\right) \]

12. Simplified 26.4%

    \[\leadsto 2 \cdot \color{blue}{\left(-0.16666666666666666 \cdot \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}\right)} \]

13. Final simplification 26.4%

    \[\leadsto 2 \cdot \left(-0.16666666666666666 \cdot \frac{\frac{\ell \cdot \ell}{k \cdot k}}{t}\right) \]

Alternative 23: 33.3% accurate, 32.4× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ -0.16666666666666666 t) (* (/ l k) (/ l k)))))

C

double code(double t, double l, double k) {
	return 2.0 * ((-0.16666666666666666 / t) * ((l / k) * (l / k)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (((-0.16666666666666666d0) / t) * ((l / k) * (l / k)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((-0.16666666666666666 / t) * ((l / k) * (l / k)));
}

Python

def code(t, l, k):
	return 2.0 * ((-0.16666666666666666 / t) * ((l / k) * (l / k)))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(-0.16666666666666666 / t) * Float64(Float64(l / k) * Float64(l / k))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((-0.16666666666666666 / t) * ((l / k) * (l / k)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(-0.16666666666666666 / t), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in t around 0 57.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
3. Step-by-step derivation

   1. associate-*r* 57.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

   2. times-frac 58.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]

   3. unpow2 58.5%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

   4. associate-*l* 60.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

   5. unpow2 60.0%

      \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\color{blue}{\ell \cdot \ell}} \cdot \frac{{\sin k}^{2}}{\cos k}} \]

4. Simplified 60.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot \frac{{\sin k}^{2}}{\cos k}}} \]
5. Step-by-step derivation

   1. associate-*r/ 60.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \ell} \cdot {\sin k}^{2}}{\cos k}}} \]

   2. times-frac 69.4%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right)} \cdot {\sin k}^{2}}{\cos k}} \]

6. Applied egg-rr 69.4%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right) \cdot {\sin k}^{2}}{\cos k}}} \]

7. Taylor expanded in k around 0 26.9%

   \[\leadsto \color{blue}{2 \cdot \frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. distribute-lft-out 26.9%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right)} \]

   2. distribute-rgt-out-- 26.9%

      \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   3. unpow2 26.9%

      \[\leadsto 2 \cdot \left(\frac{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}{\color{blue}{k \cdot k}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   4. times-frac 27.8%

      \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{{\ell}^{2}}{t}}{k} \cdot \frac{-0.5 - -0.3333333333333333}{k}} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   5. associate-/r* 27.9%

      \[\leadsto 2 \cdot \left(\color{blue}{\frac{{\ell}^{2}}{t \cdot k}} \cdot \frac{-0.5 - -0.3333333333333333}{k} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   6. unpow2 27.9%

      \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{t \cdot k} \cdot \frac{-0.5 - -0.3333333333333333}{k} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   7. times-frac 28.0%

      \[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right)} \cdot \frac{-0.5 - -0.3333333333333333}{k} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   8. metadata-eval 28.0%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right) \cdot \frac{\color{blue}{-0.16666666666666666}}{k} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) \]

   9. unpow2 28.0%

      \[\leadsto 2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right) \cdot \frac{-0.16666666666666666}{k} + \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}\right) \]

   10. *-commutative 28.0%

       \[\leadsto 2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right) \cdot \frac{-0.16666666666666666}{k} + \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}\right) \]

   11. times-frac 33.1%

       \[\leadsto 2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right) \cdot \frac{-0.16666666666666666}{k} + \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}}\right) \]

9. Simplified 33.1%

   \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right) \cdot \frac{-0.16666666666666666}{k} + \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

10. Taylor expanded in k around inf 27.1%

    \[\leadsto 2 \cdot \color{blue}{\left(-0.16666666666666666 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
11. Step-by-step derivation

    1. associate-*r/ 27.1%

       \[\leadsto 2 \cdot \color{blue}{\frac{-0.16666666666666666 \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

    2. *-commutative 27.1%

       \[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

    3. times-frac 26.4%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{t} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)} \]

    4. unpow2 26.4%

       \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \]

    5. unpow2 26.4%

       \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \]

    6. times-frac 28.1%

       \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right) \]

12. Simplified 28.1%

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)} \]

13. Final simplification 28.1%

    \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \]

Reproduce

herbie shell --seed 2023285 
(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))))