Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.2% → 84.8%

Time: 19.5s

Alternatives: 15

Speedup: 28.1×

• 27.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.2% 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: 84.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 10^{+186}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\tan k}}{\left(2 + t_1\right) \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

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+186)
     (/
      (/ (* 2.0 (* l l)) (tan k))
      (* (+ 2.0 t_1) (pow (* t (cbrt (sin k))) 3.0)))
     (/ 2.0 (* (* (/ k l) (/ t (/ l k))) (* (sin k) (tan k)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 1e+186) {
		tmp = ((2.0 * (l * l)) / tan(k)) / ((2.0 + t_1) * pow((t * cbrt(sin(k))), 3.0));
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (sin(k) * tan(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+186) {
		tmp = ((2.0 * (l * l)) / Math.tan(k)) / ((2.0 + t_1) * Math.pow((t * Math.cbrt(Math.sin(k))), 3.0));
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1)))) <= 1e+186)
		tmp = Float64(Float64(Float64(2.0 * Float64(l * l)) / tan(k)) / Float64(Float64(2.0 + t_1) * (Float64(t * cbrt(sin(k))) ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t / Float64(l / k))) * Float64(sin(k) * tan(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+186], N[(N[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + t$95$1), $MachinePrecision] * N[Power[N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 9.9999999999999998e185

   1. Initial program 84.5%

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

      1. associate-/l/ 84.5%

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

      2. associate-*l/ 85.1%

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

      3. associate-*l/ 81.9%

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

      4. associate-/r/ 83.0%

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

      5. *-commutative 83.0%

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

      6. associate-/l/ 83.0%

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

      7. associate-*r* 83.0%

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

      8. *-commutative 83.0%

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

      9. associate-*r* 83.0%

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

      10. *-commutative 83.0%

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

   3. Simplified 83.0%

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

      1. add-cube-cbrt 82.9%

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

      2. pow3 82.9%

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

      3. cbrt-prod 82.9%

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

      4. unpow3 82.9%

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

      5. add-cbrt-cube 85.6%

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

   5. Applied egg-rr 85.6%

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

      1. associate-*r/ 85.6%

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

   7. Applied egg-rr 85.6%

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

      1. associate-/r* 88.1%

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

      2. unpow2 88.1%

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

      3. *-commutative 88.1%

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

      4. unpow2 88.1%

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

   9. Simplified 88.1%

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

   if 9.9999999999999998e185 < (/.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 25.0%

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

      1. *-commutative 25.0%

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

      2. associate-*l* 25.0%

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

      3. associate-*r* 25.0%

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

      4. +-commutative 25.0%

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

      5. associate-+r+ 25.0%

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

      6. metadata-eval 25.0%

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

   3. Simplified 25.0%

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

   4. Taylor expanded in k around inf 57.1%

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

      1. *-commutative 57.1%

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

      2. unpow2 57.1%

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

      3. times-frac 67.1%

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

      4. unpow2 67.1%

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

   6. Simplified 67.1%

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

      1. associate-*l/ 67.3%

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

      2. associate-/l* 76.9%

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

   8. Applied egg-rr 76.9%

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

   9. Taylor expanded in t around 0 57.1%

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

       1. associate-/l* 56.1%

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

       2. unpow2 56.1%

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

       3. unpow2 56.1%

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

       4. associate-*r/ 61.8%

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

       5. times-frac 83.6%

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

       6. associate-/l* 84.5%

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

       7. *-commutative 84.5%

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

       8. associate-/l* 87.7%

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

   11. Simplified 87.7%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 10^{+186}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\tan k}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 2: 84.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 2 \cdot 10^{+61}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + t_1\right) \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

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))))
        2e+61)
     (*
      (* l l)
      (/ 2.0 (* (tan k) (* (+ 2.0 t_1) (pow (* t (cbrt (sin k))) 3.0)))))
     (/ 2.0 (* (* (/ k l) (/ t (/ l k))) (* (sin k) (tan k)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 2e+61) {
		tmp = (l * l) * (2.0 / (tan(k) * ((2.0 + t_1) * pow((t * cbrt(sin(k))), 3.0))));
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (sin(k) * tan(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)))) <= 2e+61) {
		tmp = (l * l) * (2.0 / (Math.tan(k) * ((2.0 + t_1) * Math.pow((t * Math.cbrt(Math.sin(k))), 3.0))));
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1)))) <= 2e+61)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * Float64(Float64(2.0 + t_1) * (Float64(t * cbrt(sin(k))) ^ 3.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t / Float64(l / k))) * Float64(sin(k) * tan(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], 2e+61], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(2.0 + t$95$1), $MachinePrecision] * N[Power[N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 1.9999999999999999e61

   1. Initial program 84.3%

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

      1. associate-/l/ 84.3%

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

      2. associate-*l/ 84.9%

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

      3. associate-*l/ 83.0%

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

      4. associate-/r/ 84.1%

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

      5. *-commutative 84.1%

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

      6. associate-/l/ 84.1%

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

      7. associate-*r* 84.1%

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

      8. *-commutative 84.1%

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

      9. associate-*r* 84.1%

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

      10. *-commutative 84.1%

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

   3. Simplified 84.1%

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

      1. add-cube-cbrt 84.0%

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

      2. pow3 84.0%

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

      3. cbrt-prod 84.0%

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

      4. unpow3 84.0%

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

      5. add-cbrt-cube 86.7%

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

   5. Applied egg-rr 86.7%

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

   if 1.9999999999999999e61 < (/.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 26.3%

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

      1. *-commutative 26.3%

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

      2. associate-*l* 26.3%

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

      3. associate-*r* 26.3%

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

      4. +-commutative 26.3%

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

      5. associate-+r+ 26.3%

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

      6. metadata-eval 26.3%

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

   3. Simplified 26.3%

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

   4. Taylor expanded in k around inf 56.2%

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

      1. *-commutative 56.2%

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

      2. unpow2 56.2%

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

      3. times-frac 66.0%

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

      4. unpow2 66.0%

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

   6. Simplified 66.0%

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

      1. associate-*l/ 66.2%

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

      2. associate-/l* 75.7%

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

   8. Applied egg-rr 75.7%

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

   9. Taylor expanded in t around 0 56.2%

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

       1. associate-/l* 55.2%

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

       2. unpow2 55.2%

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

       3. unpow2 55.2%

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

       4. associate-*r/ 60.8%

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

       5. times-frac 82.2%

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

       6. associate-/l* 83.1%

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

       7. *-commutative 83.1%

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

       8. associate-/l* 86.3%

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

   11. Simplified 86.3%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 2 \cdot 10^{+61}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 3: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{if}\;t_1 \leq 10^{+186}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1
         (/
          2.0
          (*
           (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
           (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))))))
   (if (<= t_1 1e+186)
     t_1
     (/ 2.0 (* (* (/ k l) (/ t (/ l k))) (* (sin k) (tan k)))))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + pow((k / t), 2.0))));
	double tmp;
	if (t_1 <= 1e+186) {
		tmp = t_1;
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (sin(k) * tan(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * (1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))))
    if (t_1 <= 1d+186) then
        tmp = t_1
    else
        tmp = 2.0d0 / (((k / l) * (t / (l / k))) * (sin(k) * tan(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + Math.pow((k / t), 2.0))));
	double tmp;
	if (t_1 <= 1e+186) {
		tmp = t_1;
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * (1.0 + (1.0 + math.pow((k / t), 2.0))))
	tmp = 0
	if t_1 <= 1e+186:
		tmp = t_1
	else:
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (math.sin(k) * math.tan(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))))
	tmp = 0.0
	if (t_1 <= 1e+186)
		tmp = t_1;
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t / Float64(l / k))) * Float64(sin(k) * tan(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + ((k / t) ^ 2.0))));
	tmp = 0.0;
	if (t_1 <= 1e+186)
		tmp = t_1;
	else
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (sin(k) * tan(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(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 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+186], t$95$1, N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 9.9999999999999998e185

   1. Initial program 84.5%

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

   if 9.9999999999999998e185 < (/.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 25.0%

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

      1. *-commutative 25.0%

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

      2. associate-*l* 25.0%

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

      3. associate-*r* 25.0%

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

      4. +-commutative 25.0%

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

      5. associate-+r+ 25.0%

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

      6. metadata-eval 25.0%

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

   3. Simplified 25.0%

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

   4. Taylor expanded in k around inf 57.1%

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

      1. *-commutative 57.1%

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

      2. unpow2 57.1%

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

      3. times-frac 67.1%

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

      4. unpow2 67.1%

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

   6. Simplified 67.1%

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

      1. associate-*l/ 67.3%

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

      2. associate-/l* 76.9%

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

   8. Applied egg-rr 76.9%

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

   9. Taylor expanded in t around 0 57.1%

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

       1. associate-/l* 56.1%

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

       2. unpow2 56.1%

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

       3. unpow2 56.1%

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

       4. associate-*r/ 61.8%

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

       5. times-frac 83.6%

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

       6. associate-/l* 84.5%

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

       7. *-commutative 84.5%

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

       8. associate-/l* 87.7%

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

   11. Simplified 87.7%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 10^{+186}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 4: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\\ t_2 := \frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\\ \mathbf{if}\;\frac{2}{\left(t_2 \cdot \tan k\right) \cdot t_1} \leq 10^{+186}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\tan k \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 1.0 (+ 1.0 (pow (/ k t) 2.0))))
        (t_2 (* (/ (pow t 3.0) (* l l)) (sin k))))
   (if (<= (/ 2.0 (* (* t_2 (tan k)) t_1)) 1e+186)
     (/ 2.0 (* t_2 (* (tan k) t_1)))
     (/ 2.0 (* (* (/ k l) (/ t (/ l k))) (* (sin k) (tan k)))))))

C

double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + pow((k / t), 2.0));
	double t_2 = (pow(t, 3.0) / (l * l)) * sin(k);
	double tmp;
	if ((2.0 / ((t_2 * tan(k)) * t_1)) <= 1e+186) {
		tmp = 2.0 / (t_2 * (tan(k) * t_1));
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (sin(k) * tan(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 1.0d0 + (1.0d0 + ((k / t) ** 2.0d0))
    t_2 = ((t ** 3.0d0) / (l * l)) * sin(k)
    if ((2.0d0 / ((t_2 * tan(k)) * t_1)) <= 1d+186) then
        tmp = 2.0d0 / (t_2 * (tan(k) * t_1))
    else
        tmp = 2.0d0 / (((k / l) * (t / (l / k))) * (sin(k) * tan(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 1.0 + (1.0 + Math.pow((k / t), 2.0));
	double t_2 = (Math.pow(t, 3.0) / (l * l)) * Math.sin(k);
	double tmp;
	if ((2.0 / ((t_2 * Math.tan(k)) * t_1)) <= 1e+186) {
		tmp = 2.0 / (t_2 * (Math.tan(k) * t_1));
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 1.0 + (1.0 + math.pow((k / t), 2.0))
	t_2 = (math.pow(t, 3.0) / (l * l)) * math.sin(k)
	tmp = 0
	if (2.0 / ((t_2 * math.tan(k)) * t_1)) <= 1e+186:
		tmp = 2.0 / (t_2 * (math.tan(k) * t_1))
	else:
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (math.sin(k) * math.tan(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))
	t_2 = Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k))
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(t_2 * tan(k)) * t_1)) <= 1e+186)
		tmp = Float64(2.0 / Float64(t_2 * Float64(tan(k) * t_1)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t / Float64(l / k))) * Float64(sin(k) * tan(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 1.0 + (1.0 + ((k / t) ^ 2.0));
	t_2 = ((t ^ 3.0) / (l * l)) * sin(k);
	tmp = 0.0;
	if ((2.0 / ((t_2 * tan(k)) * t_1)) <= 1e+186)
		tmp = 2.0 / (t_2 * (tan(k) * t_1));
	else
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (sin(k) * tan(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(t$95$2 * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 1e+186], N[(2.0 / N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 9.9999999999999998e185

   1. Initial program 84.5%

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

      1. associate-*l* 84.5%

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

      2. +-commutative 84.5%

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

   3. Simplified 84.5%

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

   if 9.9999999999999998e185 < (/.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 25.0%

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

      1. *-commutative 25.0%

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

      2. associate-*l* 25.0%

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

      3. associate-*r* 25.0%

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

      4. +-commutative 25.0%

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

      5. associate-+r+ 25.0%

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

      6. metadata-eval 25.0%

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

   3. Simplified 25.0%

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

   4. Taylor expanded in k around inf 57.1%

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

      1. *-commutative 57.1%

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

      2. unpow2 57.1%

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

      3. times-frac 67.1%

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

      4. unpow2 67.1%

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

   6. Simplified 67.1%

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

      1. associate-*l/ 67.3%

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

      2. associate-/l* 76.9%

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

   8. Applied egg-rr 76.9%

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

   9. Taylor expanded in t around 0 57.1%

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

       1. associate-/l* 56.1%

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

       2. unpow2 56.1%

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

       3. unpow2 56.1%

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

       4. associate-*r/ 61.8%

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

       5. times-frac 83.6%

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

       6. associate-/l* 84.5%

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

       7. *-commutative 84.5%

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

       8. associate-/l* 87.7%

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

   11. Simplified 87.7%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 10^{+186}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 5: 76.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -580000000 \lor \neg \left(k \leq 9.6 \cdot 10^{-45}\right):\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{t}}{\frac{k}{\frac{\ell}{k}}}}{\sin k \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= k -580000000.0) (not (<= k 9.6e-45)))
   (* 2.0 (/ (/ (/ l t) (/ k (/ l k))) (* (sin k) (tan k))))
   (* (/ l k) (/ (/ l k) (pow t 3.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((k <= -580000000.0) || !(k <= 9.6e-45)) {
		tmp = 2.0 * (((l / t) / (k / (l / k))) / (sin(k) * tan(k)));
	} else {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	}
	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 <= (-580000000.0d0)) .or. (.not. (k <= 9.6d-45))) then
        tmp = 2.0d0 * (((l / t) / (k / (l / k))) / (sin(k) * tan(k)))
    else
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((k <= -580000000.0) || !(k <= 9.6e-45)) {
		tmp = 2.0 * (((l / t) / (k / (l / k))) / (Math.sin(k) * Math.tan(k)));
	} else {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (k <= -580000000.0) or not (k <= 9.6e-45):
		tmp = 2.0 * (((l / t) / (k / (l / k))) / (math.sin(k) * math.tan(k)))
	else:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((k <= -580000000.0) || !(k <= 9.6e-45))
		tmp = Float64(2.0 * Float64(Float64(Float64(l / t) / Float64(k / Float64(l / k))) / Float64(sin(k) * tan(k))));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((k <= -580000000.0) || ~((k <= 9.6e-45)))
		tmp = 2.0 * (((l / t) / (k / (l / k))) / (sin(k) * tan(k)));
	else
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[k, -580000000.0], N[Not[LessEqual[k, 9.6e-45]], $MachinePrecision]], N[(2.0 * N[(N[(N[(l / t), $MachinePrecision] / N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -5.8e8 or 9.5999999999999996e-45 < k

   1. Initial program 49.7%

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

      1. *-commutative 49.7%

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

      2. associate-*l* 49.7%

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

      3. associate-*r* 49.7%

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

      4. +-commutative 49.7%

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

      5. associate-+r+ 49.7%

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

      6. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   4. Taylor expanded in k around inf 70.4%

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

      1. *-commutative 70.4%

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

      2. unpow2 70.4%

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

      3. times-frac 74.0%

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

      4. unpow2 74.0%

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

   6. Simplified 74.0%

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

      1. associate-*l/ 74.2%

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

      2. associate-/l* 81.5%

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

   8. Applied egg-rr 81.5%

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

      1. expm1-log1p-u 72.2%

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

      2. expm1-udef 64.9%

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

      3. associate-/r* 64.9%

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

      4. associate-/l* 64.9%

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

      5. associate-/r/ 64.9%

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

   10. Applied egg-rr 64.9%

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

       1. expm1-def 72.8%

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

       2. expm1-log1p 82.1%

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

       3. associate-/r* 82.1%

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

       4. metadata-eval 82.1%

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

       5. associate-*r/ 82.1%

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

       6. associate-/r* 82.1%

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

       7. associate-/r/ 80.8%

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

       8. associate-/r* 81.2%

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

       9. associate-/r/ 81.2%

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

       10. associate-*l/ 81.2%

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

       11. *-lft-identity 81.2%

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

       12. associate-*l/ 74.1%

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

       13. associate-/l* 81.2%

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

   12. Simplified 81.2%

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

   if -5.8e8 < k < 9.5999999999999996e-45

   1. Initial program 70.3%

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

      1. associate-/l/ 70.3%

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

      2. associate-*l/ 70.3%

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

      3. associate-*l/ 66.0%

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

      4. associate-/r/ 65.9%

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

      5. *-commutative 65.9%

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

      6. associate-/l/ 65.9%

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

      7. associate-*r* 65.9%

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

      8. *-commutative 65.9%

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

      9. associate-*r* 65.9%

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

      10. *-commutative 65.9%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in k around 0 62.4%

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

      1. unpow2 62.4%

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

      2. unpow2 62.4%

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

      3. associate-*l* 67.4%

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

   6. Simplified 67.4%

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

   7. Taylor expanded in l around 0 62.4%

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

      1. unpow2 62.4%

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

      2. unpow2 62.4%

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

      3. associate-*r* 67.4%

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

      4. times-frac 76.7%

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

      5. associate-/r* 79.2%

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

   9. Simplified 79.2%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -580000000 \lor \neg \left(k \leq 9.6 \cdot 10^{-45}\right):\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{t}}{\frac{k}{\frac{\ell}{k}}}}{\sin k \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \end{array} \]

Alternative 6: 78.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -580000000 \lor \neg \left(k \leq 2.3 \cdot 10^{-44}\right):\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= k -580000000.0) (not (<= k 2.3e-44)))
   (* l (* l (/ 2.0 (* (tan k) (* (sin k) (* k (* t k)))))))
   (* (/ l k) (/ (/ l k) (pow t 3.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((k <= -580000000.0) || !(k <= 2.3e-44)) {
		tmp = l * (l * (2.0 / (tan(k) * (sin(k) * (k * (t * k))))));
	} else {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	}
	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 <= (-580000000.0d0)) .or. (.not. (k <= 2.3d-44))) then
        tmp = l * (l * (2.0d0 / (tan(k) * (sin(k) * (k * (t * k))))))
    else
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((k <= -580000000.0) || !(k <= 2.3e-44)) {
		tmp = l * (l * (2.0 / (Math.tan(k) * (Math.sin(k) * (k * (t * k))))));
	} else {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (k <= -580000000.0) or not (k <= 2.3e-44):
		tmp = l * (l * (2.0 / (math.tan(k) * (math.sin(k) * (k * (t * k))))))
	else:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((k <= -580000000.0) || !(k <= 2.3e-44))
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(sin(k) * Float64(k * Float64(t * k)))))));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((k <= -580000000.0) || ~((k <= 2.3e-44)))
		tmp = l * (l * (2.0 / (tan(k) * (sin(k) * (k * (t * k))))));
	else
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[k, -580000000.0], N[Not[LessEqual[k, 2.3e-44]], $MachinePrecision]], N[(l * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -5.8e8 or 2.29999999999999998e-44 < k

   1. Initial program 49.7%

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

      1. associate-/l/ 49.7%

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

      2. associate-*l/ 49.7%

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

      3. associate-*l/ 49.7%

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

      4. associate-/r/ 51.0%

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

      5. *-commutative 51.0%

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

      6. associate-/l/ 51.0%

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

      7. associate-*r* 50.9%

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

      8. *-commutative 50.9%

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

      9. associate-*r* 51.0%

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

      10. *-commutative 51.0%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in k around inf 70.4%

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

      1. *-commutative 70.4%

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

      2. *-commutative 70.4%

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

      3. associate-*l* 70.4%

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

      4. *-commutative 70.4%

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

      5. unpow2 70.4%

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

      6. associate-*l* 70.4%

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

   6. Simplified 70.4%

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

      1. add-cbrt-cube 65.0%

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

      2. associate-*r* 65.0%

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

      3. associate-*r* 65.0%

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

      4. associate-*r* 65.0%

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

   8. Applied egg-rr 65.0%

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

      1. associate-*l* 65.0%

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

      2. cube-unmult 65.0%

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

      3. unpow2 65.0%

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

      4. associate-*r* 65.0%

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

      5. *-commutative 65.0%

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

      6. unpow2 65.0%

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

      7. associate-*r* 66.3%

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

      8. *-commutative 66.3%

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

      9. *-commutative 66.3%

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

   10. Simplified 66.3%

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

       1. expm1-log1p-u 60.2%

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

       2. expm1-udef 56.4%

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

       3. rem-cbrt-cube 61.1%

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

       4. associate-*l* 64.4%

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

       5. *-commutative 64.4%

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

   12. Applied egg-rr 64.4%

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

       1. expm1-def 73.5%

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

       2. expm1-log1p 82.7%

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

       3. *-commutative 82.7%

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

   14. Simplified 82.7%

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

   if -5.8e8 < k < 2.29999999999999998e-44

   1. Initial program 70.3%

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

      1. associate-/l/ 70.3%

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

      2. associate-*l/ 70.3%

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

      3. associate-*l/ 66.0%

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

      4. associate-/r/ 65.9%

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

      5. *-commutative 65.9%

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

      6. associate-/l/ 65.9%

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

      7. associate-*r* 65.9%

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

      8. *-commutative 65.9%

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

      9. associate-*r* 65.9%

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

      10. *-commutative 65.9%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in k around 0 62.4%

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

      1. unpow2 62.4%

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

      2. unpow2 62.4%

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

      3. associate-*l* 67.4%

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

   6. Simplified 67.4%

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

   7. Taylor expanded in l around 0 62.4%

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

      1. unpow2 62.4%

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

      2. unpow2 62.4%

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

      3. associate-*r* 67.4%

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

      4. times-frac 76.7%

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

      5. associate-/r* 79.2%

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

   9. Simplified 79.2%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -580000000 \lor \neg \left(k \leq 2.3 \cdot 10^{-44}\right):\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \end{array} \]

Alternative 7: 82.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -580000000 \lor \neg \left(k \leq 1.2 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= k -580000000.0) (not (<= k 1.2e-44)))
   (/ 2.0 (* (* (sin k) (tan k)) (* (/ k l) (* t (/ k l)))))
   (* (/ l k) (/ (/ l k) (pow t 3.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((k <= -580000000.0) || !(k <= 1.2e-44)) {
		tmp = 2.0 / ((sin(k) * tan(k)) * ((k / l) * (t * (k / l))));
	} else {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	}
	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 <= (-580000000.0d0)) .or. (.not. (k <= 1.2d-44))) then
        tmp = 2.0d0 / ((sin(k) * tan(k)) * ((k / l) * (t * (k / l))))
    else
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((k <= -580000000.0) || !(k <= 1.2e-44)) {
		tmp = 2.0 / ((Math.sin(k) * Math.tan(k)) * ((k / l) * (t * (k / l))));
	} else {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (k <= -580000000.0) or not (k <= 1.2e-44):
		tmp = 2.0 / ((math.sin(k) * math.tan(k)) * ((k / l) * (t * (k / l))))
	else:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((k <= -580000000.0) || !(k <= 1.2e-44))
		tmp = Float64(2.0 / Float64(Float64(sin(k) * tan(k)) * Float64(Float64(k / l) * Float64(t * Float64(k / l)))));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((k <= -580000000.0) || ~((k <= 1.2e-44)))
		tmp = 2.0 / ((sin(k) * tan(k)) * ((k / l) * (t * (k / l))));
	else
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[k, -580000000.0], N[Not[LessEqual[k, 1.2e-44]], $MachinePrecision]], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -5.8e8 or 1.20000000000000004e-44 < k

   1. Initial program 49.7%

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

      1. *-commutative 49.7%

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

      2. associate-*l* 49.7%

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

      3. associate-*r* 49.7%

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

      4. +-commutative 49.7%

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

      5. associate-+r+ 49.7%

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

      6. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   4. Taylor expanded in k around inf 70.4%

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

      1. *-commutative 70.4%

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

      2. unpow2 70.4%

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

      3. times-frac 74.0%

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

      4. unpow2 74.0%

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

   6. Simplified 74.0%

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

      1. associate-*l/ 74.2%

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

      2. associate-/l* 81.5%

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

   8. Applied egg-rr 81.5%

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

   9. Taylor expanded in t around 0 70.4%

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

       1. associate-/l* 69.6%

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

       2. unpow2 69.6%

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

       3. unpow2 69.6%

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

       4. associate-*r/ 71.6%

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

       5. times-frac 87.8%

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

       6. associate-/l* 89.7%

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

       7. *-commutative 89.7%

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

       8. associate-/l* 91.0%

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

   11. Simplified 91.0%

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

       1. div-inv 91.0%

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

       2. clear-num 91.0%

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

   13. Applied egg-rr 91.0%

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

   if -5.8e8 < k < 1.20000000000000004e-44

   1. Initial program 70.3%

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

      1. associate-/l/ 70.3%

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

      2. associate-*l/ 70.3%

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

      3. associate-*l/ 66.0%

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

      4. associate-/r/ 65.9%

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

      5. *-commutative 65.9%

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

      6. associate-/l/ 65.9%

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

      7. associate-*r* 65.9%

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

      8. *-commutative 65.9%

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

      9. associate-*r* 65.9%

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

      10. *-commutative 65.9%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in k around 0 62.4%

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

      1. unpow2 62.4%

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

      2. unpow2 62.4%

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

      3. associate-*l* 67.4%

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

   6. Simplified 67.4%

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

   7. Taylor expanded in l around 0 62.4%

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

      1. unpow2 62.4%

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

      2. unpow2 62.4%

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

      3. associate-*r* 67.4%

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

      4. times-frac 76.7%

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

      5. associate-/r* 79.2%

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

   9. Simplified 79.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -580000000 \lor \neg \left(k \leq 1.2 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \end{array} \]

Alternative 8: 82.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -580000000 \lor \neg \left(k \leq 4.8 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= k -580000000.0) (not (<= k 4.8e-45)))
   (/ 2.0 (* (* (/ k l) (/ t (/ l k))) (* (sin k) (tan k))))
   (* (/ l k) (/ (/ l k) (pow t 3.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((k <= -580000000.0) || !(k <= 4.8e-45)) {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (sin(k) * tan(k)));
	} else {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	}
	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 <= (-580000000.0d0)) .or. (.not. (k <= 4.8d-45))) then
        tmp = 2.0d0 / (((k / l) * (t / (l / k))) * (sin(k) * tan(k)))
    else
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((k <= -580000000.0) || !(k <= 4.8e-45)) {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (Math.sin(k) * Math.tan(k)));
	} else {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (k <= -580000000.0) or not (k <= 4.8e-45):
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (math.sin(k) * math.tan(k)))
	else:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((k <= -580000000.0) || !(k <= 4.8e-45))
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t / Float64(l / k))) * Float64(sin(k) * tan(k))));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((k <= -580000000.0) || ~((k <= 4.8e-45)))
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (sin(k) * tan(k)));
	else
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[k, -580000000.0], N[Not[LessEqual[k, 4.8e-45]], $MachinePrecision]], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -5.8e8 or 4.7999999999999998e-45 < k

   1. Initial program 49.7%

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

      1. *-commutative 49.7%

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

      2. associate-*l* 49.7%

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

      3. associate-*r* 49.7%

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

      4. +-commutative 49.7%

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

      5. associate-+r+ 49.7%

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

      6. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   4. Taylor expanded in k around inf 70.4%

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

      1. *-commutative 70.4%

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

      2. unpow2 70.4%

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

      3. times-frac 74.0%

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

      4. unpow2 74.0%

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

   6. Simplified 74.0%

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

      1. associate-*l/ 74.2%

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

      2. associate-/l* 81.5%

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

   8. Applied egg-rr 81.5%

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

   9. Taylor expanded in t around 0 70.4%

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

       1. associate-/l* 69.6%

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

       2. unpow2 69.6%

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

       3. unpow2 69.6%

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

       4. associate-*r/ 71.6%

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

       5. times-frac 87.8%

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

       6. associate-/l* 89.7%

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

       7. *-commutative 89.7%

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

       8. associate-/l* 91.0%

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

   11. Simplified 91.0%

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

   if -5.8e8 < k < 4.7999999999999998e-45

   1. Initial program 70.3%

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

      1. associate-/l/ 70.3%

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

      2. associate-*l/ 70.3%

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

      3. associate-*l/ 66.0%

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

      4. associate-/r/ 65.9%

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

      5. *-commutative 65.9%

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

      6. associate-/l/ 65.9%

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

      7. associate-*r* 65.9%

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

      8. *-commutative 65.9%

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

      9. associate-*r* 65.9%

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

      10. *-commutative 65.9%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in k around 0 62.4%

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

      1. unpow2 62.4%

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

      2. unpow2 62.4%

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

      3. associate-*l* 67.4%

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

   6. Simplified 67.4%

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

   7. Taylor expanded in l around 0 62.4%

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

      1. unpow2 62.4%

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

      2. unpow2 62.4%

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

      3. associate-*r* 67.4%

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

      4. times-frac 76.7%

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

      5. associate-/r* 79.2%

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

   9. Simplified 79.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -580000000 \lor \neg \left(k \leq 4.8 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \end{array} \]

Alternative 9: 82.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq -580000000:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot t_1}\\ \mathbf{elif}\;k \leq 5.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot t_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= k -580000000.0)
     (/ 2.0 (* (* (/ k l) (/ t (/ l k))) t_1))
     (if (<= k 5.4e-45)
       (* (/ l k) (/ (/ l k) (pow t 3.0)))
       (/ 2.0 (* (/ k l) (* t (* (/ k l) t_1))))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (k <= -580000000.0) {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * t_1);
	} else if (k <= 5.4e-45) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 / ((k / l) * (t * ((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) :: tmp
    t_1 = sin(k) * tan(k)
    if (k <= (-580000000.0d0)) then
        tmp = 2.0d0 / (((k / l) * (t / (l / k))) * t_1)
    else if (k <= 5.4d-45) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 / ((k / l) * (t * ((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.sin(k) * Math.tan(k);
	double tmp;
	if (k <= -580000000.0) {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * t_1);
	} else if (k <= 5.4e-45) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 / ((k / l) * (t * ((k / l) * t_1)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(k) * math.tan(k)
	tmp = 0
	if k <= -580000000.0:
		tmp = 2.0 / (((k / l) * (t / (l / k))) * t_1)
	elif k <= 5.4e-45:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 / ((k / l) * (t * ((k / l) * t_1)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (k <= -580000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t / Float64(l / k))) * t_1));
	elseif (k <= 5.4e-45)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(t * Float64(Float64(k / l) * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) * tan(k);
	tmp = 0.0;
	if (k <= -580000000.0)
		tmp = 2.0 / (((k / l) * (t / (l / k))) * t_1);
	elseif (k <= 5.4e-45)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 / ((k / l) * (t * ((k / l) * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -580000000.0], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.4e-45], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(t * N[(N[(k / l), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < -5.8e8

   1. Initial program 41.2%

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

      1. *-commutative 41.2%

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

      2. associate-*l* 41.2%

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

      3. associate-*r* 41.3%

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

      4. +-commutative 41.3%

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

      5. associate-+r+ 41.3%

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

      6. metadata-eval 41.3%

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

   3. Simplified 41.3%

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

   4. Taylor expanded in k around inf 69.4%

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

      1. *-commutative 69.4%

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

      2. unpow2 69.4%

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

      3. times-frac 73.2%

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

      4. unpow2 73.2%

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

   6. Simplified 73.2%

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

      1. associate-*l/ 73.3%

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

      2. associate-/l* 80.9%

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

   8. Applied egg-rr 80.9%

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

   9. Taylor expanded in t around 0 69.4%

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

       1. associate-/l* 69.4%

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

       2. unpow2 69.4%

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

       3. unpow2 69.4%

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

       4. associate-*r/ 71.1%

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

       5. times-frac 86.7%

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

       6. associate-/l* 86.6%

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

       7. *-commutative 86.6%

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

       8. associate-/l* 88.1%

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

   11. Simplified 88.1%

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

   if -5.8e8 < k < 5.3999999999999997e-45

   1. Initial program 70.3%

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

      1. associate-/l/ 70.3%

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

      2. associate-*l/ 70.3%

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

      3. associate-*l/ 66.0%

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

      4. associate-/r/ 65.9%

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

      5. *-commutative 65.9%

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

      6. associate-/l/ 65.9%

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

      7. associate-*r* 65.9%

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

      8. *-commutative 65.9%

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

      9. associate-*r* 65.9%

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

      10. *-commutative 65.9%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in k around 0 62.4%

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

      1. unpow2 62.4%

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

      2. unpow2 62.4%

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

      3. associate-*l* 67.4%

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

   6. Simplified 67.4%

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

   7. Taylor expanded in l around 0 62.4%

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

      1. unpow2 62.4%

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

      2. unpow2 62.4%

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

      3. associate-*r* 67.4%

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

      4. times-frac 76.7%

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

      5. associate-/r* 79.2%

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

   9. Simplified 79.2%

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

   if 5.3999999999999997e-45 < k

   1. Initial program 56.2%

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

      1. *-commutative 56.2%

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

      2. associate-*l* 56.2%

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

      3. associate-*r* 56.2%

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

      4. +-commutative 56.2%

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

      5. associate-+r+ 56.2%

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

      6. metadata-eval 56.2%

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

   3. Simplified 56.2%

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

   4. Taylor expanded in k around inf 71.2%

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

      1. *-commutative 71.2%

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

      2. unpow2 71.2%

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

      3. times-frac 74.7%

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

      4. unpow2 74.7%

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

   6. Simplified 74.7%

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

      1. associate-*l/ 74.9%

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

      2. associate-/l* 82.0%

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

   8. Applied egg-rr 82.0%

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

   9. Taylor expanded in t around 0 71.2%

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

       1. associate-/l* 69.7%

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

       2. unpow2 69.7%

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

       3. unpow2 69.7%

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

       4. associate-*r/ 72.1%

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

       5. times-frac 88.7%

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

       6. associate-/l* 92.1%

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

       7. *-commutative 92.1%

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

       8. associate-/l* 93.2%

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

   11. Simplified 93.2%

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

       1. pow1 93.2%

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

       2. associate-*l* 93.2%

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

       3. div-inv 93.1%

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

       4. clear-num 93.2%

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

   13. Applied egg-rr 93.2%

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

       1. unpow1 93.2%

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

       2. associate-*l* 93.2%

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

   15. Simplified 93.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -580000000:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{elif}\;k \leq 5.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)\right)}\\ \end{array} \]

Alternative 10: 82.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq -580000000:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot t_1}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\ell}}}{t \cdot \left(\frac{k}{\ell} \cdot t_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= k -580000000.0)
     (/ 2.0 (* (* (/ k l) (/ t (/ l k))) t_1))
     (if (<= k 2.2e-44)
       (* (/ l k) (/ (/ l k) (pow t 3.0)))
       (/ (/ 2.0 (/ k l)) (* t (* (/ k l) t_1)))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (k <= -580000000.0) {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * t_1);
	} else if (k <= 2.2e-44) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = (2.0 / (k / l)) / (t * ((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) :: tmp
    t_1 = sin(k) * tan(k)
    if (k <= (-580000000.0d0)) then
        tmp = 2.0d0 / (((k / l) * (t / (l / k))) * t_1)
    else if (k <= 2.2d-44) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = (2.0d0 / (k / l)) / (t * ((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.sin(k) * Math.tan(k);
	double tmp;
	if (k <= -580000000.0) {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * t_1);
	} else if (k <= 2.2e-44) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = (2.0 / (k / l)) / (t * ((k / l) * t_1));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(k) * math.tan(k)
	tmp = 0
	if k <= -580000000.0:
		tmp = 2.0 / (((k / l) * (t / (l / k))) * t_1)
	elif k <= 2.2e-44:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = (2.0 / (k / l)) / (t * ((k / l) * t_1))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (k <= -580000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t / Float64(l / k))) * t_1));
	elseif (k <= 2.2e-44)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(Float64(2.0 / Float64(k / l)) / Float64(t * Float64(Float64(k / l) * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) * tan(k);
	tmp = 0.0;
	if (k <= -580000000.0)
		tmp = 2.0 / (((k / l) * (t / (l / k))) * t_1);
	elseif (k <= 2.2e-44)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = (2.0 / (k / l)) / (t * ((k / l) * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -580000000.0], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.2e-44], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(t * N[(N[(k / l), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < -5.8e8

   1. Initial program 41.2%

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

      1. *-commutative 41.2%

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

      2. associate-*l* 41.2%

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

      3. associate-*r* 41.3%

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

      4. +-commutative 41.3%

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

      5. associate-+r+ 41.3%

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

      6. metadata-eval 41.3%

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

   3. Simplified 41.3%

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

   4. Taylor expanded in k around inf 69.4%

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

      1. *-commutative 69.4%

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

      2. unpow2 69.4%

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

      3. times-frac 73.2%

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

      4. unpow2 73.2%

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

   6. Simplified 73.2%

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

      1. associate-*l/ 73.3%

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

      2. associate-/l* 80.9%

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

   8. Applied egg-rr 80.9%

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

   9. Taylor expanded in t around 0 69.4%

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

       1. associate-/l* 69.4%

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

       2. unpow2 69.4%

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

       3. unpow2 69.4%

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

       4. associate-*r/ 71.1%

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

       5. times-frac 86.7%

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

       6. associate-/l* 86.6%

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

       7. *-commutative 86.6%

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

       8. associate-/l* 88.1%

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

   11. Simplified 88.1%

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

   if -5.8e8 < k < 2.20000000000000012e-44

   1. Initial program 70.3%

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

      1. associate-/l/ 70.3%

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

      2. associate-*l/ 70.3%

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

      3. associate-*l/ 66.0%

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

      4. associate-/r/ 65.9%

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

      5. *-commutative 65.9%

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

      6. associate-/l/ 65.9%

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

      7. associate-*r* 65.9%

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

      8. *-commutative 65.9%

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

      9. associate-*r* 65.9%

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

      10. *-commutative 65.9%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in k around 0 62.4%

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

      1. unpow2 62.4%

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

      2. unpow2 62.4%

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

      3. associate-*l* 67.4%

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

   6. Simplified 67.4%

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

   7. Taylor expanded in l around 0 62.4%

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

      1. unpow2 62.4%

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

      2. unpow2 62.4%

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

      3. associate-*r* 67.4%

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

      4. times-frac 76.7%

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

      5. associate-/r* 79.2%

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

   9. Simplified 79.2%

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

   if 2.20000000000000012e-44 < k

   1. Initial program 56.2%

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

      1. *-commutative 56.2%

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

      2. associate-*l* 56.2%

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

      3. associate-*r* 56.2%

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

      4. +-commutative 56.2%

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

      5. associate-+r+ 56.2%

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

      6. metadata-eval 56.2%

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

   3. Simplified 56.2%

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

   4. Taylor expanded in k around inf 71.2%

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

      1. *-commutative 71.2%

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

      2. unpow2 71.2%

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

      3. times-frac 74.7%

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

      4. unpow2 74.7%

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

   6. Simplified 74.7%

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

      1. associate-*l/ 74.9%

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

      2. associate-/l* 82.0%

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

   8. Applied egg-rr 82.0%

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

   9. Taylor expanded in t around 0 71.2%

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

       1. associate-/l* 69.7%

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

       2. unpow2 69.7%

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

       3. unpow2 69.7%

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

       4. associate-*r/ 72.1%

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

       5. times-frac 88.7%

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

       6. associate-/l* 92.1%

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

       7. *-commutative 92.1%

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

       8. associate-/l* 93.2%

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

   11. Simplified 93.2%

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

       1. div-inv 93.2%

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

       2. associate-*l* 93.2%

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

       3. div-inv 93.1%

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

       4. clear-num 93.2%

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

   13. Applied egg-rr 93.2%

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

       1. associate-*r/ 93.2%

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

       2. metadata-eval 93.2%

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

       3. associate-/r* 93.9%

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

       4. associate-*l* 93.9%

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

   15. Simplified 93.9%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -580000000:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\ell}}}{t \cdot \left(\frac{k}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \end{array} \]

Alternative 11: 68.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-27} \lor \neg \left(t \leq 8.6 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.35e-27) (not (<= t 8.6e-44)))
   (* (/ l k) (/ l (* (pow t 3.0) k)))
   (/ 2.0 (* (* (/ k l) (/ t (/ l k))) (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.35e-27) || !(t <= 8.6e-44)) {
		tmp = (l / k) * (l / (pow(t, 3.0) * k));
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (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 <= (-1.35d-27)) .or. (.not. (t <= 8.6d-44))) then
        tmp = (l / k) * (l / ((t ** 3.0d0) * k))
    else
        tmp = 2.0d0 / (((k / l) * (t / (l / k))) * (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.35e-27) || !(t <= 8.6e-44)) {
		tmp = (l / k) * (l / (Math.pow(t, 3.0) * k));
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -1.35e-27) or not (t <= 8.6e-44):
		tmp = (l / k) * (l / (math.pow(t, 3.0) * k))
	else:
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.35e-27) || !(t <= 8.6e-44))
		tmp = Float64(Float64(l / k) * Float64(l / Float64((t ^ 3.0) * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t / Float64(l / k))) * Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.35e-27) || ~((t <= 8.6e-44)))
		tmp = (l / k) * (l / ((t ^ 3.0) * k));
	else
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -1.35e-27], N[Not[LessEqual[t, 8.6e-44]], $MachinePrecision]], N[(N[(l / k), $MachinePrecision] * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.34999999999999994e-27 or 8.60000000000000027e-44 < t

   1. Initial program 69.8%

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

      1. associate-/l/ 69.8%

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

      2. associate-*l/ 70.5%

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

      3. associate-*l/ 67.3%

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

      4. associate-/r/ 67.8%

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

      5. *-commutative 67.8%

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

      6. associate-/l/ 67.8%

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

      7. associate-*r* 67.8%

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

      8. *-commutative 67.8%

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

      9. associate-*r* 67.8%

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

      10. *-commutative 67.8%

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

   3. Simplified 67.8%

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

   4. Taylor expanded in k around 0 59.8%

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

      1. unpow2 59.8%

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

      2. unpow2 59.8%

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

      3. associate-*l* 63.5%

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

   6. Simplified 63.5%

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

      1. times-frac 69.6%

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

   8. Applied egg-rr 69.6%

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

   if -1.34999999999999994e-27 < t < 8.60000000000000027e-44

   1. Initial program 43.2%

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

      1. *-commutative 43.2%

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

      2. associate-*l* 43.2%

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

      3. associate-*r* 43.2%

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

      4. +-commutative 43.2%

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

      5. associate-+r+ 43.2%

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

      6. metadata-eval 43.2%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in k around inf 73.0%

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

      1. *-commutative 73.0%

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

      2. unpow2 73.0%

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

      3. times-frac 79.4%

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

      4. unpow2 79.4%

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

   6. Simplified 79.4%

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

      1. associate-*l/ 79.6%

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

      2. associate-/l* 88.6%

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

   8. Applied egg-rr 88.6%

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

   9. Taylor expanded in t around 0 73.0%

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

       1. associate-/l* 71.8%

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

       2. unpow2 71.8%

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

       3. unpow2 71.8%

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

       4. associate-*r/ 75.8%

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

       5. times-frac 92.1%

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

       6. associate-/l* 94.7%

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

       7. *-commutative 94.7%

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

       8. associate-/l* 96.3%

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

   11. Simplified 96.3%

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

   12. Taylor expanded in k around 0 67.6%

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

       1. unpow2 67.6%

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

   14. Simplified 67.6%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-27} \lor \neg \left(t \leq 8.6 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(k \cdot k\right)}\\ \end{array} \]

Alternative 12: 69.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.76 \cdot 10^{-71} \lor \neg \left(t \leq 7.4 \cdot 10^{-96}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.76e-71) (not (<= t 7.4e-96)))
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (/ 2.0 (* (* (/ k l) (/ t (/ l k))) (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.76e-71) || !(t <= 7.4e-96)) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (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 <= (-1.76d-71)) .or. (.not. (t <= 7.4d-96))) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 / (((k / l) * (t / (l / k))) * (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.76e-71) || !(t <= 7.4e-96)) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -1.76e-71) or not (t <= 7.4e-96):
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.76e-71) || !(t <= 7.4e-96))
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t / Float64(l / k))) * Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.76e-71) || ~((t <= 7.4e-96)))
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -1.76e-71], N[Not[LessEqual[t, 7.4e-96]], $MachinePrecision]], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.76000000000000002e-71 or 7.39999999999999972e-96 < t

   1. Initial program 69.8%

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

      1. associate-/l/ 69.8%

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

      2. associate-*l/ 70.4%

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

      3. associate-*l/ 67.6%

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

      4. associate-/r/ 68.6%

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

      5. *-commutative 68.6%

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

      6. associate-/l/ 68.7%

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

      7. associate-*r* 68.6%

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

      8. *-commutative 68.6%

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

      9. associate-*r* 68.7%

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

      10. *-commutative 68.7%

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

   3. Simplified 68.7%

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

   4. Taylor expanded in k around 0 60.3%

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

      1. unpow2 60.3%

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

      2. unpow2 60.3%

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

      3. associate-*l* 63.5%

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

   6. Simplified 63.5%

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

   7. Taylor expanded in l around 0 60.3%

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

      1. unpow2 60.3%

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

      2. unpow2 60.3%

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

      3. associate-*r* 63.5%

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

      4. times-frac 68.4%

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

      5. associate-/r* 70.0%

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

   9. Simplified 70.0%

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

   if -1.76000000000000002e-71 < t < 7.39999999999999972e-96

   1. Initial program 36.1%

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

      1. *-commutative 36.1%

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

      2. associate-*l* 36.1%

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

      3. associate-*r* 36.1%

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

      4. +-commutative 36.1%

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

      5. associate-+r+ 36.1%

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

      6. metadata-eval 36.1%

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

   3. Simplified 36.1%

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

   4. Taylor expanded in k around inf 73.7%

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

      1. *-commutative 73.7%

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

      2. unpow2 73.7%

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

      3. times-frac 81.6%

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

      4. unpow2 81.6%

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

   6. Simplified 81.6%

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

      1. associate-*l/ 81.8%

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

      2. associate-/l* 92.1%

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

   8. Applied egg-rr 92.1%

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

   9. Taylor expanded in t around 0 73.7%

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

       1. associate-/l* 72.2%

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

       2. unpow2 72.2%

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

       3. unpow2 72.2%

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

       4. associate-*r/ 77.3%

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

       5. times-frac 95.4%

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

       6. associate-/l* 97.7%

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

       7. *-commutative 97.7%

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

       8. associate-/l* 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in k around 0 68.9%

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

       1. unpow2 68.9%

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

   14. Simplified 68.9%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.76 \cdot 10^{-71} \lor \neg \left(t \leq 7.4 \cdot 10^{-96}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(k \cdot k\right)}\\ \end{array} \]

Alternative 13: 58.5% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.5%

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

   1. *-commutative 58.5%

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

   2. associate-*l* 55.6%

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

   3. associate-*r* 55.6%

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

   4. +-commutative 55.6%

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

   5. associate-+r+ 55.6%

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

   6. metadata-eval 55.6%

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

3. Simplified 55.6%

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

4. Taylor expanded in k around inf 65.5%

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

   1. *-commutative 65.5%

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

   2. unpow2 65.5%

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

   3. times-frac 70.3%

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

   4. unpow2 70.3%

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

6. Simplified 70.3%

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

   1. associate-*l/ 70.1%

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

   2. associate-/l* 74.3%

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

8. Applied egg-rr 74.3%

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

9. Taylor expanded in t around 0 65.5%

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

    1. associate-/l* 66.2%

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

    2. unpow2 66.2%

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

    3. unpow2 66.2%

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

    4. associate-*r/ 68.7%

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

    5. times-frac 77.9%

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

    6. associate-/l* 78.3%

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

    7. *-commutative 78.3%

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

    8. associate-/l* 79.7%

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

11. Simplified 79.7%

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

12. Taylor expanded in k around 0 62.0%

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

    1. unpow2 62.0%

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

14. Simplified 62.0%

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

15. Final simplification 62.0%

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

Alternative 14: 24.0% accurate, 60.1× speedup

Math

\[\begin{array}{l} \\ -0.044444444444444446 \cdot \frac{\ell}{\frac{t}{\ell}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* -0.044444444444444446 (/ l (/ t l))))

C

double code(double t, double l, double k) {
	return -0.044444444444444446 * (l / (t / 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 = (-0.044444444444444446d0) * (l / (t / l))
end function

Java

public static double code(double t, double l, double k) {
	return -0.044444444444444446 * (l / (t / l));
}

Python

def code(t, l, k):
	return -0.044444444444444446 * (l / (t / l))

Julia

function code(t, l, k)
	return Float64(-0.044444444444444446 * Float64(l / Float64(t / l)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = -0.044444444444444446 * (l / (t / l));
end

Wolfram

code[t_, l_, k_] := N[(-0.044444444444444446 * N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.5%

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

   1. associate-/l/ 58.5%

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

   2. associate-*l/ 58.5%

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

   3. associate-*l/ 56.6%

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

   4. associate-/r/ 57.3%

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

   5. *-commutative 57.3%

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

   6. associate-/l/ 57.3%

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

   7. associate-*r* 57.3%

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

   8. *-commutative 57.3%

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

   9. associate-*r* 57.3%

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

   10. *-commutative 57.3%

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

3. Simplified 57.3%

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

4. Taylor expanded in k around inf 64.4%

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

   1. *-commutative 64.4%

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

   2. *-commutative 64.4%

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

   3. associate-*l* 64.0%

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

   4. *-commutative 64.0%

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

   5. unpow2 64.0%

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

   6. associate-*l* 64.0%

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

6. Simplified 64.0%

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

7. Taylor expanded in k around 0 56.6%

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

   1. unpow2 56.6%

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

9. Simplified 56.6%

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

10. Taylor expanded in k around 0 30.6%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(2 \cdot \frac{1}{{k}^{4} \cdot t} - \left(0.6666666666666666 \cdot \frac{1}{{k}^{2} \cdot t} + 0.044444444444444446 \cdot \frac{1}{t}\right)\right)} \]
11. Step-by-step derivation

    1. associate-*r/ 30.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot t}} - \left(0.6666666666666666 \cdot \frac{1}{{k}^{2} \cdot t} + 0.044444444444444446 \cdot \frac{1}{t}\right)\right) \]

    2. metadata-eval 30.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\color{blue}{2}}{{k}^{4} \cdot t} - \left(0.6666666666666666 \cdot \frac{1}{{k}^{2} \cdot t} + 0.044444444444444446 \cdot \frac{1}{t}\right)\right) \]

    3. *-commutative 30.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{\color{blue}{t \cdot {k}^{4}}} - \left(0.6666666666666666 \cdot \frac{1}{{k}^{2} \cdot t} + 0.044444444444444446 \cdot \frac{1}{t}\right)\right) \]

    4. associate-*r/ 30.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \left(\color{blue}{\frac{0.6666666666666666 \cdot 1}{{k}^{2} \cdot t}} + 0.044444444444444446 \cdot \frac{1}{t}\right)\right) \]

    5. metadata-eval 30.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \left(\frac{\color{blue}{0.6666666666666666}}{{k}^{2} \cdot t} + 0.044444444444444446 \cdot \frac{1}{t}\right)\right) \]

    6. *-commutative 30.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \left(\frac{0.6666666666666666}{\color{blue}{t \cdot {k}^{2}}} + 0.044444444444444446 \cdot \frac{1}{t}\right)\right) \]

    7. associate-/r* 30.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \left(\color{blue}{\frac{\frac{0.6666666666666666}{t}}{{k}^{2}}} + 0.044444444444444446 \cdot \frac{1}{t}\right)\right) \]

    8. unpow2 30.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \left(\frac{\frac{0.6666666666666666}{t}}{\color{blue}{k \cdot k}} + 0.044444444444444446 \cdot \frac{1}{t}\right)\right) \]

    9. associate-*r/ 30.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \left(\frac{\frac{0.6666666666666666}{t}}{k \cdot k} + \color{blue}{\frac{0.044444444444444446 \cdot 1}{t}}\right)\right) \]

    10. metadata-eval 30.6%

        \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \left(\frac{\frac{0.6666666666666666}{t}}{k \cdot k} + \frac{\color{blue}{0.044444444444444446}}{t}\right)\right) \]

12. Simplified 30.6%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{2}{t \cdot {k}^{4}} - \left(\frac{\frac{0.6666666666666666}{t}}{k \cdot k} + \frac{0.044444444444444446}{t}\right)\right)} \]

13. Taylor expanded in k around inf 31.6%

    \[\leadsto \color{blue}{-0.044444444444444446 \cdot \frac{{\ell}^{2}}{t}} \]
14. Step-by-step derivation

    1. unpow2 31.6%

       \[\leadsto -0.044444444444444446 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t} \]

    2. associate-/l* 28.6%

       \[\leadsto -0.044444444444444446 \cdot \color{blue}{\frac{\ell}{\frac{t}{\ell}}} \]

15. Simplified 28.6%

    \[\leadsto \color{blue}{-0.044444444444444446 \cdot \frac{\ell}{\frac{t}{\ell}}} \]

16. Final simplification 28.6%

    \[\leadsto -0.044444444444444446 \cdot \frac{\ell}{\frac{t}{\ell}} \]

Alternative 15: 26.0% accurate, 60.1× speedup

Math

\[\begin{array}{l} \\ \left(\ell \cdot \ell\right) \cdot \frac{-0.044444444444444446}{t} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (* l l) (/ -0.044444444444444446 t)))

C

double code(double t, double l, double k) {
	return (l * l) * (-0.044444444444444446 / t);
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l * l) * ((-0.044444444444444446d0) / t)
end function

Java

public static double code(double t, double l, double k) {
	return (l * l) * (-0.044444444444444446 / t);
}

Python

def code(t, l, k):
	return (l * l) * (-0.044444444444444446 / t)

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(-0.044444444444444446 / t))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * l) * (-0.044444444444444446 / t);
end

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(-0.044444444444444446 / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.5%

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

   1. associate-/l/ 58.5%

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

   2. associate-*l/ 58.5%

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

   3. associate-*l/ 56.6%

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

   4. associate-/r/ 57.3%

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

   5. *-commutative 57.3%

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

   6. associate-/l/ 57.3%

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

   7. associate-*r* 57.3%

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

   8. *-commutative 57.3%

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

   9. associate-*r* 57.3%

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

   10. *-commutative 57.3%

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

3. Simplified 57.3%

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

4. Taylor expanded in k around inf 64.4%

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

   1. *-commutative 64.4%

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

   2. *-commutative 64.4%

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

   3. associate-*l* 64.0%

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

   4. *-commutative 64.0%

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

   5. unpow2 64.0%

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

   6. associate-*l* 64.0%

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

6. Simplified 64.0%

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

7. Taylor expanded in k around 0 56.6%

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

   1. unpow2 56.6%

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

9. Simplified 56.6%

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

10. Taylor expanded in k around 0 30.6%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(2 \cdot \frac{1}{{k}^{4} \cdot t} - \left(0.6666666666666666 \cdot \frac{1}{{k}^{2} \cdot t} + 0.044444444444444446 \cdot \frac{1}{t}\right)\right)} \]
11. Step-by-step derivation

    1. associate-*r/ 30.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot t}} - \left(0.6666666666666666 \cdot \frac{1}{{k}^{2} \cdot t} + 0.044444444444444446 \cdot \frac{1}{t}\right)\right) \]

    2. metadata-eval 30.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\color{blue}{2}}{{k}^{4} \cdot t} - \left(0.6666666666666666 \cdot \frac{1}{{k}^{2} \cdot t} + 0.044444444444444446 \cdot \frac{1}{t}\right)\right) \]

    3. *-commutative 30.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{\color{blue}{t \cdot {k}^{4}}} - \left(0.6666666666666666 \cdot \frac{1}{{k}^{2} \cdot t} + 0.044444444444444446 \cdot \frac{1}{t}\right)\right) \]

    4. associate-*r/ 30.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \left(\color{blue}{\frac{0.6666666666666666 \cdot 1}{{k}^{2} \cdot t}} + 0.044444444444444446 \cdot \frac{1}{t}\right)\right) \]

    5. metadata-eval 30.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \left(\frac{\color{blue}{0.6666666666666666}}{{k}^{2} \cdot t} + 0.044444444444444446 \cdot \frac{1}{t}\right)\right) \]

    6. *-commutative 30.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \left(\frac{0.6666666666666666}{\color{blue}{t \cdot {k}^{2}}} + 0.044444444444444446 \cdot \frac{1}{t}\right)\right) \]

    7. associate-/r* 30.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \left(\color{blue}{\frac{\frac{0.6666666666666666}{t}}{{k}^{2}}} + 0.044444444444444446 \cdot \frac{1}{t}\right)\right) \]

    8. unpow2 30.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \left(\frac{\frac{0.6666666666666666}{t}}{\color{blue}{k \cdot k}} + 0.044444444444444446 \cdot \frac{1}{t}\right)\right) \]

    9. associate-*r/ 30.6%

       \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \left(\frac{\frac{0.6666666666666666}{t}}{k \cdot k} + \color{blue}{\frac{0.044444444444444446 \cdot 1}{t}}\right)\right) \]

    10. metadata-eval 30.6%

        \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{t \cdot {k}^{4}} - \left(\frac{\frac{0.6666666666666666}{t}}{k \cdot k} + \frac{\color{blue}{0.044444444444444446}}{t}\right)\right) \]

12. Simplified 30.6%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{2}{t \cdot {k}^{4}} - \left(\frac{\frac{0.6666666666666666}{t}}{k \cdot k} + \frac{0.044444444444444446}{t}\right)\right)} \]

13. Taylor expanded in k around inf 31.6%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{-0.044444444444444446}{t}} \]

14. Final simplification 31.6%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.044444444444444446}{t} \]

Reproduce

herbie shell --seed 2023171 
(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))))