Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.5% → 85.2%

Time: 18.2s

Alternatives: 25

Speedup: 3.8×

• 39.2% 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: 34.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\sin k \cdot \tan k}\\ t_2 := \frac{\sqrt{2}}{\frac{k}{t}}\\ t_3 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_4 := t\_1 \cdot t\_3\\ \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\left(t \cdot \left(\frac{\sqrt{2}}{k} \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot t\_1\right)\right)}^{-2}\right)\right) \cdot \frac{t\_2}{t\_4}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+243}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(-{\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{{t\_4}^{2}} \cdot \frac{t\_2}{t\_3 \cdot {\left(\sqrt[3]{t\_1}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cbrt (* (sin k) (tan k))))
        (t_2 (/ (sqrt 2.0) (/ k t)))
        (t_3 (/ t (pow (cbrt l) 2.0)))
        (t_4 (* t_1 t_3)))
   (if (<= (* l l) 0.0)
     (*
      (* t (* (/ (sqrt 2.0) k) (pow (* t (* (pow (cbrt l) -2.0) t_1)) -2.0)))
      (/ t_2 t_4))
     (if (<= (* l l) 5e+243)
       (*
        -2.0
        (/
         (/ (* (pow l 2.0) (cos k)) (pow k 2.0))
         (* t (- (pow (sin k) 2.0)))))
       (* (/ t_2 (pow t_4 2.0)) (/ t_2 (* t_3 (pow (cbrt t_1) 3.0))))))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt((sin(k) * tan(k)));
	double t_2 = sqrt(2.0) / (k / t);
	double t_3 = t / pow(cbrt(l), 2.0);
	double t_4 = t_1 * t_3;
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (t * ((sqrt(2.0) / k) * pow((t * (pow(cbrt(l), -2.0) * t_1)), -2.0))) * (t_2 / t_4);
	} else if ((l * l) <= 5e+243) {
		tmp = -2.0 * (((pow(l, 2.0) * cos(k)) / pow(k, 2.0)) / (t * -pow(sin(k), 2.0)));
	} else {
		tmp = (t_2 / pow(t_4, 2.0)) * (t_2 / (t_3 * pow(cbrt(t_1), 3.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_2 = Math.sqrt(2.0) / (k / t);
	double t_3 = t / Math.pow(Math.cbrt(l), 2.0);
	double t_4 = t_1 * t_3;
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (t * ((Math.sqrt(2.0) / k) * Math.pow((t * (Math.pow(Math.cbrt(l), -2.0) * t_1)), -2.0))) * (t_2 / t_4);
	} else if ((l * l) <= 5e+243) {
		tmp = -2.0 * (((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow(k, 2.0)) / (t * -Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = (t_2 / Math.pow(t_4, 2.0)) * (t_2 / (t_3 * Math.pow(Math.cbrt(t_1), 3.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(Float64(sin(k) * tan(k)))
	t_2 = Float64(sqrt(2.0) / Float64(k / t))
	t_3 = Float64(t / (cbrt(l) ^ 2.0))
	t_4 = Float64(t_1 * t_3)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(Float64(t * Float64(Float64(sqrt(2.0) / k) * (Float64(t * Float64((cbrt(l) ^ -2.0) * t_1)) ^ -2.0))) * Float64(t_2 / t_4));
	elseif (Float64(l * l) <= 5e+243)
		tmp = Float64(-2.0 * Float64(Float64(Float64((l ^ 2.0) * cos(k)) / (k ^ 2.0)) / Float64(t * Float64(-(sin(k) ^ 2.0)))));
	else
		tmp = Float64(Float64(t_2 / (t_4 ^ 2.0)) * Float64(t_2 / Float64(t_3 * (cbrt(t_1) ^ 3.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / N[(k / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * t$95$3), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(N[(t * N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * N[Power[N[(t * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 5e+243], N[(-2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * (-N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 / N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[(t$95$3 * N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 20.2%

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

      1. *-commutative 20.2%

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

      2. associate-/r* 20.2%

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

   3. Simplified 42.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 42.0%

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

      2. add-cube-cbrt 42.0%

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

      3. times-frac 42.0%

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

   6. Applied egg-rr 86.6%

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

   7. Taylor expanded in k around 0 86.7%

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

      1. div-inv 86.7%

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

      2. associate-/l* 86.7%

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

      3. pow-flip 86.6%

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

      4. div-inv 86.6%

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

      5. pow-flip 86.6%

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

      6. metadata-eval 86.6%

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

      7. metadata-eval 86.6%

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

   9. Applied egg-rr 86.6%

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

       1. associate-*l* 86.7%

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

       2. associate-*l* 86.7%

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

   11. Simplified 86.7%

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

   if 0.0 < (*.f64 l l) < 5.00000000000000037e243

   1. Initial program 40.8%

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

      1. *-commutative 40.8%

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

      2. associate-/r* 41.1%

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

   3. Simplified 53.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 53.8%

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

      2. unpow2 53.8%

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

      3. clear-num 53.8%

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

      4. un-div-inv 53.8%

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

   6. Applied egg-rr 53.8%

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

      1. add-sqr-sqrt 30.6%

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

      2. pow2 30.6%

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

      3. sqrt-div 30.6%

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

      4. sqrt-pow1 33.9%

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

      5. metadata-eval 33.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 19.0%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 33.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 33.9%

      \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around -inf 0.0%

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

       1. associate-/r* 0.0%

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

       2. *-commutative 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 92.1%

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

       5. associate-*l* 92.1%

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

       6. neg-mul-1 92.1%

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

   11. Simplified 92.1%

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

   if 5.00000000000000037e243 < (*.f64 l l)

   1. Initial program 38.0%

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

      1. *-commutative 38.0%

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

      2. associate-/r* 38.1%

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

   3. Simplified 38.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 38.1%

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

      2. add-cube-cbrt 38.1%

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

      3. times-frac 38.1%

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

   6. Applied egg-rr 81.8%

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

      1. add-cube-cbrt 81.9%

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

      2. pow3 81.9%

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

   8. Applied egg-rr 81.9%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\left(t \cdot \left(\frac{\sqrt{2}}{k} \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right) \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+243}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(-{\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin k \cdot \tan k}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\sin k \cdot \tan k}\\ \mathbf{if}\;\ell \cdot \ell \leq 0 \lor \neg \left(\ell \cdot \ell \leq 5 \cdot 10^{+243}\right):\\ \;\;\;\;\left(t \cdot \left(\frac{\sqrt{2}}{k} \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot t\_1\right)\right)}^{-2}\right)\right) \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{t\_1 \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(-{\sin k}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cbrt (* (sin k) (tan k)))))
   (if (or (<= (* l l) 0.0) (not (<= (* l l) 5e+243)))
     (*
      (* t (* (/ (sqrt 2.0) k) (pow (* t (* (pow (cbrt l) -2.0) t_1)) -2.0)))
      (/ (/ (sqrt 2.0) (/ k t)) (* t_1 (/ t (pow (cbrt l) 2.0)))))
     (*
      -2.0
      (/
       (/ (* (pow l 2.0) (cos k)) (pow k 2.0))
       (* t (- (pow (sin k) 2.0))))))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt((sin(k) * tan(k)));
	double tmp;
	if (((l * l) <= 0.0) || !((l * l) <= 5e+243)) {
		tmp = (t * ((sqrt(2.0) / k) * pow((t * (pow(cbrt(l), -2.0) * t_1)), -2.0))) * ((sqrt(2.0) / (k / t)) / (t_1 * (t / pow(cbrt(l), 2.0))));
	} else {
		tmp = -2.0 * (((pow(l, 2.0) * cos(k)) / pow(k, 2.0)) / (t * -pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double tmp;
	if (((l * l) <= 0.0) || !((l * l) <= 5e+243)) {
		tmp = (t * ((Math.sqrt(2.0) / k) * Math.pow((t * (Math.pow(Math.cbrt(l), -2.0) * t_1)), -2.0))) * ((Math.sqrt(2.0) / (k / t)) / (t_1 * (t / Math.pow(Math.cbrt(l), 2.0))));
	} else {
		tmp = -2.0 * (((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow(k, 2.0)) / (t * -Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(Float64(sin(k) * tan(k)))
	tmp = 0.0
	if ((Float64(l * l) <= 0.0) || !(Float64(l * l) <= 5e+243))
		tmp = Float64(Float64(t * Float64(Float64(sqrt(2.0) / k) * (Float64(t * Float64((cbrt(l) ^ -2.0) * t_1)) ^ -2.0))) * Float64(Float64(sqrt(2.0) / Float64(k / t)) / Float64(t_1 * Float64(t / (cbrt(l) ^ 2.0)))));
	else
		tmp = Float64(-2.0 * Float64(Float64(Float64((l ^ 2.0) * cos(k)) / (k ^ 2.0)) / Float64(t * Float64(-(sin(k) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[Or[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(l * l), $MachinePrecision], 5e+243]], $MachinePrecision]], N[(N[(t * N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * N[Power[N[(t * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(k / t), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * (-N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0 or 5.00000000000000037e243 < (*.f64 l l)

   1. Initial program 30.1%

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

      1. *-commutative 30.1%

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

      2. associate-/r* 30.1%

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

   3. Simplified 39.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 39.8%

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

      2. add-cube-cbrt 39.8%

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

      3. times-frac 39.8%

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

   6. Applied egg-rr 84.0%

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

   7. Taylor expanded in k around 0 84.0%

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

      1. div-inv 84.0%

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

      2. associate-/l* 84.0%

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

      3. pow-flip 84.0%

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

      4. div-inv 84.0%

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

      5. pow-flip 84.0%

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

      6. metadata-eval 84.0%

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

      7. metadata-eval 84.0%

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

   9. Applied egg-rr 84.0%

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

       1. associate-*l* 82.7%

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

       2. associate-*l* 82.7%

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

   11. Simplified 82.7%

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

   if 0.0 < (*.f64 l l) < 5.00000000000000037e243

   1. Initial program 40.8%

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

      1. *-commutative 40.8%

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

      2. associate-/r* 41.1%

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

   3. Simplified 53.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 53.8%

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

      2. unpow2 53.8%

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

      3. clear-num 53.8%

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

      4. un-div-inv 53.8%

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

   6. Applied egg-rr 53.8%

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

      1. add-sqr-sqrt 30.6%

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

      2. pow2 30.6%

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

      3. sqrt-div 30.6%

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

      4. sqrt-pow1 33.9%

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

      5. metadata-eval 33.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 19.0%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 33.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 33.9%

      \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around -inf 0.0%

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

       1. associate-/r* 0.0%

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

       2. *-commutative 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 92.1%

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

       5. associate-*l* 92.1%

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

       6. neg-mul-1 92.1%

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

   11. Simplified 92.1%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0 \lor \neg \left(\ell \cdot \ell \leq 5 \cdot 10^{+243}\right):\\ \;\;\;\;\left(t \cdot \left(\frac{\sqrt{2}}{k} \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right) \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(-{\sin k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\sin k \cdot \tan k}\\ t_2 := \frac{\sqrt{2}}{\frac{k}{t}}\\ t_3 := t\_1 \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_4 := \frac{t\_2}{t\_3}\\ \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\left(t \cdot \left(\frac{\sqrt{2}}{k} \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot t\_1\right)\right)}^{-2}\right)\right) \cdot t\_4\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+243}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(-{\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_4 \cdot \frac{t\_2}{{t\_3}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cbrt (* (sin k) (tan k))))
        (t_2 (/ (sqrt 2.0) (/ k t)))
        (t_3 (* t_1 (/ t (pow (cbrt l) 2.0))))
        (t_4 (/ t_2 t_3)))
   (if (<= (* l l) 0.0)
     (*
      (* t (* (/ (sqrt 2.0) k) (pow (* t (* (pow (cbrt l) -2.0) t_1)) -2.0)))
      t_4)
     (if (<= (* l l) 5e+243)
       (*
        -2.0
        (/
         (/ (* (pow l 2.0) (cos k)) (pow k 2.0))
         (* t (- (pow (sin k) 2.0)))))
       (* t_4 (/ t_2 (pow t_3 2.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt((sin(k) * tan(k)));
	double t_2 = sqrt(2.0) / (k / t);
	double t_3 = t_1 * (t / pow(cbrt(l), 2.0));
	double t_4 = t_2 / t_3;
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (t * ((sqrt(2.0) / k) * pow((t * (pow(cbrt(l), -2.0) * t_1)), -2.0))) * t_4;
	} else if ((l * l) <= 5e+243) {
		tmp = -2.0 * (((pow(l, 2.0) * cos(k)) / pow(k, 2.0)) / (t * -pow(sin(k), 2.0)));
	} else {
		tmp = t_4 * (t_2 / pow(t_3, 2.0));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_2 = Math.sqrt(2.0) / (k / t);
	double t_3 = t_1 * (t / Math.pow(Math.cbrt(l), 2.0));
	double t_4 = t_2 / t_3;
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (t * ((Math.sqrt(2.0) / k) * Math.pow((t * (Math.pow(Math.cbrt(l), -2.0) * t_1)), -2.0))) * t_4;
	} else if ((l * l) <= 5e+243) {
		tmp = -2.0 * (((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow(k, 2.0)) / (t * -Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = t_4 * (t_2 / Math.pow(t_3, 2.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(Float64(sin(k) * tan(k)))
	t_2 = Float64(sqrt(2.0) / Float64(k / t))
	t_3 = Float64(t_1 * Float64(t / (cbrt(l) ^ 2.0)))
	t_4 = Float64(t_2 / t_3)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(Float64(t * Float64(Float64(sqrt(2.0) / k) * (Float64(t * Float64((cbrt(l) ^ -2.0) * t_1)) ^ -2.0))) * t_4);
	elseif (Float64(l * l) <= 5e+243)
		tmp = Float64(-2.0 * Float64(Float64(Float64((l ^ 2.0) * cos(k)) / (k ^ 2.0)) / Float64(t * Float64(-(sin(k) ^ 2.0)))));
	else
		tmp = Float64(t_4 * Float64(t_2 / (t_3 ^ 2.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / N[(k / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 / t$95$3), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(N[(t * N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * N[Power[N[(t * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 5e+243], N[(-2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * (-N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 * N[(t$95$2 / N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 20.2%

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

      1. *-commutative 20.2%

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

      2. associate-/r* 20.2%

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

   3. Simplified 42.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 42.0%

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

      2. add-cube-cbrt 42.0%

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

      3. times-frac 42.0%

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

   6. Applied egg-rr 86.6%

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

   7. Taylor expanded in k around 0 86.7%

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

      1. div-inv 86.7%

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

      2. associate-/l* 86.7%

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

      3. pow-flip 86.6%

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

      4. div-inv 86.6%

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

      5. pow-flip 86.6%

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

      6. metadata-eval 86.6%

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

      7. metadata-eval 86.6%

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

   9. Applied egg-rr 86.6%

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

       1. associate-*l* 86.7%

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

       2. associate-*l* 86.7%

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

   11. Simplified 86.7%

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

   if 0.0 < (*.f64 l l) < 5.00000000000000037e243

   1. Initial program 40.8%

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

      1. *-commutative 40.8%

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

      2. associate-/r* 41.1%

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

   3. Simplified 53.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 53.8%

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

      2. unpow2 53.8%

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

      3. clear-num 53.8%

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

      4. un-div-inv 53.8%

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

   6. Applied egg-rr 53.8%

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

      1. add-sqr-sqrt 30.6%

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

      2. pow2 30.6%

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

      3. sqrt-div 30.6%

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

      4. sqrt-pow1 33.9%

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

      5. metadata-eval 33.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 19.0%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 33.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 33.9%

      \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around -inf 0.0%

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

       1. associate-/r* 0.0%

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

       2. *-commutative 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 92.1%

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

       5. associate-*l* 92.1%

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

       6. neg-mul-1 92.1%

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

   11. Simplified 92.1%

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

   if 5.00000000000000037e243 < (*.f64 l l)

   1. Initial program 38.0%

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

      1. *-commutative 38.0%

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

      2. associate-/r* 38.1%

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

   3. Simplified 38.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 38.1%

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

      2. add-cube-cbrt 38.1%

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

      3. times-frac 38.1%

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

   6. Applied egg-rr 81.8%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\left(t \cdot \left(\frac{\sqrt{2}}{k} \cdot {\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\right) \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+243}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(-{\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_2 := \sqrt[3]{\sin k \cdot \tan k} \cdot t\_1\\ t_3 := t \cdot \frac{\sqrt{2}}{k}\\ t_4 := \sqrt{2} \cdot \frac{t}{k}\\ \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\left(t\_3 \cdot {t\_2}^{-2}\right) \cdot t\_4}{t\_1 \cdot \sqrt[3]{{k}^{2}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 4 \cdot 10^{+283}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(-{\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{t\_3 \cdot t\_4}}{t\_2}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* t (pow (cbrt l) -2.0)))
        (t_2 (* (cbrt (* (sin k) (tan k))) t_1))
        (t_3 (* t (/ (sqrt 2.0) k)))
        (t_4 (* (sqrt 2.0) (/ t k))))
   (if (<= (* l l) 0.0)
     (/ (* (* t_3 (pow t_2 -2.0)) t_4) (* t_1 (cbrt (pow k 2.0))))
     (if (<= (* l l) 4e+283)
       (*
        -2.0
        (/
         (/ (* (pow l 2.0) (cos k)) (pow k 2.0))
         (* t (- (pow (sin k) 2.0)))))
       (pow (/ (cbrt (* t_3 t_4)) t_2) 3.0)))))

C

double code(double t, double l, double k) {
	double t_1 = t * pow(cbrt(l), -2.0);
	double t_2 = cbrt((sin(k) * tan(k))) * t_1;
	double t_3 = t * (sqrt(2.0) / k);
	double t_4 = sqrt(2.0) * (t / k);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = ((t_3 * pow(t_2, -2.0)) * t_4) / (t_1 * cbrt(pow(k, 2.0)));
	} else if ((l * l) <= 4e+283) {
		tmp = -2.0 * (((pow(l, 2.0) * cos(k)) / pow(k, 2.0)) / (t * -pow(sin(k), 2.0)));
	} else {
		tmp = pow((cbrt((t_3 * t_4)) / t_2), 3.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = t * Math.pow(Math.cbrt(l), -2.0);
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k))) * t_1;
	double t_3 = t * (Math.sqrt(2.0) / k);
	double t_4 = Math.sqrt(2.0) * (t / k);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = ((t_3 * Math.pow(t_2, -2.0)) * t_4) / (t_1 * Math.cbrt(Math.pow(k, 2.0)));
	} else if ((l * l) <= 4e+283) {
		tmp = -2.0 * (((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow(k, 2.0)) / (t * -Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = Math.pow((Math.cbrt((t_3 * t_4)) / t_2), 3.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(t * (cbrt(l) ^ -2.0))
	t_2 = Float64(cbrt(Float64(sin(k) * tan(k))) * t_1)
	t_3 = Float64(t * Float64(sqrt(2.0) / k))
	t_4 = Float64(sqrt(2.0) * Float64(t / k))
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(Float64(Float64(t_3 * (t_2 ^ -2.0)) * t_4) / Float64(t_1 * cbrt((k ^ 2.0))));
	elseif (Float64(l * l) <= 4e+283)
		tmp = Float64(-2.0 * Float64(Float64(Float64((l ^ 2.0) * cos(k)) / (k ^ 2.0)) / Float64(t * Float64(-(sin(k) ^ 2.0)))));
	else
		tmp = Float64(cbrt(Float64(t_3 * t_4)) / t_2) ^ 3.0;
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(N[(N[(t$95$3 * N[Power[t$95$2, -2.0], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] / N[(t$95$1 * N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 4e+283], N[(-2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * (-N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[(t$95$3 * t$95$4), $MachinePrecision], 1/3], $MachinePrecision] / t$95$2), $MachinePrecision], 3.0], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 20.2%

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

      1. *-commutative 20.2%

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

      2. associate-/r* 20.2%

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

   3. Simplified 42.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 42.0%

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

      2. add-cube-cbrt 42.0%

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

      3. times-frac 42.0%

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

   6. Applied egg-rr 86.6%

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

   7. Taylor expanded in k around 0 86.7%

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

      1. associate-*r/ 86.7%

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

   9. Applied egg-rr 86.7%

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

   10. Taylor expanded in k around 0 86.7%

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

   if 0.0 < (*.f64 l l) < 3.99999999999999982e283

   1. Initial program 40.8%

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

      1. *-commutative 40.8%

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

      2. associate-/r* 41.1%

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

   3. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 52.6%

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

      2. unpow2 52.6%

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

      3. clear-num 52.7%

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

      4. un-div-inv 52.6%

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

   6. Applied egg-rr 52.6%

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

      1. add-sqr-sqrt 29.4%

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

      2. pow2 29.4%

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

      3. sqrt-div 29.4%

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

      4. sqrt-pow1 32.4%

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

      5. metadata-eval 32.4%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 18.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 32.4%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 32.4%

      \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around -inf 0.0%

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

       1. associate-/r* 0.0%

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

       2. *-commutative 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 89.4%

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

       5. associate-*l* 89.4%

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

       6. neg-mul-1 89.4%

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

   11. Simplified 89.4%

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

   if 3.99999999999999982e283 < (*.f64 l l)

   1. Initial program 37.3%

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

      1. *-commutative 37.3%

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

      2. associate-/r* 37.3%

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

   3. Simplified 37.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 37.3%

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

      2. add-cube-cbrt 37.3%

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

      3. times-frac 37.3%

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

   6. Applied egg-rr 83.1%

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

   7. Taylor expanded in k around 0 83.1%

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

      1. add-cube-cbrt 83.1%

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

      2. pow3 83.1%

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

   9. Applied egg-rr 72.3%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\left(\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot {\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{-2}\right) \cdot \left(\sqrt{2} \cdot \frac{t}{k}\right)}{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{{k}^{2}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 4 \cdot 10^{+283}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(-{\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left(\sqrt{2} \cdot \frac{t}{k}\right)}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\sin k \cdot \tan k}\\ t_2 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{t \cdot \sqrt{2}}{k}}{{\left(t\_1 \cdot t\_2\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{t\_2 \cdot \sqrt[3]{{k}^{2}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 4 \cdot 10^{+283}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(-{\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left(\sqrt{2} \cdot \frac{t}{k}\right)}}{t\_1 \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cbrt (* (sin k) (tan k)))) (t_2 (/ t (pow (cbrt l) 2.0))))
   (if (<= (* l l) 0.0)
     (*
      (/ (/ (* t (sqrt 2.0)) k) (pow (* t_1 t_2) 2.0))
      (/ (/ (sqrt 2.0) (/ k t)) (* t_2 (cbrt (pow k 2.0)))))
     (if (<= (* l l) 4e+283)
       (*
        -2.0
        (/
         (/ (* (pow l 2.0) (cos k)) (pow k 2.0))
         (* t (- (pow (sin k) 2.0)))))
       (pow
        (/
         (cbrt (* (* t (/ (sqrt 2.0) k)) (* (sqrt 2.0) (/ t k))))
         (* t_1 (* t (pow (cbrt l) -2.0))))
        3.0)))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt((sin(k) * tan(k)));
	double t_2 = t / pow(cbrt(l), 2.0);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (((t * sqrt(2.0)) / k) / pow((t_1 * t_2), 2.0)) * ((sqrt(2.0) / (k / t)) / (t_2 * cbrt(pow(k, 2.0))));
	} else if ((l * l) <= 4e+283) {
		tmp = -2.0 * (((pow(l, 2.0) * cos(k)) / pow(k, 2.0)) / (t * -pow(sin(k), 2.0)));
	} else {
		tmp = pow((cbrt(((t * (sqrt(2.0) / k)) * (sqrt(2.0) * (t / k)))) / (t_1 * (t * pow(cbrt(l), -2.0)))), 3.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_2 = t / Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (((t * Math.sqrt(2.0)) / k) / Math.pow((t_1 * t_2), 2.0)) * ((Math.sqrt(2.0) / (k / t)) / (t_2 * Math.cbrt(Math.pow(k, 2.0))));
	} else if ((l * l) <= 4e+283) {
		tmp = -2.0 * (((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow(k, 2.0)) / (t * -Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = Math.pow((Math.cbrt(((t * (Math.sqrt(2.0) / k)) * (Math.sqrt(2.0) * (t / k)))) / (t_1 * (t * Math.pow(Math.cbrt(l), -2.0)))), 3.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(Float64(sin(k) * tan(k)))
	t_2 = Float64(t / (cbrt(l) ^ 2.0))
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(t * sqrt(2.0)) / k) / (Float64(t_1 * t_2) ^ 2.0)) * Float64(Float64(sqrt(2.0) / Float64(k / t)) / Float64(t_2 * cbrt((k ^ 2.0)))));
	elseif (Float64(l * l) <= 4e+283)
		tmp = Float64(-2.0 * Float64(Float64(Float64((l ^ 2.0) * cos(k)) / (k ^ 2.0)) / Float64(t * Float64(-(sin(k) ^ 2.0)))));
	else
		tmp = Float64(cbrt(Float64(Float64(t * Float64(sqrt(2.0) / k)) * Float64(sqrt(2.0) * Float64(t / k)))) / Float64(t_1 * Float64(t * (cbrt(l) ^ -2.0)))) ^ 3.0;
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(N[(N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[Power[N[(t$95$1 * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(k / t), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 4e+283], N[(-2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * (-N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[(N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t$95$1 * N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 20.2%

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

      1. *-commutative 20.2%

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

      2. associate-/r* 20.2%

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

   3. Simplified 42.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 42.0%

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

      2. add-cube-cbrt 42.0%

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

      3. times-frac 42.0%

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

   6. Applied egg-rr 86.6%

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

   7. Taylor expanded in k around 0 86.7%

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

   8. Taylor expanded in k around 0 86.7%

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

   if 0.0 < (*.f64 l l) < 3.99999999999999982e283

   1. Initial program 40.8%

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

      1. *-commutative 40.8%

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

      2. associate-/r* 41.1%

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

   3. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 52.6%

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

      2. unpow2 52.6%

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

      3. clear-num 52.7%

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

      4. un-div-inv 52.6%

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

   6. Applied egg-rr 52.6%

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

      1. add-sqr-sqrt 29.4%

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

      2. pow2 29.4%

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

      3. sqrt-div 29.4%

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

      4. sqrt-pow1 32.4%

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

      5. metadata-eval 32.4%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 18.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 32.4%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 32.4%

      \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around -inf 0.0%

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

       1. associate-/r* 0.0%

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

       2. *-commutative 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 89.4%

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

       5. associate-*l* 89.4%

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

       6. neg-mul-1 89.4%

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

   11. Simplified 89.4%

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

   if 3.99999999999999982e283 < (*.f64 l l)

   1. Initial program 37.3%

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

      1. *-commutative 37.3%

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

      2. associate-/r* 37.3%

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

   3. Simplified 37.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 37.3%

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

      2. add-cube-cbrt 37.3%

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

      3. times-frac 37.3%

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

   6. Applied egg-rr 83.1%

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

   7. Taylor expanded in k around 0 83.1%

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

      1. add-cube-cbrt 83.1%

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

      2. pow3 83.1%

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

   9. Applied egg-rr 72.3%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{t \cdot \sqrt{2}}{k}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{{k}^{2}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 4 \cdot 10^{+283}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(-{\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left(\sqrt{2} \cdot \frac{t}{k}\right)}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt[3]{\sin k \cdot \tan k}\\ t_2 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_3 := \frac{\sqrt{2}}{\frac{k}{t}}\\ \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{t\_3}{{\left(t\_1 \cdot t\_2\right)}^{2}} \cdot \frac{t\_3}{t\_2 \cdot \sqrt[3]{{k}^{2}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 4 \cdot 10^{+283}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(-{\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left(\sqrt{2} \cdot \frac{t}{k}\right)}}{t\_1 \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cbrt (* (sin k) (tan k))))
        (t_2 (/ t (pow (cbrt l) 2.0)))
        (t_3 (/ (sqrt 2.0) (/ k t))))
   (if (<= (* l l) 0.0)
     (* (/ t_3 (pow (* t_1 t_2) 2.0)) (/ t_3 (* t_2 (cbrt (pow k 2.0)))))
     (if (<= (* l l) 4e+283)
       (*
        -2.0
        (/
         (/ (* (pow l 2.0) (cos k)) (pow k 2.0))
         (* t (- (pow (sin k) 2.0)))))
       (pow
        (/
         (cbrt (* (* t (/ (sqrt 2.0) k)) (* (sqrt 2.0) (/ t k))))
         (* t_1 (* t (pow (cbrt l) -2.0))))
        3.0)))))

C

double code(double t, double l, double k) {
	double t_1 = cbrt((sin(k) * tan(k)));
	double t_2 = t / pow(cbrt(l), 2.0);
	double t_3 = sqrt(2.0) / (k / t);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (t_3 / pow((t_1 * t_2), 2.0)) * (t_3 / (t_2 * cbrt(pow(k, 2.0))));
	} else if ((l * l) <= 4e+283) {
		tmp = -2.0 * (((pow(l, 2.0) * cos(k)) / pow(k, 2.0)) / (t * -pow(sin(k), 2.0)));
	} else {
		tmp = pow((cbrt(((t * (sqrt(2.0) / k)) * (sqrt(2.0) * (t / k)))) / (t_1 * (t * pow(cbrt(l), -2.0)))), 3.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_2 = t / Math.pow(Math.cbrt(l), 2.0);
	double t_3 = Math.sqrt(2.0) / (k / t);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = (t_3 / Math.pow((t_1 * t_2), 2.0)) * (t_3 / (t_2 * Math.cbrt(Math.pow(k, 2.0))));
	} else if ((l * l) <= 4e+283) {
		tmp = -2.0 * (((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow(k, 2.0)) / (t * -Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = Math.pow((Math.cbrt(((t * (Math.sqrt(2.0) / k)) * (Math.sqrt(2.0) * (t / k)))) / (t_1 * (t * Math.pow(Math.cbrt(l), -2.0)))), 3.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(Float64(sin(k) * tan(k)))
	t_2 = Float64(t / (cbrt(l) ^ 2.0))
	t_3 = Float64(sqrt(2.0) / Float64(k / t))
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(Float64(t_3 / (Float64(t_1 * t_2) ^ 2.0)) * Float64(t_3 / Float64(t_2 * cbrt((k ^ 2.0)))));
	elseif (Float64(l * l) <= 4e+283)
		tmp = Float64(-2.0 * Float64(Float64(Float64((l ^ 2.0) * cos(k)) / (k ^ 2.0)) / Float64(t * Float64(-(sin(k) ^ 2.0)))));
	else
		tmp = Float64(cbrt(Float64(Float64(t * Float64(sqrt(2.0) / k)) * Float64(sqrt(2.0) * Float64(t / k)))) / Float64(t_1 * Float64(t * (cbrt(l) ^ -2.0)))) ^ 3.0;
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / N[(k / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(N[(t$95$3 / N[Power[N[(t$95$1 * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$3 / N[(t$95$2 * N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 4e+283], N[(-2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * (-N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[(N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t$95$1 * N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 20.2%

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

      1. *-commutative 20.2%

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

      2. associate-/r* 20.2%

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

   3. Simplified 42.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 42.0%

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

      2. add-cube-cbrt 42.0%

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

      3. times-frac 42.0%

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

   6. Applied egg-rr 86.6%

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

   7. Taylor expanded in k around 0 86.6%

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

   if 0.0 < (*.f64 l l) < 3.99999999999999982e283

   1. Initial program 40.8%

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

      1. *-commutative 40.8%

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

      2. associate-/r* 41.1%

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

   3. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 52.6%

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

      2. unpow2 52.6%

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

      3. clear-num 52.7%

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

      4. un-div-inv 52.6%

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

   6. Applied egg-rr 52.6%

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

      1. add-sqr-sqrt 29.4%

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

      2. pow2 29.4%

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

      3. sqrt-div 29.4%

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

      4. sqrt-pow1 32.4%

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

      5. metadata-eval 32.4%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 18.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 32.4%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 32.4%

      \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around -inf 0.0%

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

       1. associate-/r* 0.0%

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

       2. *-commutative 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 89.4%

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

       5. associate-*l* 89.4%

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

       6. neg-mul-1 89.4%

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

   11. Simplified 89.4%

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

   if 3.99999999999999982e283 < (*.f64 l l)

   1. Initial program 37.3%

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

      1. *-commutative 37.3%

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

      2. associate-/r* 37.3%

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

   3. Simplified 37.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 37.3%

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

      2. add-cube-cbrt 37.3%

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

      3. times-frac 37.3%

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

   6. Applied egg-rr 83.1%

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

   7. Taylor expanded in k around 0 83.1%

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

      1. add-cube-cbrt 83.1%

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

      2. pow3 83.1%

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

   9. Applied egg-rr 72.3%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{{k}^{2}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 4 \cdot 10^{+283}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(-{\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left(\sqrt{2} \cdot \frac{t}{k}\right)}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;t \leq 4.4 \cdot 10^{-124}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \left(t \cdot \frac{t\_1}{\cos k}\right)}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+140}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left(\sqrt{2} \cdot \frac{t}{k}\right)}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t\_1 \cdot \left(-t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= t 4.4e-124)
     (* (* l l) (/ 2.0 (* (pow k 2.0) (* t (/ t_1 (cos k))))))
     (if (<= t 1.35e+140)
       (pow
        (/
         (cbrt (* (* t (/ (sqrt 2.0) k)) (* (sqrt 2.0) (/ t k))))
         (* (cbrt (* (sin k) (tan k))) (* t (pow (cbrt l) -2.0))))
        3.0)
       (* -2.0 (/ (/ (* (pow l 2.0) (cos k)) (pow k 2.0)) (* t_1 (- t))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (t <= 4.4e-124) {
		tmp = (l * l) * (2.0 / (pow(k, 2.0) * (t * (t_1 / cos(k)))));
	} else if (t <= 1.35e+140) {
		tmp = pow((cbrt(((t * (sqrt(2.0) / k)) * (sqrt(2.0) * (t / k)))) / (cbrt((sin(k) * tan(k))) * (t * pow(cbrt(l), -2.0)))), 3.0);
	} else {
		tmp = -2.0 * (((pow(l, 2.0) * cos(k)) / pow(k, 2.0)) / (t_1 * -t));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (t <= 4.4e-124) {
		tmp = (l * l) * (2.0 / (Math.pow(k, 2.0) * (t * (t_1 / Math.cos(k)))));
	} else if (t <= 1.35e+140) {
		tmp = Math.pow((Math.cbrt(((t * (Math.sqrt(2.0) / k)) * (Math.sqrt(2.0) * (t / k)))) / (Math.cbrt((Math.sin(k) * Math.tan(k))) * (t * Math.pow(Math.cbrt(l), -2.0)))), 3.0);
	} else {
		tmp = -2.0 * (((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow(k, 2.0)) / (t_1 * -t));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (t <= 4.4e-124)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64((k ^ 2.0) * Float64(t * Float64(t_1 / cos(k))))));
	elseif (t <= 1.35e+140)
		tmp = Float64(cbrt(Float64(Float64(t * Float64(sqrt(2.0) / k)) * Float64(sqrt(2.0) * Float64(t / k)))) / Float64(cbrt(Float64(sin(k) * tan(k))) * Float64(t * (cbrt(l) ^ -2.0)))) ^ 3.0;
	else
		tmp = Float64(-2.0 * Float64(Float64(Float64((l ^ 2.0) * cos(k)) / (k ^ 2.0)) / Float64(t_1 * Float64(-t))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, 4.4e-124], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[(t$95$1 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e+140], N[Power[N[(N[Power[N[(N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(-2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.3999999999999998e-124

   1. Initial program 32.6%

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

   3. Simplified 41.0%

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

      1. add-sqr-sqrt 14.8%

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

      2. pow2 14.8%

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

      3. *-commutative 14.8%

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

      4. sqrt-prod 6.4%

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

      5. associate-*r* 6.4%

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

      6. sqrt-prod 6.4%

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

      7. sqrt-pow1 8.4%

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

      8. metadata-eval 8.4%

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

      9. pow1 8.4%

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

      10. sqrt-pow1 10.3%

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

      11. metadata-eval 10.3%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{\color{blue}{1.5}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 10.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

   7. Taylor expanded in t around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. associate-/l* 0.0%

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

      3. distribute-lft-neg-in 0.0%

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

      4. associate-/l* 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 73.7%

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

   9. Simplified 73.7%

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

   if 4.3999999999999998e-124 < t < 1.35000000000000009e140

   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-/r* 56.2%

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

   3. Simplified 67.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 67.4%

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

      2. add-cube-cbrt 67.3%

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

      3. times-frac 67.4%

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

   6. Applied egg-rr 93.4%

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

   7. Taylor expanded in k around 0 93.4%

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

      1. add-cube-cbrt 93.3%

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

      2. pow3 93.3%

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

   9. Applied egg-rr 89.1%

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

   if 1.35000000000000009e140 < t

   1. Initial program 5.9%

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

      1. *-commutative 5.9%

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

      2. associate-/r* 5.9%

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

   3. Simplified 35.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 35.3%

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

      2. unpow2 35.3%

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

      3. clear-num 35.3%

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

      4. un-div-inv 35.3%

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

   6. Applied egg-rr 35.3%

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

      1. add-sqr-sqrt 35.3%

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

      2. pow2 35.3%

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

      3. sqrt-div 35.3%

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

      4. sqrt-pow1 41.5%

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

      5. metadata-eval 41.5%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 17.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 44.4%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 44.4%

      \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around -inf 0.0%

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

       1. associate-/r* 0.0%

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

       2. *-commutative 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 77.6%

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

       5. associate-*l* 77.6%

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

       6. neg-mul-1 77.6%

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

   11. Simplified 77.6%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{-124}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+140}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left(\sqrt{2} \cdot \frac{t}{k}\right)}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{{\sin k}^{2} \cdot \left(-t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(-1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<=
      (*
       (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))
       (+ -1.0 (+ 1.0 (pow (/ k t) 2.0))))
      INFINITY)
   (/
    (/ 2.0 (/ (/ k t) (/ t k)))
    (* (tan k) (* (sin k) (pow (/ (pow t 1.5) l) 2.0))))
   (* (* l l) (* 2.0 (/ (/ (cos k) (pow k 2.0)) (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (((tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))) * (-1.0 + (1.0 + pow((k / t), 2.0)))) <= ((double) INFINITY)) {
		tmp = (2.0 / ((k / t) / (t / k))) / (tan(k) * (sin(k) * pow((pow(t, 1.5) / l), 2.0)));
	} else {
		tmp = (l * l) * (2.0 * ((cos(k) / pow(k, 2.0)) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (((Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))) * (-1.0 + (1.0 + Math.pow((k / t), 2.0)))) <= Double.POSITIVE_INFINITY) {
		tmp = (2.0 / ((k / t) / (t / k))) / (Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0)));
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / Math.pow(k, 2.0)) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if ((math.tan(k) * (math.sin(k) * (math.pow(t, 3.0) / (l * l)))) * (-1.0 + (1.0 + math.pow((k / t), 2.0)))) <= math.inf:
		tmp = (2.0 / ((k / t) / (t / k))) / (math.tan(k) * (math.sin(k) * math.pow((math.pow(t, 1.5) / l), 2.0)))
	else:
		tmp = (l * l) * (2.0 * ((math.cos(k) / math.pow(k, 2.0)) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(Float64(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))) * Float64(-1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0)))) <= Inf)
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) / Float64(t / k))) / Float64(tan(k) * Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (((tan(k) * (sin(k) * ((t ^ 3.0) / (l * l)))) * (-1.0 + (1.0 + ((k / t) ^ 2.0)))) <= Inf)
		tmp = (2.0 / ((k / t) / (t / k))) / (tan(k) * (sin(k) * (((t ^ 1.5) / l) ^ 2.0)));
	else
		tmp = (l * l) * (2.0 * ((cos(k) / (k ^ 2.0)) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 82.5%

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

      1. *-commutative 82.5%

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

      2. associate-/r* 82.9%

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

   3. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 88.6%

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

      2. unpow2 88.6%

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

      3. clear-num 88.7%

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

      4. un-div-inv 88.6%

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

   6. Applied egg-rr 88.6%

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

      1. add-sqr-sqrt 48.2%

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

      2. pow2 48.2%

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

      3. sqrt-div 48.2%

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

      4. sqrt-pow1 50.0%

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

      5. metadata-eval 50.0%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 31.4%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 55.2%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 55.2%

      \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]
   9. Step-by-step derivation

      1. pow1 55.2%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{{\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)\right)}^{1}}} \]

      2. *-commutative 55.2%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\color{blue}{\left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}}^{1}} \]

   10. Applied egg-rr 55.2%

       \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{{\left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}^{1}}} \]
   11. Step-by-step derivation

       1. unpow1 55.2%

          \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}} \]

       2. *-commutative 55.2%

          \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{\left(\tan k \cdot \sin k\right)} \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \]

       3. associate-*l* 55.3%

          \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]

   12. Simplified 55.3%

       \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}} \]

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   3. Simplified 15.9%

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

   5. Taylor expanded in t around 0 62.4%

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

      1. associate-/r* 62.4%

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

   7. Simplified 62.4%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(-1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7 \cdot 10^{-170} \lor \neg \left(\ell \leq 1.75 \cdot 10^{+145}\right):\\ \;\;\;\;\frac{\frac{t}{k} \cdot \left(2 \cdot \frac{t}{k}\right)}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(-{\sin k}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= l 7e-170) (not (<= l 1.75e+145)))
   (/
    (* (/ t k) (* 2.0 (/ t k)))
    (pow (* (cbrt (* (sin k) (tan k))) (/ t (pow (cbrt l) 2.0))) 3.0))
   (*
    -2.0
    (/ (/ (* (pow l 2.0) (cos k)) (pow k 2.0)) (* t (- (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l <= 7e-170) || !(l <= 1.75e+145)) {
		tmp = ((t / k) * (2.0 * (t / k))) / pow((cbrt((sin(k) * tan(k))) * (t / pow(cbrt(l), 2.0))), 3.0);
	} else {
		tmp = -2.0 * (((pow(l, 2.0) * cos(k)) / pow(k, 2.0)) / (t * -pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l <= 7e-170) || !(l <= 1.75e+145)) {
		tmp = ((t / k) * (2.0 * (t / k))) / Math.pow((Math.cbrt((Math.sin(k) * Math.tan(k))) * (t / Math.pow(Math.cbrt(l), 2.0))), 3.0);
	} else {
		tmp = -2.0 * (((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow(k, 2.0)) / (t * -Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if ((l <= 7e-170) || !(l <= 1.75e+145))
		tmp = Float64(Float64(Float64(t / k) * Float64(2.0 * Float64(t / k))) / (Float64(cbrt(Float64(sin(k) * tan(k))) * Float64(t / (cbrt(l) ^ 2.0))) ^ 3.0));
	else
		tmp = Float64(-2.0 * Float64(Float64(Float64((l ^ 2.0) * cos(k)) / (k ^ 2.0)) / Float64(t * Float64(-(sin(k) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[l, 7e-170], N[Not[LessEqual[l, 1.75e+145]], $MachinePrecision]], N[(N[(N[(t / k), $MachinePrecision] * N[(2.0 * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * (-N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.9999999999999997e-170 or 1.7500000000000001e145 < l

   1. Initial program 30.3%

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

      1. *-commutative 30.3%

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

      2. associate-/r* 30.3%

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

   3. Simplified 42.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 42.3%

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

      2. unpow2 42.3%

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

      3. clear-num 42.3%

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

      4. un-div-inv 42.3%

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

   6. Applied egg-rr 42.3%

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

      1. add-cube-cbrt 42.3%

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

      2. pow3 42.3%

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

      3. cbrt-prod 42.3%

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

      4. cbrt-div 42.3%

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

      5. rem-cbrt-cube 54.0%

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

      6. cbrt-prod 67.5%

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

      7. pow2 67.5%

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

   8. Applied egg-rr 67.5%

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

      1. associate-/r/ 68.1%

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

   10. Applied egg-rr 68.1%

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

       1. *-commutative 68.1%

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

       2. associate-/r/ 68.1%

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

       3. associate-*l/ 68.1%

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

       4. associate-*r/ 68.1%

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

   12. Simplified 68.1%

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

   if 6.9999999999999997e-170 < l < 1.7500000000000001e145

   1. Initial program 47.2%

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

      1. *-commutative 47.2%

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

      2. associate-/r* 47.7%

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

   3. Simplified 56.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 56.7%

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

      2. unpow2 56.7%

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

      3. clear-num 56.7%

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

      4. un-div-inv 56.7%

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

   6. Applied egg-rr 56.7%

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

      1. add-sqr-sqrt 32.0%

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

      2. pow2 32.0%

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

      3. sqrt-div 32.0%

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

      4. sqrt-pow1 36.1%

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

      5. metadata-eval 36.1%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 36.1%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 36.1%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 36.1%

      \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around -inf 0.0%

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

       1. associate-/r* 0.0%

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

       2. *-commutative 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 89.5%

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

       5. associate-*l* 89.5%

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

       6. neg-mul-1 89.5%

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

   11. Simplified 89.5%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7 \cdot 10^{-170} \lor \neg \left(\ell \leq 1.75 \cdot 10^{+145}\right):\\ \;\;\;\;\frac{\frac{t}{k} \cdot \left(2 \cdot \frac{t}{k}\right)}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(-{\sin k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_2 := {\sin k}^{2}\\ t_3 := \frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{if}\;\ell \leq 7.6 \cdot 10^{-170}:\\ \;\;\;\;\frac{t\_3}{{\left(t\_1 \cdot \sqrt[3]{{k}^{2}}\right)}^{3}}\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+148}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(-t\_2\right)}\\ \mathbf{elif}\;\ell \leq 10^{+280}:\\ \;\;\;\;\frac{t\_3}{\left(\sin k \cdot \tan k\right) \cdot {t\_1}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot t\_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (pow (cbrt l) 2.0)))
        (t_2 (pow (sin k) 2.0))
        (t_3 (/ 2.0 (/ (/ k t) (/ t k)))))
   (if (<= l 7.6e-170)
     (/ t_3 (pow (* t_1 (cbrt (pow k 2.0))) 3.0))
     (if (<= l 6.6e+148)
       (* -2.0 (/ (/ (* (pow l 2.0) (cos k)) (pow k 2.0)) (* t (- t_2))))
       (if (<= l 1e+280)
         (/ t_3 (* (* (sin k) (tan k)) (pow t_1 3.0)))
         (* (* l l) (* 2.0 (/ (/ (cos k) (pow k 2.0)) (* t t_2)))))))))

C

double code(double t, double l, double k) {
	double t_1 = t / pow(cbrt(l), 2.0);
	double t_2 = pow(sin(k), 2.0);
	double t_3 = 2.0 / ((k / t) / (t / k));
	double tmp;
	if (l <= 7.6e-170) {
		tmp = t_3 / pow((t_1 * cbrt(pow(k, 2.0))), 3.0);
	} else if (l <= 6.6e+148) {
		tmp = -2.0 * (((pow(l, 2.0) * cos(k)) / pow(k, 2.0)) / (t * -t_2));
	} else if (l <= 1e+280) {
		tmp = t_3 / ((sin(k) * tan(k)) * pow(t_1, 3.0));
	} else {
		tmp = (l * l) * (2.0 * ((cos(k) / pow(k, 2.0)) / (t * t_2)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = t / Math.pow(Math.cbrt(l), 2.0);
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double t_3 = 2.0 / ((k / t) / (t / k));
	double tmp;
	if (l <= 7.6e-170) {
		tmp = t_3 / Math.pow((t_1 * Math.cbrt(Math.pow(k, 2.0))), 3.0);
	} else if (l <= 6.6e+148) {
		tmp = -2.0 * (((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow(k, 2.0)) / (t * -t_2));
	} else if (l <= 1e+280) {
		tmp = t_3 / ((Math.sin(k) * Math.tan(k)) * Math.pow(t_1, 3.0));
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / Math.pow(k, 2.0)) / (t * t_2)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(t / (cbrt(l) ^ 2.0))
	t_2 = sin(k) ^ 2.0
	t_3 = Float64(2.0 / Float64(Float64(k / t) / Float64(t / k)))
	tmp = 0.0
	if (l <= 7.6e-170)
		tmp = Float64(t_3 / (Float64(t_1 * cbrt((k ^ 2.0))) ^ 3.0));
	elseif (l <= 6.6e+148)
		tmp = Float64(-2.0 * Float64(Float64(Float64((l ^ 2.0) * cos(k)) / (k ^ 2.0)) / Float64(t * Float64(-t_2))));
	elseif (l <= 1e+280)
		tmp = Float64(t_3 / Float64(Float64(sin(k) * tan(k)) * (t_1 ^ 3.0)));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64(t * t_2))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 / N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 7.6e-170], N[(t$95$3 / N[Power[N[(t$95$1 * N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.6e+148], N[(-2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * (-t$95$2)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1e+280], N[(t$95$3 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < 7.5999999999999995e-170

   1. Initial program 29.7%

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

      1. *-commutative 29.7%

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

      2. associate-/r* 29.7%

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

   3. Simplified 44.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 44.0%

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

      2. unpow2 44.0%

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

      3. clear-num 44.0%

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

      4. un-div-inv 44.0%

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

   6. Applied egg-rr 44.0%

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

      1. add-cube-cbrt 44.0%

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

      2. pow3 44.0%

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

      3. cbrt-prod 44.0%

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

      4. cbrt-div 44.0%

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

      5. rem-cbrt-cube 54.6%

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

      6. cbrt-prod 66.6%

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

      7. pow2 66.6%

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

   8. Applied egg-rr 66.6%

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

   9. Taylor expanded in k around 0 65.1%

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

   if 7.5999999999999995e-170 < l < 6.60000000000000021e148

   1. Initial program 47.2%

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

      1. *-commutative 47.2%

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

      2. associate-/r* 47.7%

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

   3. Simplified 56.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 56.7%

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

      2. unpow2 56.7%

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

      3. clear-num 56.7%

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

      4. un-div-inv 56.7%

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

   6. Applied egg-rr 56.7%

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

      1. add-sqr-sqrt 32.0%

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

      2. pow2 32.0%

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

      3. sqrt-div 32.0%

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

      4. sqrt-pow1 36.1%

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

      5. metadata-eval 36.1%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 36.1%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 36.1%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 36.1%

      \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around -inf 0.0%

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

       1. associate-/r* 0.0%

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

       2. *-commutative 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 89.5%

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

       5. associate-*l* 89.5%

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

       6. neg-mul-1 89.5%

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

   11. Simplified 89.5%

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

   if 6.60000000000000021e148 < l < 1e280

   1. Initial program 37.8%

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

      1. *-commutative 37.8%

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

      2. associate-/r* 37.8%

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

   3. Simplified 37.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 37.8%

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

      2. unpow2 37.8%

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

      3. clear-num 37.8%

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

      4. un-div-inv 37.8%

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

   6. Applied egg-rr 37.8%

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

      1. add-cube-cbrt 37.8%

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

      2. associate-*l* 37.8%

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

      3. cbrt-div 37.8%

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

      4. rem-cbrt-cube 37.8%

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

      5. cbrt-prod 37.8%

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

      6. pow2 37.8%

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

      7. pow2 37.8%

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

      8. cbrt-div 37.8%

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

      9. rem-cbrt-cube 50.8%

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

      10. cbrt-prod 74.2%

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

      11. pow2 74.2%

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

   8. Applied egg-rr 74.2%

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

      1. unpow2 74.2%

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

      2. cube-mult 74.1%

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

   10. Simplified 74.1%

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

   if 1e280 < l

   1. Initial program 16.7%

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

   3. Simplified 16.7%

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

   5. Taylor expanded in t around 0 83.3%

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

      1. associate-/r* 84.9%

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

   7. Simplified 84.9%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.6 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{{k}^{2}}\right)}^{3}}\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+148}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(-{\sin k}^{2}\right)}\\ \mathbf{elif}\;\ell \leq 10^{+280}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.1 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{{k}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(-{\sin k}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 4.1e-170)
   (/
    (/ 2.0 (/ (/ k t) (/ t k)))
    (pow (* (/ t (pow (cbrt l) 2.0)) (cbrt (pow k 2.0))) 3.0))
   (*
    -2.0
    (/ (/ (* (pow l 2.0) (cos k)) (pow k 2.0)) (* t (- (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 4.1e-170) {
		tmp = (2.0 / ((k / t) / (t / k))) / pow(((t / pow(cbrt(l), 2.0)) * cbrt(pow(k, 2.0))), 3.0);
	} else {
		tmp = -2.0 * (((pow(l, 2.0) * cos(k)) / pow(k, 2.0)) / (t * -pow(sin(k), 2.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 4.1e-170) {
		tmp = (2.0 / ((k / t) / (t / k))) / Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.pow(k, 2.0))), 3.0);
	} else {
		tmp = -2.0 * (((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow(k, 2.0)) / (t * -Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 4.1e-170)
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) / Float64(t / k))) / (Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt((k ^ 2.0))) ^ 3.0));
	else
		tmp = Float64(-2.0 * Float64(Float64(Float64((l ^ 2.0) * cos(k)) / (k ^ 2.0)) / Float64(t * Float64(-(sin(k) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 4.1e-170], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * (-N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.09999999999999966e-170

   1. Initial program 29.7%

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

      1. *-commutative 29.7%

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

      2. associate-/r* 29.7%

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

   3. Simplified 44.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 44.0%

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

      2. unpow2 44.0%

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

      3. clear-num 44.0%

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

      4. un-div-inv 44.0%

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

   6. Applied egg-rr 44.0%

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

      1. add-cube-cbrt 44.0%

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

      2. pow3 44.0%

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

      3. cbrt-prod 44.0%

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

      4. cbrt-div 44.0%

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

      5. rem-cbrt-cube 54.6%

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

      6. cbrt-prod 66.6%

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

      7. pow2 66.6%

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

   8. Applied egg-rr 66.6%

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

   9. Taylor expanded in k around 0 65.1%

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

   if 4.09999999999999966e-170 < l

   1. Initial program 43.2%

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

      1. *-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-/r* 43.6%

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

   3. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 50.0%

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

      2. unpow2 50.0%

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

      3. clear-num 50.0%

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

      4. un-div-inv 50.0%

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

   6. Applied egg-rr 50.0%

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

      1. add-sqr-sqrt 25.6%

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

      2. pow2 25.6%

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

      3. sqrt-div 25.6%

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

      4. sqrt-pow1 30.5%

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

      5. metadata-eval 30.5%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 33.3%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 33.4%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 33.4%

      \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around -inf 0.0%

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

       1. associate-/r* 0.0%

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

       2. *-commutative 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 81.0%

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

       5. associate-*l* 81.0%

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

       6. neg-mul-1 81.0%

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

   11. Simplified 81.0%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.1 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{{k}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(-{\sin k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{t}{k} \cdot \left(2 \cdot \frac{t}{k}\right)}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(-{\sin k}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 5.8e-170)
   (/
    (* (/ t k) (* 2.0 (/ t k)))
    (* (* (sin k) (tan k)) (pow (/ (pow t 1.5) l) 2.0)))
   (*
    -2.0
    (/ (/ (* (pow l 2.0) (cos k)) (pow k 2.0)) (* t (- (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 5.8e-170) {
		tmp = ((t / k) * (2.0 * (t / k))) / ((sin(k) * tan(k)) * pow((pow(t, 1.5) / l), 2.0));
	} else {
		tmp = -2.0 * (((pow(l, 2.0) * cos(k)) / pow(k, 2.0)) / (t * -pow(sin(k), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 5.8d-170) then
        tmp = ((t / k) * (2.0d0 * (t / k))) / ((sin(k) * tan(k)) * (((t ** 1.5d0) / l) ** 2.0d0))
    else
        tmp = (-2.0d0) * ((((l ** 2.0d0) * cos(k)) / (k ** 2.0d0)) / (t * -(sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 5.8e-170) {
		tmp = ((t / k) * (2.0 * (t / k))) / ((Math.sin(k) * Math.tan(k)) * Math.pow((Math.pow(t, 1.5) / l), 2.0));
	} else {
		tmp = -2.0 * (((Math.pow(l, 2.0) * Math.cos(k)) / Math.pow(k, 2.0)) / (t * -Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 5.8e-170:
		tmp = ((t / k) * (2.0 * (t / k))) / ((math.sin(k) * math.tan(k)) * math.pow((math.pow(t, 1.5) / l), 2.0))
	else:
		tmp = -2.0 * (((math.pow(l, 2.0) * math.cos(k)) / math.pow(k, 2.0)) / (t * -math.pow(math.sin(k), 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 5.8e-170)
		tmp = Float64(Float64(Float64(t / k) * Float64(2.0 * Float64(t / k))) / Float64(Float64(sin(k) * tan(k)) * (Float64((t ^ 1.5) / l) ^ 2.0)));
	else
		tmp = Float64(-2.0 * Float64(Float64(Float64((l ^ 2.0) * cos(k)) / (k ^ 2.0)) / Float64(t * Float64(-(sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 5.8e-170)
		tmp = ((t / k) * (2.0 * (t / k))) / ((sin(k) * tan(k)) * (((t ^ 1.5) / l) ^ 2.0));
	else
		tmp = -2.0 * ((((l ^ 2.0) * cos(k)) / (k ^ 2.0)) / (t * -(sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 5.8e-170], N[(N[(N[(t / k), $MachinePrecision] * N[(2.0 * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * (-N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.8000000000000001e-170

   1. Initial program 29.7%

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

      1. *-commutative 29.7%

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

      2. associate-/r* 29.7%

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

   3. Simplified 44.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 44.0%

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

      2. unpow2 44.0%

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

      3. clear-num 44.0%

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

      4. un-div-inv 44.0%

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

   6. Applied egg-rr 44.0%

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

      1. add-sqr-sqrt 26.3%

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

      2. pow2 26.3%

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

      3. sqrt-div 26.3%

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

      4. sqrt-pow1 29.6%

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

      5. metadata-eval 29.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 7.2%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 35.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 35.9%

      \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]
   9. Step-by-step derivation

      1. associate-/r/ 67.2%

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

   10. Applied egg-rr 35.9%

       \[\leadsto \frac{\color{blue}{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]
   11. Step-by-step derivation

       1. *-commutative 67.2%

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

       2. associate-/r/ 67.2%

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

       3. associate-*l/ 67.2%

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

       4. associate-*r/ 67.2%

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

   12. Simplified 35.9%

       \[\leadsto \frac{\color{blue}{\frac{t}{k} \cdot \left(2 \cdot \frac{t}{k}\right)}}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   if 5.8000000000000001e-170 < l

   1. Initial program 43.2%

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

      1. *-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-/r* 43.6%

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

   3. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 50.0%

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

      2. unpow2 50.0%

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

      3. clear-num 50.0%

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

      4. un-div-inv 50.0%

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

   6. Applied egg-rr 50.0%

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

      1. add-sqr-sqrt 25.6%

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

      2. pow2 25.6%

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

      3. sqrt-div 25.6%

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

      4. sqrt-pow1 30.5%

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

      5. metadata-eval 30.5%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 33.3%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 33.4%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 33.4%

      \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   9. Taylor expanded in t around -inf 0.0%

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

       1. associate-/r* 0.0%

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

       2. *-commutative 0.0%

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

       3. unpow2 0.0%

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

       4. rem-square-sqrt 81.0%

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

       5. associate-*l* 81.0%

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

       6. neg-mul-1 81.0%

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

   11. Simplified 81.0%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{t}{k} \cdot \left(2 \cdot \frac{t}{k}\right)}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{{\ell}^{2} \cdot \cos k}{{k}^{2}}}{t \cdot \left(-{\sin k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8.6 \cdot 10^{-132} \lor \neg \left(t \leq 1.25 \cdot 10^{+140}\right):\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t 8.6e-132) (not (<= t 1.25e+140)))
   (* (* l l) (* 2.0 (/ (/ (cos k) (pow k 2.0)) (* t (pow (sin k) 2.0)))))
   (/
    (/ 2.0 (/ (/ k t) (/ t k)))
    (* (* (sin k) (tan k)) (* (/ (pow t 2.0) l) (/ t l))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= 8.6e-132) || !(t <= 1.25e+140)) {
		tmp = (l * l) * (2.0 * ((cos(k) / pow(k, 2.0)) / (t * pow(sin(k), 2.0))));
	} else {
		tmp = (2.0 / ((k / t) / (t / k))) / ((sin(k) * tan(k)) * ((pow(t, 2.0) / l) * (t / l)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= 8.6d-132) .or. (.not. (t <= 1.25d+140))) then
        tmp = (l * l) * (2.0d0 * ((cos(k) / (k ** 2.0d0)) / (t * (sin(k) ** 2.0d0))))
    else
        tmp = (2.0d0 / ((k / t) / (t / k))) / ((sin(k) * tan(k)) * (((t ** 2.0d0) / l) * (t / l)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= 8.6e-132) || !(t <= 1.25e+140)) {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / Math.pow(k, 2.0)) / (t * Math.pow(Math.sin(k), 2.0))));
	} else {
		tmp = (2.0 / ((k / t) / (t / k))) / ((Math.sin(k) * Math.tan(k)) * ((Math.pow(t, 2.0) / l) * (t / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= 8.6e-132) or not (t <= 1.25e+140):
		tmp = (l * l) * (2.0 * ((math.cos(k) / math.pow(k, 2.0)) / (t * math.pow(math.sin(k), 2.0))))
	else:
		tmp = (2.0 / ((k / t) / (t / k))) / ((math.sin(k) * math.tan(k)) * ((math.pow(t, 2.0) / l) * (t / l)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= 8.6e-132) || !(t <= 1.25e+140))
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64(t * (sin(k) ^ 2.0)))));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) / Float64(t / k))) / Float64(Float64(sin(k) * tan(k)) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= 8.6e-132) || ~((t <= 1.25e+140)))
		tmp = (l * l) * (2.0 * ((cos(k) / (k ^ 2.0)) / (t * (sin(k) ^ 2.0))));
	else
		tmp = (2.0 / ((k / t) / (t / k))) / ((sin(k) * tan(k)) * (((t ^ 2.0) / l) * (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, 8.6e-132], N[Not[LessEqual[t, 1.25e+140]], $MachinePrecision]], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.5999999999999994e-132 or 1.25000000000000002e140 < t

   1. Initial program 27.7%

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

   3. Simplified 40.1%

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

   5. Taylor expanded in t around 0 74.4%

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

      1. associate-/r* 74.5%

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

   7. Simplified 74.5%

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

   if 8.5999999999999994e-132 < t < 1.25000000000000002e140

   1. Initial program 55.3%

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

      1. *-commutative 55.3%

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

      2. associate-/r* 55.3%

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

   3. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 66.1%

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

      2. unpow2 66.1%

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

      3. clear-num 66.1%

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

      4. un-div-inv 66.1%

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

   6. Applied egg-rr 66.1%

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

      1. unpow3 66.1%

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

      2. times-frac 85.9%

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

      3. pow2 85.9%

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

   8. Applied egg-rr 85.9%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.6 \cdot 10^{-132} \lor \neg \left(t \leq 1.25 \cdot 10^{+140}\right):\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;t \leq 7.6 \cdot 10^{-131}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \left(t \cdot \frac{t\_1}{\cos k}\right)}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot t\_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= t 7.6e-131)
     (* (* l l) (/ 2.0 (* (pow k 2.0) (* t (/ t_1 (cos k))))))
     (if (<= t 1.4e+140)
       (/
        (/ 2.0 (/ (/ k t) (/ t k)))
        (* (* (sin k) (tan k)) (* (/ (pow t 2.0) l) (/ t l))))
       (* (* l l) (* 2.0 (/ (/ (cos k) (pow k 2.0)) (* t t_1))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (t <= 7.6e-131) {
		tmp = (l * l) * (2.0 / (pow(k, 2.0) * (t * (t_1 / cos(k)))));
	} else if (t <= 1.4e+140) {
		tmp = (2.0 / ((k / t) / (t / k))) / ((sin(k) * tan(k)) * ((pow(t, 2.0) / l) * (t / l)));
	} else {
		tmp = (l * l) * (2.0 * ((cos(k) / pow(k, 2.0)) / (t * 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) ** 2.0d0
    if (t <= 7.6d-131) then
        tmp = (l * l) * (2.0d0 / ((k ** 2.0d0) * (t * (t_1 / cos(k)))))
    else if (t <= 1.4d+140) then
        tmp = (2.0d0 / ((k / t) / (t / k))) / ((sin(k) * tan(k)) * (((t ** 2.0d0) / l) * (t / l)))
    else
        tmp = (l * l) * (2.0d0 * ((cos(k) / (k ** 2.0d0)) / (t * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (t <= 7.6e-131) {
		tmp = (l * l) * (2.0 / (Math.pow(k, 2.0) * (t * (t_1 / Math.cos(k)))));
	} else if (t <= 1.4e+140) {
		tmp = (2.0 / ((k / t) / (t / k))) / ((Math.sin(k) * Math.tan(k)) * ((Math.pow(t, 2.0) / l) * (t / l)));
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / Math.pow(k, 2.0)) / (t * t_1)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if t <= 7.6e-131:
		tmp = (l * l) * (2.0 / (math.pow(k, 2.0) * (t * (t_1 / math.cos(k)))))
	elif t <= 1.4e+140:
		tmp = (2.0 / ((k / t) / (t / k))) / ((math.sin(k) * math.tan(k)) * ((math.pow(t, 2.0) / l) * (t / l)))
	else:
		tmp = (l * l) * (2.0 * ((math.cos(k) / math.pow(k, 2.0)) / (t * t_1)))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (t <= 7.6e-131)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64((k ^ 2.0) * Float64(t * Float64(t_1 / cos(k))))));
	elseif (t <= 1.4e+140)
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) / Float64(t / k))) / Float64(Float64(sin(k) * tan(k)) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64(t * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (t <= 7.6e-131)
		tmp = (l * l) * (2.0 / ((k ^ 2.0) * (t * (t_1 / cos(k)))));
	elseif (t <= 1.4e+140)
		tmp = (2.0 / ((k / t) / (t / k))) / ((sin(k) * tan(k)) * (((t ^ 2.0) / l) * (t / l)));
	else
		tmp = (l * l) * (2.0 * ((cos(k) / (k ^ 2.0)) / (t * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, 7.6e-131], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[(t$95$1 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.4e+140], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.59999999999999989e-131

   1. Initial program 32.6%

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

   3. Simplified 41.1%

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

      1. add-sqr-sqrt 14.5%

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

      2. pow2 14.5%

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

      3. *-commutative 14.5%

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

      4. sqrt-prod 5.9%

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

      5. associate-*r* 5.9%

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

      6. sqrt-prod 5.9%

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

      7. sqrt-pow1 7.9%

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

      8. metadata-eval 7.9%

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

      9. pow1 7.9%

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

      10. sqrt-pow1 9.2%

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

      11. metadata-eval 9.2%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{\color{blue}{1.5}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 9.2%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

   7. Taylor expanded in t around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. associate-/l* 0.0%

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

      3. distribute-lft-neg-in 0.0%

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

      4. associate-/l* 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 73.2%

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

   9. Simplified 73.2%

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

   if 7.59999999999999989e-131 < t < 1.39999999999999991e140

   1. Initial program 55.3%

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

      1. *-commutative 55.3%

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

      2. associate-/r* 55.3%

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

   3. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 66.1%

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

      2. unpow2 66.1%

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

      3. clear-num 66.1%

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

      4. un-div-inv 66.1%

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

   6. Applied egg-rr 66.1%

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

      1. unpow3 66.1%

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

      2. times-frac 85.9%

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

      3. pow2 85.9%

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

   8. Applied egg-rr 85.9%

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

   if 1.39999999999999991e140 < t

   1. Initial program 5.9%

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

   3. Simplified 35.4%

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

   5. Taylor expanded in t around 0 79.6%

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

      1. associate-/r* 79.7%

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

   7. Simplified 79.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{-131}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;t \leq 2.1 \cdot 10^{-132}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\cos k \cdot \frac{2}{t\_1 \cdot \left(t \cdot {k}^{2}\right)}\right)\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot t\_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= t 2.1e-132)
     (* (* l l) (* (cos k) (/ 2.0 (* t_1 (* t (pow k 2.0))))))
     (if (<= t 1.45e+140)
       (/
        (/ 2.0 (/ (/ k t) (/ t k)))
        (* (* (sin k) (tan k)) (* (/ (pow t 2.0) l) (/ t l))))
       (* (* l l) (* 2.0 (/ (/ (cos k) (pow k 2.0)) (* t t_1))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (t <= 2.1e-132) {
		tmp = (l * l) * (cos(k) * (2.0 / (t_1 * (t * pow(k, 2.0)))));
	} else if (t <= 1.45e+140) {
		tmp = (2.0 / ((k / t) / (t / k))) / ((sin(k) * tan(k)) * ((pow(t, 2.0) / l) * (t / l)));
	} else {
		tmp = (l * l) * (2.0 * ((cos(k) / pow(k, 2.0)) / (t * 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) ** 2.0d0
    if (t <= 2.1d-132) then
        tmp = (l * l) * (cos(k) * (2.0d0 / (t_1 * (t * (k ** 2.0d0)))))
    else if (t <= 1.45d+140) then
        tmp = (2.0d0 / ((k / t) / (t / k))) / ((sin(k) * tan(k)) * (((t ** 2.0d0) / l) * (t / l)))
    else
        tmp = (l * l) * (2.0d0 * ((cos(k) / (k ** 2.0d0)) / (t * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (t <= 2.1e-132) {
		tmp = (l * l) * (Math.cos(k) * (2.0 / (t_1 * (t * Math.pow(k, 2.0)))));
	} else if (t <= 1.45e+140) {
		tmp = (2.0 / ((k / t) / (t / k))) / ((Math.sin(k) * Math.tan(k)) * ((Math.pow(t, 2.0) / l) * (t / l)));
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / Math.pow(k, 2.0)) / (t * t_1)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if t <= 2.1e-132:
		tmp = (l * l) * (math.cos(k) * (2.0 / (t_1 * (t * math.pow(k, 2.0)))))
	elif t <= 1.45e+140:
		tmp = (2.0 / ((k / t) / (t / k))) / ((math.sin(k) * math.tan(k)) * ((math.pow(t, 2.0) / l) * (t / l)))
	else:
		tmp = (l * l) * (2.0 * ((math.cos(k) / math.pow(k, 2.0)) / (t * t_1)))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (t <= 2.1e-132)
		tmp = Float64(Float64(l * l) * Float64(cos(k) * Float64(2.0 / Float64(t_1 * Float64(t * (k ^ 2.0))))));
	elseif (t <= 1.45e+140)
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) / Float64(t / k))) / Float64(Float64(sin(k) * tan(k)) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64(t * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (t <= 2.1e-132)
		tmp = (l * l) * (cos(k) * (2.0 / (t_1 * (t * (k ^ 2.0)))));
	elseif (t <= 1.45e+140)
		tmp = (2.0 / ((k / t) / (t / k))) / ((sin(k) * tan(k)) * (((t ^ 2.0) / l) * (t / l)));
	else
		tmp = (l * l) * (2.0 * ((cos(k) / (k ^ 2.0)) / (t * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, 2.1e-132], N[(N[(l * l), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[(2.0 / N[(t$95$1 * N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.45e+140], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.1000000000000001e-132

   1. Initial program 32.6%

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

   3. Simplified 41.1%

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

      1. add-sqr-sqrt 14.5%

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

      2. pow2 14.5%

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

      3. *-commutative 14.5%

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

      4. sqrt-prod 5.9%

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

      5. associate-*r* 5.9%

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

      6. sqrt-prod 5.9%

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

      7. sqrt-pow1 7.9%

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

      8. metadata-eval 7.9%

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

      9. pow1 7.9%

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

      10. sqrt-pow1 9.2%

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

      11. metadata-eval 9.2%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{\color{blue}{1.5}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 9.2%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

   7. Taylor expanded in k around inf 73.2%

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

      1. *-commutative 73.2%

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

      2. associate-*l/ 73.2%

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

      3. associate-/l* 73.2%

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

      4. associate-*r* 73.2%

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

   9. Simplified 73.2%

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

   if 2.1000000000000001e-132 < t < 1.4499999999999999e140

   1. Initial program 55.3%

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

      1. *-commutative 55.3%

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

      2. associate-/r* 55.3%

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

   3. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 66.1%

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

      2. unpow2 66.1%

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

      3. clear-num 66.1%

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

      4. un-div-inv 66.1%

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

   6. Applied egg-rr 66.1%

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

      1. unpow3 66.1%

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

      2. times-frac 85.9%

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

      3. pow2 85.9%

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

   8. Applied egg-rr 85.9%

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

   if 1.4499999999999999e140 < t

   1. Initial program 5.9%

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

   3. Simplified 35.4%

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

   5. Taylor expanded in t around 0 79.6%

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

      1. associate-/r* 79.7%

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

   7. Simplified 79.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-132}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\cos k \cdot \frac{2}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\right)\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 48.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{t}{k} \cdot \left(2 \cdot \frac{t}{k}\right)}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 6.5e-170)
   (/
    (* (/ t k) (* 2.0 (/ t k)))
    (* (* (sin k) (tan k)) (pow (/ (pow t 1.5) l) 2.0)))
   (* (* l l) (* 2.0 (/ (/ (cos k) (pow k 2.0)) (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 6.5e-170) {
		tmp = ((t / k) * (2.0 * (t / k))) / ((sin(k) * tan(k)) * pow((pow(t, 1.5) / l), 2.0));
	} else {
		tmp = (l * l) * (2.0 * ((cos(k) / pow(k, 2.0)) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 6.5d-170) then
        tmp = ((t / k) * (2.0d0 * (t / k))) / ((sin(k) * tan(k)) * (((t ** 1.5d0) / l) ** 2.0d0))
    else
        tmp = (l * l) * (2.0d0 * ((cos(k) / (k ** 2.0d0)) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 6.5e-170) {
		tmp = ((t / k) * (2.0 * (t / k))) / ((Math.sin(k) * Math.tan(k)) * Math.pow((Math.pow(t, 1.5) / l), 2.0));
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / Math.pow(k, 2.0)) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 6.5e-170:
		tmp = ((t / k) * (2.0 * (t / k))) / ((math.sin(k) * math.tan(k)) * math.pow((math.pow(t, 1.5) / l), 2.0))
	else:
		tmp = (l * l) * (2.0 * ((math.cos(k) / math.pow(k, 2.0)) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 6.5e-170)
		tmp = Float64(Float64(Float64(t / k) * Float64(2.0 * Float64(t / k))) / Float64(Float64(sin(k) * tan(k)) * (Float64((t ^ 1.5) / l) ^ 2.0)));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 6.5e-170)
		tmp = ((t / k) * (2.0 * (t / k))) / ((sin(k) * tan(k)) * (((t ^ 1.5) / l) ^ 2.0));
	else
		tmp = (l * l) * (2.0 * ((cos(k) / (k ^ 2.0)) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 6.5e-170], N[(N[(N[(t / k), $MachinePrecision] * N[(2.0 * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.50000000000000035e-170

   1. Initial program 29.7%

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

      1. *-commutative 29.7%

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

      2. associate-/r* 29.7%

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

   3. Simplified 44.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 44.0%

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

      2. unpow2 44.0%

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

      3. clear-num 44.0%

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

      4. un-div-inv 44.0%

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

   6. Applied egg-rr 44.0%

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

      1. add-sqr-sqrt 26.3%

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

      2. pow2 26.3%

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

      3. sqrt-div 26.3%

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

      4. sqrt-pow1 29.6%

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

      5. metadata-eval 29.6%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 7.2%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 35.9%

         \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   8. Applied egg-rr 35.9%

      \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]
   9. Step-by-step derivation

      1. associate-/r/ 67.2%

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

   10. Applied egg-rr 35.9%

       \[\leadsto \frac{\color{blue}{\frac{2}{\frac{k}{t}} \cdot \frac{t}{k}}}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]
   11. Step-by-step derivation

       1. *-commutative 67.2%

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

       2. associate-/r/ 67.2%

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

       3. associate-*l/ 67.2%

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

       4. associate-*r/ 67.2%

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

   12. Simplified 35.9%

       \[\leadsto \frac{\color{blue}{\frac{t}{k} \cdot \left(2 \cdot \frac{t}{k}\right)}}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   if 6.50000000000000035e-170 < l

   1. Initial program 43.2%

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

   3. Simplified 50.4%

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

   5. Taylor expanded in t around 0 79.6%

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

      1. associate-/r* 79.7%

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

   7. Simplified 79.7%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{t}{k} \cdot \left(2 \cdot \frac{t}{k}\right)}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.42 \cdot 10^{-152}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(k \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left({k}^{2} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1.42e-152)
   (* (* l l) (/ 2.0 (pow (* k (* (sin k) (sqrt (/ t (cos k))))) 2.0)))
   (if (<= t 1.55e+140)
     (/
      (/ 2.0 (/ (/ k t) (/ t k)))
      (* (* (sin k) (tan k)) (* (/ (pow t 2.0) l) (/ t l))))
     (* (* l l) (/ 2.0 (pow (* (pow k 2.0) (sqrt t)) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.42e-152) {
		tmp = (l * l) * (2.0 / pow((k * (sin(k) * sqrt((t / cos(k))))), 2.0));
	} else if (t <= 1.55e+140) {
		tmp = (2.0 / ((k / t) / (t / k))) / ((sin(k) * tan(k)) * ((pow(t, 2.0) / l) * (t / l)));
	} else {
		tmp = (l * l) * (2.0 / pow((pow(k, 2.0) * sqrt(t)), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 1.42d-152) then
        tmp = (l * l) * (2.0d0 / ((k * (sin(k) * sqrt((t / cos(k))))) ** 2.0d0))
    else if (t <= 1.55d+140) then
        tmp = (2.0d0 / ((k / t) / (t / k))) / ((sin(k) * tan(k)) * (((t ** 2.0d0) / l) * (t / l)))
    else
        tmp = (l * l) * (2.0d0 / (((k ** 2.0d0) * sqrt(t)) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.42e-152) {
		tmp = (l * l) * (2.0 / Math.pow((k * (Math.sin(k) * Math.sqrt((t / Math.cos(k))))), 2.0));
	} else if (t <= 1.55e+140) {
		tmp = (2.0 / ((k / t) / (t / k))) / ((Math.sin(k) * Math.tan(k)) * ((Math.pow(t, 2.0) / l) * (t / l)));
	} else {
		tmp = (l * l) * (2.0 / Math.pow((Math.pow(k, 2.0) * Math.sqrt(t)), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 1.42e-152:
		tmp = (l * l) * (2.0 / math.pow((k * (math.sin(k) * math.sqrt((t / math.cos(k))))), 2.0))
	elif t <= 1.55e+140:
		tmp = (2.0 / ((k / t) / (t / k))) / ((math.sin(k) * math.tan(k)) * ((math.pow(t, 2.0) / l) * (t / l)))
	else:
		tmp = (l * l) * (2.0 / math.pow((math.pow(k, 2.0) * math.sqrt(t)), 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 1.42e-152)
		tmp = Float64(Float64(l * l) * Float64(2.0 / (Float64(k * Float64(sin(k) * sqrt(Float64(t / cos(k))))) ^ 2.0)));
	elseif (t <= 1.55e+140)
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) / Float64(t / k))) / Float64(Float64(sin(k) * tan(k)) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / (Float64((k ^ 2.0) * sqrt(t)) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1.42e-152)
		tmp = (l * l) * (2.0 / ((k * (sin(k) * sqrt((t / cos(k))))) ^ 2.0));
	elseif (t <= 1.55e+140)
		tmp = (2.0 / ((k / t) / (t / k))) / ((sin(k) * tan(k)) * (((t ^ 2.0) / l) * (t / l)));
	else
		tmp = (l * l) * (2.0 / (((k ^ 2.0) * sqrt(t)) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1.42e-152], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[Power[N[(k * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.55e+140], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[Power[N[(N[Power[k, 2.0], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.42e-152

   1. Initial program 33.4%

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

   3. Simplified 42.2%

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

      1. add-sqr-sqrt 14.8%

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

      2. pow2 14.8%

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

      3. *-commutative 14.8%

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

      4. sqrt-prod 6.0%

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

      5. associate-*r* 6.0%

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

      6. sqrt-prod 6.0%

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

      7. sqrt-pow1 8.1%

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

      8. metadata-eval 8.1%

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

      9. pow1 8.1%

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

      10. sqrt-pow1 9.4%

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

      11. metadata-eval 9.4%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{\color{blue}{1.5}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 9.4%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

   7. Taylor expanded in k around inf 29.6%

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

      1. associate-*l* 29.6%

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

   9. Simplified 29.6%

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

   if 1.42e-152 < t < 1.55e140

   1. Initial program 52.2%

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

      1. *-commutative 52.2%

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

      2. associate-/r* 52.3%

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

   3. Simplified 62.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 62.5%

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

      2. unpow2 62.5%

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

      3. clear-num 62.5%

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

      4. un-div-inv 62.5%

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

   6. Applied egg-rr 62.5%

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

      1. unpow3 62.4%

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

      2. times-frac 82.5%

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

      3. pow2 82.5%

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

   8. Applied egg-rr 82.5%

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

   if 1.55e140 < t

   1. Initial program 5.9%

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

   3. Simplified 35.4%

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

      1. add-sqr-sqrt 20.6%

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

      2. pow2 20.6%

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

      3. *-commutative 20.6%

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

      4. sqrt-prod 20.6%

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

      5. associate-*r* 20.6%

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

      6. sqrt-prod 23.5%

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

      7. sqrt-pow1 26.5%

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

      8. metadata-eval 26.5%

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

      9. pow1 26.5%

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

      10. sqrt-pow1 38.3%

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

      11. metadata-eval 38.3%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{\color{blue}{1.5}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 38.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

   7. Taylor expanded in k around 0 76.8%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.42 \cdot 10^{-152}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(k \cdot \left(\sin k \cdot \sqrt{\frac{t}{\cos k}}\right)\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left({k}^{2} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 64.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-132}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left({k}^{2} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 9.5e-132)
   (* (* l l) (/ 2.0 (* t (pow k 4.0))))
   (if (<= t 1.25e+140)
     (/
      (/ 2.0 (/ (/ k t) (/ t k)))
      (* (* (sin k) (tan k)) (* (/ (pow t 2.0) l) (/ t l))))
     (* (* l l) (/ 2.0 (pow (* (pow k 2.0) (sqrt t)) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 9.5e-132) {
		tmp = (l * l) * (2.0 / (t * pow(k, 4.0)));
	} else if (t <= 1.25e+140) {
		tmp = (2.0 / ((k / t) / (t / k))) / ((sin(k) * tan(k)) * ((pow(t, 2.0) / l) * (t / l)));
	} else {
		tmp = (l * l) * (2.0 / pow((pow(k, 2.0) * sqrt(t)), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 9.5d-132) then
        tmp = (l * l) * (2.0d0 / (t * (k ** 4.0d0)))
    else if (t <= 1.25d+140) then
        tmp = (2.0d0 / ((k / t) / (t / k))) / ((sin(k) * tan(k)) * (((t ** 2.0d0) / l) * (t / l)))
    else
        tmp = (l * l) * (2.0d0 / (((k ** 2.0d0) * sqrt(t)) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 9.5e-132) {
		tmp = (l * l) * (2.0 / (t * Math.pow(k, 4.0)));
	} else if (t <= 1.25e+140) {
		tmp = (2.0 / ((k / t) / (t / k))) / ((Math.sin(k) * Math.tan(k)) * ((Math.pow(t, 2.0) / l) * (t / l)));
	} else {
		tmp = (l * l) * (2.0 / Math.pow((Math.pow(k, 2.0) * Math.sqrt(t)), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 9.5e-132:
		tmp = (l * l) * (2.0 / (t * math.pow(k, 4.0)))
	elif t <= 1.25e+140:
		tmp = (2.0 / ((k / t) / (t / k))) / ((math.sin(k) * math.tan(k)) * ((math.pow(t, 2.0) / l) * (t / l)))
	else:
		tmp = (l * l) * (2.0 / math.pow((math.pow(k, 2.0) * math.sqrt(t)), 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 9.5e-132)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t * (k ^ 4.0))));
	elseif (t <= 1.25e+140)
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) / Float64(t / k))) / Float64(Float64(sin(k) * tan(k)) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / (Float64((k ^ 2.0) * sqrt(t)) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 9.5e-132)
		tmp = (l * l) * (2.0 / (t * (k ^ 4.0)));
	elseif (t <= 1.25e+140)
		tmp = (2.0 / ((k / t) / (t / k))) / ((sin(k) * tan(k)) * (((t ^ 2.0) / l) * (t / l)));
	else
		tmp = (l * l) * (2.0 / (((k ^ 2.0) * sqrt(t)) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 9.5e-132], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.25e+140], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[Power[N[(N[Power[k, 2.0], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 9.49999999999999987e-132

   1. Initial program 32.6%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in k around 0 60.7%

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

   if 9.49999999999999987e-132 < t < 1.25000000000000002e140

   1. Initial program 55.3%

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

      1. *-commutative 55.3%

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

      2. associate-/r* 55.3%

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

   3. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 66.1%

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

      2. unpow2 66.1%

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

      3. clear-num 66.1%

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

      4. un-div-inv 66.1%

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

   6. Applied egg-rr 66.1%

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

      1. unpow3 66.1%

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

      2. times-frac 85.9%

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

      3. pow2 85.9%

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

   8. Applied egg-rr 85.9%

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

   if 1.25000000000000002e140 < t

   1. Initial program 5.9%

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

   3. Simplified 35.4%

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

      1. add-sqr-sqrt 20.6%

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

      2. pow2 20.6%

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

      3. *-commutative 20.6%

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

      4. sqrt-prod 20.6%

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

      5. associate-*r* 20.6%

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

      6. sqrt-prod 23.5%

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

      7. sqrt-pow1 26.5%

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

      8. metadata-eval 26.5%

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

      9. pow1 26.5%

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

      10. sqrt-pow1 38.3%

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

      11. metadata-eval 38.3%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{\color{blue}{1.5}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 38.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

   7. Taylor expanded in k around 0 76.8%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-132}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left({k}^{2} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 64.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-132}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left({k}^{2} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 3.7e-132)
   (* (* l l) (/ 2.0 (* t (pow k 4.0))))
   (if (<= t 1.45e+140)
     (/
      (/ 2.0 (* (/ k t) (/ k t)))
      (* (* (sin k) (tan k)) (* (/ (pow t 2.0) l) (/ t l))))
     (* (* l l) (/ 2.0 (pow (* (pow k 2.0) (sqrt t)) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.7e-132) {
		tmp = (l * l) * (2.0 / (t * pow(k, 4.0)));
	} else if (t <= 1.45e+140) {
		tmp = (2.0 / ((k / t) * (k / t))) / ((sin(k) * tan(k)) * ((pow(t, 2.0) / l) * (t / l)));
	} else {
		tmp = (l * l) * (2.0 / pow((pow(k, 2.0) * sqrt(t)), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 3.7d-132) then
        tmp = (l * l) * (2.0d0 / (t * (k ** 4.0d0)))
    else if (t <= 1.45d+140) then
        tmp = (2.0d0 / ((k / t) * (k / t))) / ((sin(k) * tan(k)) * (((t ** 2.0d0) / l) * (t / l)))
    else
        tmp = (l * l) * (2.0d0 / (((k ** 2.0d0) * sqrt(t)) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 3.7e-132) {
		tmp = (l * l) * (2.0 / (t * Math.pow(k, 4.0)));
	} else if (t <= 1.45e+140) {
		tmp = (2.0 / ((k / t) * (k / t))) / ((Math.sin(k) * Math.tan(k)) * ((Math.pow(t, 2.0) / l) * (t / l)));
	} else {
		tmp = (l * l) * (2.0 / Math.pow((Math.pow(k, 2.0) * Math.sqrt(t)), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 3.7e-132:
		tmp = (l * l) * (2.0 / (t * math.pow(k, 4.0)))
	elif t <= 1.45e+140:
		tmp = (2.0 / ((k / t) * (k / t))) / ((math.sin(k) * math.tan(k)) * ((math.pow(t, 2.0) / l) * (t / l)))
	else:
		tmp = (l * l) * (2.0 / math.pow((math.pow(k, 2.0) * math.sqrt(t)), 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 3.7e-132)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t * (k ^ 4.0))));
	elseif (t <= 1.45e+140)
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) * Float64(k / t))) / Float64(Float64(sin(k) * tan(k)) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / (Float64((k ^ 2.0) * sqrt(t)) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 3.7e-132)
		tmp = (l * l) * (2.0 / (t * (k ^ 4.0)));
	elseif (t <= 1.45e+140)
		tmp = (2.0 / ((k / t) * (k / t))) / ((sin(k) * tan(k)) * (((t ^ 2.0) / l) * (t / l)));
	else
		tmp = (l * l) * (2.0 / (((k ^ 2.0) * sqrt(t)) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 3.7e-132], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.45e+140], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[Power[N[(N[Power[k, 2.0], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.7000000000000002e-132

   1. Initial program 32.6%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in k around 0 60.7%

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

   if 3.7000000000000002e-132 < t < 1.4499999999999999e140

   1. Initial program 55.3%

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

      1. *-commutative 55.3%

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

      2. associate-/r* 55.3%

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

   3. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 66.1%

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

      2. unpow2 66.1%

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

   6. Applied egg-rr 66.1%

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

      1. unpow3 66.1%

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

      2. times-frac 85.9%

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

      3. pow2 85.9%

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

   8. Applied egg-rr 85.8%

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

   if 1.4499999999999999e140 < t

   1. Initial program 5.9%

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

   3. Simplified 35.4%

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

      1. add-sqr-sqrt 20.6%

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

      2. pow2 20.6%

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

      3. *-commutative 20.6%

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

      4. sqrt-prod 20.6%

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

      5. associate-*r* 20.6%

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

      6. sqrt-prod 23.5%

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

      7. sqrt-pow1 26.5%

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

      8. metadata-eval 26.5%

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

      9. pow1 26.5%

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

      10. sqrt-pow1 38.3%

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

      11. metadata-eval 38.3%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{\color{blue}{1.5}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 38.3%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

   7. Taylor expanded in k around 0 76.8%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-132}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left({k}^{2} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 64.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8.4 \cdot 10^{-113}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot \left(t + \left(t \cdot {k}^{2}\right) \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\frac{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left({k}^{2} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 8.4e-113)
   (*
    (* l l)
    (/ 2.0 (* (pow k 4.0) (+ t (* (* t (pow k 2.0)) 0.16666666666666666)))))
   (if (<= t 5e+102)
     (/
      (/ 2.0 (/ (/ k t) (/ t k)))
      (/ (* (* (sin k) (tan k)) (/ (pow t 3.0) l)) l))
     (* (* l l) (/ 2.0 (pow (* (pow k 2.0) (sqrt t)) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 8.4e-113) {
		tmp = (l * l) * (2.0 / (pow(k, 4.0) * (t + ((t * pow(k, 2.0)) * 0.16666666666666666))));
	} else if (t <= 5e+102) {
		tmp = (2.0 / ((k / t) / (t / k))) / (((sin(k) * tan(k)) * (pow(t, 3.0) / l)) / l);
	} else {
		tmp = (l * l) * (2.0 / pow((pow(k, 2.0) * sqrt(t)), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 8.4d-113) then
        tmp = (l * l) * (2.0d0 / ((k ** 4.0d0) * (t + ((t * (k ** 2.0d0)) * 0.16666666666666666d0))))
    else if (t <= 5d+102) then
        tmp = (2.0d0 / ((k / t) / (t / k))) / (((sin(k) * tan(k)) * ((t ** 3.0d0) / l)) / l)
    else
        tmp = (l * l) * (2.0d0 / (((k ** 2.0d0) * sqrt(t)) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 8.4e-113) {
		tmp = (l * l) * (2.0 / (Math.pow(k, 4.0) * (t + ((t * Math.pow(k, 2.0)) * 0.16666666666666666))));
	} else if (t <= 5e+102) {
		tmp = (2.0 / ((k / t) / (t / k))) / (((Math.sin(k) * Math.tan(k)) * (Math.pow(t, 3.0) / l)) / l);
	} else {
		tmp = (l * l) * (2.0 / Math.pow((Math.pow(k, 2.0) * Math.sqrt(t)), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 8.4e-113:
		tmp = (l * l) * (2.0 / (math.pow(k, 4.0) * (t + ((t * math.pow(k, 2.0)) * 0.16666666666666666))))
	elif t <= 5e+102:
		tmp = (2.0 / ((k / t) / (t / k))) / (((math.sin(k) * math.tan(k)) * (math.pow(t, 3.0) / l)) / l)
	else:
		tmp = (l * l) * (2.0 / math.pow((math.pow(k, 2.0) * math.sqrt(t)), 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 8.4e-113)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64((k ^ 4.0) * Float64(t + Float64(Float64(t * (k ^ 2.0)) * 0.16666666666666666)))));
	elseif (t <= 5e+102)
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) / Float64(t / k))) / Float64(Float64(Float64(sin(k) * tan(k)) * Float64((t ^ 3.0) / l)) / l));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / (Float64((k ^ 2.0) * sqrt(t)) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 8.4e-113)
		tmp = (l * l) * (2.0 / ((k ^ 4.0) * (t + ((t * (k ^ 2.0)) * 0.16666666666666666))));
	elseif (t <= 5e+102)
		tmp = (2.0 / ((k / t) / (t / k))) / (((sin(k) * tan(k)) * ((t ^ 3.0) / l)) / l);
	else
		tmp = (l * l) * (2.0 / (((k ^ 2.0) * sqrt(t)) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 8.4e-113], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t + N[(N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5e+102], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] / N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[Power[N[(N[Power[k, 2.0], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 8.39999999999999999e-113

   1. Initial program 32.6%

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

   3. Simplified 41.0%

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

   5. Taylor expanded in k around 0 60.8%

      \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \left(t + 0.16666666666666666 \cdot \left({k}^{2} \cdot t\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]

   if 8.39999999999999999e-113 < t < 5e102

   1. Initial program 61.5%

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

      1. *-commutative 61.5%

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

      2. associate-/r* 61.6%

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

   3. Simplified 72.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 72.9%

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

      2. unpow2 72.9%

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

      3. clear-num 72.9%

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

      4. un-div-inv 72.9%

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

   6. Applied egg-rr 72.9%

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

      1. associate-/r* 83.4%

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

      2. associate-*l/ 84.2%

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

   8. Applied egg-rr 84.2%

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

   if 5e102 < t

   1. Initial program 9.3%

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

   3. Simplified 34.9%

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

      1. add-sqr-sqrt 18.6%

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

      2. pow2 18.6%

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

      3. *-commutative 18.6%

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

      4. sqrt-prod 18.6%

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

      5. associate-*r* 18.6%

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

      6. sqrt-prod 20.9%

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

      7. sqrt-pow1 23.3%

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

      8. metadata-eval 23.3%

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

      9. pow1 23.3%

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

      10. sqrt-pow1 39.6%

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

      11. metadata-eval 39.6%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{\color{blue}{1.5}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 39.6%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

   7. Taylor expanded in k around 0 74.7%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.4 \cdot 10^{-113}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot \left(t + \left(t \cdot {k}^{2}\right) \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{k}{t}}{\frac{t}{k}}}}{\frac{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left({k}^{2} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 62.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-85}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot \left(t + \left(t \cdot {k}^{2}\right) \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t}}}{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left({k}^{2} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 5.8e-85)
   (*
    (* l l)
    (/ 2.0 (* (pow k 4.0) (+ t (* (* t (pow k 2.0)) 0.16666666666666666)))))
   (if (<= t 2.1e+102)
     (/
      (/ 2.0 (* (/ k t) (/ k t)))
      (* (* (sin k) (tan k)) (/ (pow t 3.0) (* l l))))
     (* (* l l) (/ 2.0 (pow (* (pow k 2.0) (sqrt t)) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 5.8e-85) {
		tmp = (l * l) * (2.0 / (pow(k, 4.0) * (t + ((t * pow(k, 2.0)) * 0.16666666666666666))));
	} else if (t <= 2.1e+102) {
		tmp = (2.0 / ((k / t) * (k / t))) / ((sin(k) * tan(k)) * (pow(t, 3.0) / (l * l)));
	} else {
		tmp = (l * l) * (2.0 / pow((pow(k, 2.0) * sqrt(t)), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 5.8d-85) then
        tmp = (l * l) * (2.0d0 / ((k ** 4.0d0) * (t + ((t * (k ** 2.0d0)) * 0.16666666666666666d0))))
    else if (t <= 2.1d+102) then
        tmp = (2.0d0 / ((k / t) * (k / t))) / ((sin(k) * tan(k)) * ((t ** 3.0d0) / (l * l)))
    else
        tmp = (l * l) * (2.0d0 / (((k ** 2.0d0) * sqrt(t)) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 5.8e-85) {
		tmp = (l * l) * (2.0 / (Math.pow(k, 4.0) * (t + ((t * Math.pow(k, 2.0)) * 0.16666666666666666))));
	} else if (t <= 2.1e+102) {
		tmp = (2.0 / ((k / t) * (k / t))) / ((Math.sin(k) * Math.tan(k)) * (Math.pow(t, 3.0) / (l * l)));
	} else {
		tmp = (l * l) * (2.0 / Math.pow((Math.pow(k, 2.0) * Math.sqrt(t)), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 5.8e-85:
		tmp = (l * l) * (2.0 / (math.pow(k, 4.0) * (t + ((t * math.pow(k, 2.0)) * 0.16666666666666666))))
	elif t <= 2.1e+102:
		tmp = (2.0 / ((k / t) * (k / t))) / ((math.sin(k) * math.tan(k)) * (math.pow(t, 3.0) / (l * l)))
	else:
		tmp = (l * l) * (2.0 / math.pow((math.pow(k, 2.0) * math.sqrt(t)), 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 5.8e-85)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64((k ^ 4.0) * Float64(t + Float64(Float64(t * (k ^ 2.0)) * 0.16666666666666666)))));
	elseif (t <= 2.1e+102)
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) * Float64(k / t))) / Float64(Float64(sin(k) * tan(k)) * Float64((t ^ 3.0) / Float64(l * l))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / (Float64((k ^ 2.0) * sqrt(t)) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 5.8e-85)
		tmp = (l * l) * (2.0 / ((k ^ 4.0) * (t + ((t * (k ^ 2.0)) * 0.16666666666666666))));
	elseif (t <= 2.1e+102)
		tmp = (2.0 / ((k / t) * (k / t))) / ((sin(k) * tan(k)) * ((t ^ 3.0) / (l * l)));
	else
		tmp = (l * l) * (2.0 / (((k ^ 2.0) * sqrt(t)) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 5.8e-85], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t + N[(N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.1e+102], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[Power[N[(N[Power[k, 2.0], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 5.8000000000000004e-85

   1. Initial program 34.3%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in k around 0 61.2%

      \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \left(t + 0.16666666666666666 \cdot \left({k}^{2} \cdot t\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]

   if 5.8000000000000004e-85 < t < 2.10000000000000001e102

   1. Initial program 60.1%

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

      1. *-commutative 60.1%

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

      2. associate-/r* 60.2%

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

   3. Simplified 73.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 73.0%

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

      2. unpow2 73.0%

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

   6. Applied egg-rr 73.0%

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

   if 2.10000000000000001e102 < t

   1. Initial program 9.3%

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

   3. Simplified 34.9%

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

      1. add-sqr-sqrt 18.6%

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

      2. pow2 18.6%

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

      3. *-commutative 18.6%

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

      4. sqrt-prod 18.6%

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

      5. associate-*r* 18.6%

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

      6. sqrt-prod 20.9%

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

      7. sqrt-pow1 23.3%

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

      8. metadata-eval 23.3%

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

      9. pow1 23.3%

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

      10. sqrt-pow1 39.6%

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

      11. metadata-eval 39.6%

          \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{\color{blue}{1.5}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

   6. Applied egg-rr 39.6%

      \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

   7. Taylor expanded in k around 0 74.7%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-85}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot \left(t + \left(t \cdot {k}^{2}\right) \cdot 0.16666666666666666\right)}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t}}}{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left({k}^{2} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 31.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

3. Simplified 46.3%

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

   1. add-sqr-sqrt 24.8%

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

   2. pow2 24.8%

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

   3. *-commutative 24.8%

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

   4. sqrt-prod 19.7%

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

   5. associate-*r* 19.7%

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

   6. sqrt-prod 20.5%

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

   7. sqrt-pow1 22.5%

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

   8. metadata-eval 22.5%

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

   9. pow1 22.5%

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

   10. sqrt-pow1 26.3%

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

   11. metadata-eval 26.3%

       \[\leadsto \frac{2}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{\color{blue}{1.5}}\right)}^{2}} \cdot \left(\ell \cdot \ell\right) \]

6. Applied egg-rr 26.3%

   \[\leadsto \frac{2}{\color{blue}{{\left(\left(\sqrt{\sin k \cdot \tan k} \cdot \frac{k}{t}\right) \cdot {t}^{1.5}\right)}^{2}}} \cdot \left(\ell \cdot \ell\right) \]

7. Taylor expanded in k around 0 34.3%

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

8. Final simplification 34.3%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{{\left({k}^{2} \cdot \sqrt{t}\right)}^{2}} \]
9. Add Preprocessing

Alternative 23: 61.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot \left(t + \left(t \cdot {k}^{2}\right) \cdot 0.16666666666666666\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  (* l l)
  (/ 2.0 (* (pow k 4.0) (+ t (* (* t (pow k 2.0)) 0.16666666666666666))))))

C

double code(double t, double l, double k) {
	return (l * l) * (2.0 / (pow(k, 4.0) * (t + ((t * pow(k, 2.0)) * 0.16666666666666666))));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l * l) * (2.0d0 / ((k ** 4.0d0) * (t + ((t * (k ** 2.0d0)) * 0.16666666666666666d0))))
end function

Java

public static double code(double t, double l, double k) {
	return (l * l) * (2.0 / (Math.pow(k, 4.0) * (t + ((t * Math.pow(k, 2.0)) * 0.16666666666666666))));
}

Python

def code(t, l, k):
	return (l * l) * (2.0 / (math.pow(k, 4.0) * (t + ((t * math.pow(k, 2.0)) * 0.16666666666666666))))

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(2.0 / Float64((k ^ 4.0) * Float64(t + Float64(Float64(t * (k ^ 2.0)) * 0.16666666666666666)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * l) * (2.0 / ((k ^ 4.0) * (t + ((t * (k ^ 2.0)) * 0.16666666666666666))));
end

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t + N[(N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.1%

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

3. Simplified 46.3%

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

5. Taylor expanded in k around 0 62.9%

   \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \left(t + 0.16666666666666666 \cdot \left({k}^{2} \cdot t\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]

6. Final simplification 62.9%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot \left(t + \left(t \cdot {k}^{2}\right) \cdot 0.16666666666666666\right)} \]
7. Add Preprocessing

Alternative 24: 61.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 46.3%

   \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 62.7%

   \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]
6. Step-by-step derivation

   1. *-commutative 62.7%

      \[\leadsto \frac{2}{\color{blue}{t \cdot {k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   2. associate-/r* 62.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

7. Simplified 62.7%

   \[\leadsto \color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

8. Final simplification 62.7%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{t}}{{k}^{4}} \]
9. Add Preprocessing

Alternative 25: 61.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (* l l) (/ 2.0 (* t (pow k 4.0)))))

C

double code(double t, double l, double k) {
	return (l * l) * (2.0 / (t * pow(k, 4.0)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l * l) * (2.0d0 / (t * (k ** 4.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return (l * l) * (2.0 / (t * Math.pow(k, 4.0)));
}

Python

def code(t, l, k):
	return (l * l) * (2.0 / (t * math.pow(k, 4.0)))

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(2.0 / Float64(t * (k ^ 4.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * l) * (2.0 / (t * (k ^ 4.0)));
end

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.1%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 46.3%

   \[\leadsto \color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 62.7%

   \[\leadsto \color{blue}{\frac{2}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]

6. Final simplification 62.7%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024177 
(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))))