Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 94.3%

Time: 30.7s

Alternatives: 25

Speedup: 3.8×

• 39.5% 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: 35.6% 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: 94.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[t_, l_, k_] := N[Power[N[(N[(N[Power[2.0, 1/3], $MachinePrecision] / N[Power[N[(N[Power[k, 1/3], $MachinePrecision] / N[Power[t, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t * N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 33.4%

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

   1. *-commutative 33.4%

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

   2. associate-/r* 33.6%

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

3. Simplified 41.0%

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

   1. add-cube-cbrt 41.0%

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

   2. pow3 41.0%

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

   3. cbrt-prod 41.0%

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

   4. cbrt-div 41.8%

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

   5. rem-cbrt-cube 53.8%

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

   6. cbrt-prod 63.5%

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

   7. pow2 63.5%

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

6. Applied egg-rr 63.5%

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

   1. add-cube-cbrt 63.5%

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

8. Applied egg-rr 68.4%

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

   1. unpow2 68.4%

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

   2. unpow3 68.4%

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

   3. associate-*r* 70.0%

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

   4. *-commutative 70.0%

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

10. Simplified 70.0%

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

    1. metadata-eval 70.0%

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

    2. pow-flip 69.9%

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

    3. +-rgt-identity 69.9%

       \[\leadsto {\left(\frac{\sqrt[3]{2 \cdot \frac{1}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\left(t \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3} \]

    4. div-inv 69.9%

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

    5. cbrt-div 70.8%

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

    6. +-rgt-identity 70.8%

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

    7. unpow2 70.8%

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

    8. cbrt-prod 89.1%

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

    9. pow2 89.1%

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

12. Applied egg-rr 89.1%

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

    1. cbrt-div 95.5%

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

14. Applied egg-rr 95.5%

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

15. Final simplification 95.5%

    \[\leadsto {\left(\frac{\frac{\sqrt[3]{2}}{{\left(\frac{\sqrt[3]{k}}{\sqrt[3]{t}}\right)}^{2}}}{\left(t \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3} \]
16. Add Preprocessing

Alternative 2: 75.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ \mathbf{if}\;k \leq 9.3 \cdot 10^{-101}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}}{t\_1 \cdot \left(t \cdot \sqrt[3]{{k}^{2}}\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot t\_1\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) -2.0)))
   (if (<= k 9.3e-101)
     (pow
      (/ (cbrt (* 2.0 (pow (/ k t) -2.0))) (* t_1 (* t (cbrt (pow k 2.0)))))
      3.0)
     (pow
      (/
       (/ (cbrt 2.0) (pow (cbrt (/ k t)) 2.0))
       (* (cbrt (* (sin k) (tan k))) (* t t_1)))
      3.0))))

C

double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), -2.0);
	double tmp;
	if (k <= 9.3e-101) {
		tmp = pow((cbrt((2.0 * pow((k / t), -2.0))) / (t_1 * (t * cbrt(pow(k, 2.0))))), 3.0);
	} else {
		tmp = pow(((cbrt(2.0) / pow(cbrt((k / t)), 2.0)) / (cbrt((sin(k) * tan(k))) * (t * t_1))), 3.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), -2.0);
	double tmp;
	if (k <= 9.3e-101) {
		tmp = Math.pow((Math.cbrt((2.0 * Math.pow((k / t), -2.0))) / (t_1 * (t * Math.cbrt(Math.pow(k, 2.0))))), 3.0);
	} else {
		tmp = Math.pow(((Math.cbrt(2.0) / Math.pow(Math.cbrt((k / t)), 2.0)) / (Math.cbrt((Math.sin(k) * Math.tan(k))) * (t * t_1))), 3.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(l) ^ -2.0
	tmp = 0.0
	if (k <= 9.3e-101)
		tmp = Float64(cbrt(Float64(2.0 * (Float64(k / t) ^ -2.0))) / Float64(t_1 * Float64(t * cbrt((k ^ 2.0))))) ^ 3.0;
	else
		tmp = Float64(Float64(cbrt(2.0) / (cbrt(Float64(k / t)) ^ 2.0)) / Float64(cbrt(Float64(sin(k) * tan(k))) * Float64(t * t_1))) ^ 3.0;
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, If[LessEqual[k, 9.3e-101], N[Power[N[(N[Power[N[(2.0 * N[Power[N[(k / t), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t$95$1 * N[(t * N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[Power[N[(N[(N[Power[2.0, 1/3], $MachinePrecision] / N[Power[N[Power[N[(k / t), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 9.29999999999999983e-101

   1. Initial program 36.3%

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

      1. *-commutative 36.3%

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

      2. associate-/r* 36.3%

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

   3. Simplified 40.9%

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

      1. add-cube-cbrt 40.9%

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

      2. pow3 40.9%

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

      3. cbrt-prod 40.9%

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

      4. cbrt-div 42.0%

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

      5. rem-cbrt-cube 56.7%

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

      6. cbrt-prod 70.1%

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

      7. pow2 70.1%

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

   6. Applied egg-rr 70.1%

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

      1. add-cube-cbrt 70.0%

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

   8. Applied egg-rr 74.0%

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

      1. unpow2 74.0%

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

      2. unpow3 74.0%

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

      3. associate-*r* 76.4%

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

      4. *-commutative 76.4%

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

   10. Simplified 76.4%

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

   11. Taylor expanded in k around 0 69.6%

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

   if 9.29999999999999983e-101 < k

   1. Initial program 27.2%

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

      1. *-commutative 27.2%

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

      2. associate-/r* 28.0%

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

   3. Simplified 41.3%

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

      1. add-cube-cbrt 41.3%

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

      2. pow3 41.3%

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

      3. cbrt-prod 41.2%

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

      4. cbrt-div 41.2%

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

      5. rem-cbrt-cube 47.8%

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

      6. cbrt-prod 49.9%

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

      7. pow2 49.9%

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

   6. Applied egg-rr 49.9%

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

      1. add-cube-cbrt 49.9%

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

   8. Applied egg-rr 56.8%

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

      1. unpow2 56.8%

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

      2. unpow3 56.8%

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

      3. associate-*r* 56.8%

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

      4. *-commutative 56.8%

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

   10. Simplified 56.8%

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

       1. metadata-eval 56.8%

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

       2. pow-flip 56.8%

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

       3. +-rgt-identity 56.8%

          \[\leadsto {\left(\frac{\sqrt[3]{2 \cdot \frac{1}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\left(t \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3} \]

       4. div-inv 56.8%

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

       5. cbrt-div 56.9%

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

       6. +-rgt-identity 56.9%

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

       7. unpow2 56.9%

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

       8. cbrt-prod 83.0%

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

       9. pow2 83.0%

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

   12. Applied egg-rr 83.0%

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

       1. pow1 83.0%

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

       2. associate-*l* 83.1%

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

   14. Applied egg-rr 83.1%

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

       1. unpow1 83.1%

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

       2. associate-*r* 83.0%

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

       3. *-commutative 83.0%

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

       4. associate-*l* 83.0%

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

   16. Simplified 83.0%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.3 \cdot 10^{-101}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}}{{\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{{k}^{2}}\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[t_, l_, k_] := N[Power[N[(N[(N[Power[2.0, 1/3], $MachinePrecision] / N[Power[N[(1.0 / N[Power[N[(t / k), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t * N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 33.4%

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

   1. *-commutative 33.4%

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

   2. associate-/r* 33.6%

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

3. Simplified 41.0%

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

   1. add-cube-cbrt 41.0%

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

   2. pow3 41.0%

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

   3. cbrt-prod 41.0%

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

   4. cbrt-div 41.8%

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

   5. rem-cbrt-cube 53.8%

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

   6. cbrt-prod 63.5%

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

   7. pow2 63.5%

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

6. Applied egg-rr 63.5%

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

   1. add-cube-cbrt 63.5%

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

8. Applied egg-rr 68.4%

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

   1. unpow2 68.4%

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

   2. unpow3 68.4%

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

   3. associate-*r* 70.0%

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

   4. *-commutative 70.0%

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

10. Simplified 70.0%

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

    1. metadata-eval 70.0%

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

    2. pow-flip 69.9%

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

    3. +-rgt-identity 69.9%

       \[\leadsto {\left(\frac{\sqrt[3]{2 \cdot \frac{1}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\left(t \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3} \]

    4. div-inv 69.9%

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

    5. cbrt-div 70.8%

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

    6. +-rgt-identity 70.8%

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

    7. unpow2 70.8%

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

    8. cbrt-prod 89.1%

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

    9. pow2 89.1%

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

12. Applied egg-rr 89.1%

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

    1. clear-num 89.0%

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

    2. cbrt-div 90.4%

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

    3. metadata-eval 90.4%

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

14. Applied egg-rr 90.4%

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

15. Final simplification 90.4%

    \[\leadsto {\left(\frac{\frac{\sqrt[3]{2}}{{\left(\frac{1}{\sqrt[3]{\frac{t}{k}}}\right)}^{2}}}{\left(t \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3} \]
16. Add Preprocessing

Alternative 4: 88.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[t_, l_, k_] := N[Power[N[(N[(N[Power[2.0, 1/3], $MachinePrecision] / N[Power[N[Power[N[(k / t), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t * N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 33.4%

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

   1. *-commutative 33.4%

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

   2. associate-/r* 33.6%

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

3. Simplified 41.0%

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

   1. add-cube-cbrt 41.0%

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

   2. pow3 41.0%

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

   3. cbrt-prod 41.0%

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

   4. cbrt-div 41.8%

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

   5. rem-cbrt-cube 53.8%

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

   6. cbrt-prod 63.5%

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

   7. pow2 63.5%

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

6. Applied egg-rr 63.5%

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

   1. add-cube-cbrt 63.5%

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

8. Applied egg-rr 68.4%

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

   1. unpow2 68.4%

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

   2. unpow3 68.4%

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

   3. associate-*r* 70.0%

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

   4. *-commutative 70.0%

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

10. Simplified 70.0%

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

    1. metadata-eval 70.0%

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

    2. pow-flip 69.9%

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

    3. +-rgt-identity 69.9%

       \[\leadsto {\left(\frac{\sqrt[3]{2 \cdot \frac{1}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\left(t \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3} \]

    4. div-inv 69.9%

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

    5. cbrt-div 70.8%

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

    6. +-rgt-identity 70.8%

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

    7. unpow2 70.8%

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

    8. cbrt-prod 89.1%

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

    9. pow2 89.1%

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

12. Applied egg-rr 89.1%

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

13. Final simplification 89.1%

    \[\leadsto {\left(\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}}}{\left(t \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3} \]
14. Add Preprocessing

Alternative 5: 78.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ \mathbf{if}\;k \leq 1.02 \cdot 10^{-5}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}}}{t\_1 \cdot \left(t \cdot \sqrt[3]{{k}^{2}}\right)}\right)}^{3}\\ \mathbf{elif}\;k \leq 2.95 \cdot 10^{+153}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\cos k}{t \cdot {k}^{2}}}{{\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}}{t \cdot \left(\sqrt[3]{\sin k \cdot \tan k} \cdot t\_1\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) -2.0)))
   (if (<= k 1.02e-5)
     (pow
      (/
       (/ (cbrt 2.0) (pow (cbrt (/ k t)) 2.0))
       (* t_1 (* t (cbrt (pow k 2.0)))))
      3.0)
     (if (<= k 2.95e+153)
       (* (* 2.0 (/ (/ (cos k) (* t (pow k 2.0))) (pow (sin k) 2.0))) (* l l))
       (pow
        (/
         (cbrt (* 2.0 (pow (/ k t) -2.0)))
         (* t (* (cbrt (* (sin k) (tan k))) t_1)))
        3.0)))))

C

double code(double t, double l, double k) {
	double t_1 = pow(cbrt(l), -2.0);
	double tmp;
	if (k <= 1.02e-5) {
		tmp = pow(((cbrt(2.0) / pow(cbrt((k / t)), 2.0)) / (t_1 * (t * cbrt(pow(k, 2.0))))), 3.0);
	} else if (k <= 2.95e+153) {
		tmp = (2.0 * ((cos(k) / (t * pow(k, 2.0))) / pow(sin(k), 2.0))) * (l * l);
	} else {
		tmp = pow((cbrt((2.0 * pow((k / t), -2.0))) / (t * (cbrt((sin(k) * tan(k))) * t_1))), 3.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.cbrt(l), -2.0);
	double tmp;
	if (k <= 1.02e-5) {
		tmp = Math.pow(((Math.cbrt(2.0) / Math.pow(Math.cbrt((k / t)), 2.0)) / (t_1 * (t * Math.cbrt(Math.pow(k, 2.0))))), 3.0);
	} else if (k <= 2.95e+153) {
		tmp = (2.0 * ((Math.cos(k) / (t * Math.pow(k, 2.0))) / Math.pow(Math.sin(k), 2.0))) * (l * l);
	} else {
		tmp = Math.pow((Math.cbrt((2.0 * Math.pow((k / t), -2.0))) / (t * (Math.cbrt((Math.sin(k) * Math.tan(k))) * t_1))), 3.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = cbrt(l) ^ -2.0
	tmp = 0.0
	if (k <= 1.02e-5)
		tmp = Float64(Float64(cbrt(2.0) / (cbrt(Float64(k / t)) ^ 2.0)) / Float64(t_1 * Float64(t * cbrt((k ^ 2.0))))) ^ 3.0;
	elseif (k <= 2.95e+153)
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k) / Float64(t * (k ^ 2.0))) / (sin(k) ^ 2.0))) * Float64(l * l));
	else
		tmp = Float64(cbrt(Float64(2.0 * (Float64(k / t) ^ -2.0))) / Float64(t * Float64(cbrt(Float64(sin(k) * tan(k))) * t_1))) ^ 3.0;
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, If[LessEqual[k, 1.02e-5], N[Power[N[(N[(N[Power[2.0, 1/3], $MachinePrecision] / N[Power[N[Power[N[(k / t), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(t * N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[k, 2.95e+153], N[(N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[(2.0 * N[Power[N[(k / t), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t * N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.0200000000000001e-5

   1. Initial program 34.9%

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

      1. *-commutative 34.9%

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

      2. associate-/r* 34.9%

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

   3. Simplified 39.6%

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

      1. add-cube-cbrt 39.6%

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

      2. pow3 39.6%

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

      3. cbrt-prod 39.5%

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

      4. cbrt-div 40.6%

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

      5. rem-cbrt-cube 54.8%

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

      6. cbrt-prod 66.9%

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

      7. pow2 66.9%

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

   6. Applied egg-rr 66.9%

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

      1. add-cube-cbrt 66.9%

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

   8. Applied egg-rr 71.9%

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

      1. unpow2 71.9%

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

      2. unpow3 71.9%

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

      3. associate-*r* 74.0%

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

      4. *-commutative 74.0%

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

   10. Simplified 74.0%

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

       1. metadata-eval 74.0%

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

       2. pow-flip 73.8%

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

       3. +-rgt-identity 73.8%

          \[\leadsto {\left(\frac{\sqrt[3]{2 \cdot \frac{1}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\left(t \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3} \]

       4. div-inv 73.8%

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

       5. cbrt-div 75.0%

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

       6. +-rgt-identity 75.0%

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

       7. unpow2 75.0%

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

       8. cbrt-prod 91.6%

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

       9. pow2 91.6%

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

   12. Applied egg-rr 91.6%

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

   13. Taylor expanded in k around 0 78.5%

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

   if 1.0200000000000001e-5 < k < 2.9500000000000001e153

   1. Initial program 22.9%

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

   3. Simplified 48.7%

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

   5. Taylor expanded in t around 0 89.1%

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

      1. associate-*r* 89.2%

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

      2. associate-/r* 89.2%

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

   7. Simplified 89.2%

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

   if 2.9500000000000001e153 < k

   1. Initial program 36.0%

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

      1. *-commutative 36.0%

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

      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 41.6%

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

      1. add-cube-cbrt 41.6%

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

      2. pow3 41.6%

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

      3. cbrt-prod 41.6%

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

      4. cbrt-div 41.6%

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

      5. rem-cbrt-cube 52.2%

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

      6. cbrt-prod 55.2%

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

      7. pow2 55.2%

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

   6. Applied egg-rr 55.2%

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

      1. add-cube-cbrt 55.1%

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

   8. Applied egg-rr 64.2%

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

      1. unpow2 64.2%

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

      2. unpow3 64.2%

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

      3. associate-*r* 64.2%

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

      4. *-commutative 64.2%

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

   10. Simplified 64.2%

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

       1. div-inv 64.0%

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

       2. associate-*l* 64.0%

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

   12. Applied egg-rr 64.0%

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

       1. associate-*r/ 64.2%

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

       2. *-rgt-identity 64.2%

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

   14. Simplified 64.2%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{-5}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}}}{{\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{{k}^{2}}\right)}\right)}^{3}\\ \mathbf{elif}\;k \leq 2.95 \cdot 10^{+153}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\cos k}{t \cdot {k}^{2}}}{{\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}}{t \cdot \left(\sqrt[3]{\sin k \cdot \tan k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}}}{{\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{{k}^{2}}\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e-323)
   (pow
    (/
     (/ (cbrt 2.0) (pow (cbrt (/ k t)) 2.0))
     (* (pow (cbrt l) -2.0) (* t (cbrt (pow k 2.0)))))
    3.0)
   (*
    2.0
    (/ (* (cos k) (pow l 2.0)) (* (pow k 2.0) (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-323) {
		tmp = pow(((cbrt(2.0) / pow(cbrt((k / t)), 2.0)) / (pow(cbrt(l), -2.0) * (t * cbrt(pow(k, 2.0))))), 3.0);
	} else {
		tmp = 2.0 * ((cos(k) * pow(l, 2.0)) / (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 * l) <= 5e-323) {
		tmp = Math.pow(((Math.cbrt(2.0) / Math.pow(Math.cbrt((k / t)), 2.0)) / (Math.pow(Math.cbrt(l), -2.0) * (t * Math.cbrt(Math.pow(k, 2.0))))), 3.0);
	} else {
		tmp = 2.0 * ((Math.cos(k) * Math.pow(l, 2.0)) / (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(l * l) <= 5e-323)
		tmp = Float64(Float64(cbrt(2.0) / (cbrt(Float64(k / t)) ^ 2.0)) / Float64((cbrt(l) ^ -2.0) * Float64(t * cbrt((k ^ 2.0))))) ^ 3.0;
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * (l ^ 2.0)) / Float64((k ^ 2.0) * Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e-323], N[Power[N[(N[(N[Power[2.0, 1/3], $MachinePrecision] / N[Power[N[Power[N[(k / t), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[(t * N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $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 l l) < 4.94066e-323

   1. Initial program 20.9%

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

      1. *-commutative 20.9%

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

      2. associate-/r* 20.9%

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

   3. Simplified 34.1%

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

      1. add-cube-cbrt 34.1%

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

      2. pow3 34.1%

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

      3. cbrt-prod 34.1%

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

      4. cbrt-div 34.1%

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

      5. rem-cbrt-cube 40.2%

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

      6. cbrt-prod 69.0%

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

      7. pow2 69.0%

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

   6. Applied egg-rr 69.0%

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

      1. add-cube-cbrt 68.9%

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

   8. Applied egg-rr 76.2%

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

      1. unpow2 76.2%

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

      2. unpow3 76.2%

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

      3. associate-*r* 81.9%

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

      4. *-commutative 81.9%

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

   10. Simplified 81.9%

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

       1. metadata-eval 81.9%

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

       2. pow-flip 81.9%

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

       3. +-rgt-identity 81.9%

          \[\leadsto {\left(\frac{\sqrt[3]{2 \cdot \frac{1}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + 0}}}}{\left(t \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3} \]

       4. div-inv 81.9%

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

       5. cbrt-div 83.8%

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

       6. +-rgt-identity 83.8%

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

       7. unpow2 83.8%

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

       8. cbrt-prod 94.0%

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

       9. pow2 94.0%

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

   12. Applied egg-rr 94.0%

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

   13. Taylor expanded in k around 0 90.8%

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

   if 4.94066e-323 < (*.f64 l l)

   1. Initial program 36.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in t around 0 79.9%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}}}{{\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{{k}^{2}}\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}}{{\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{{k}^{2}}\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e-323)
   (pow
    (/
     (cbrt (* 2.0 (pow (/ k t) -2.0)))
     (* (pow (cbrt l) -2.0) (* t (cbrt (pow k 2.0)))))
    3.0)
   (*
    2.0
    (/ (* (cos k) (pow l 2.0)) (* (pow k 2.0) (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-323) {
		tmp = pow((cbrt((2.0 * pow((k / t), -2.0))) / (pow(cbrt(l), -2.0) * (t * cbrt(pow(k, 2.0))))), 3.0);
	} else {
		tmp = 2.0 * ((cos(k) * pow(l, 2.0)) / (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 * l) <= 5e-323) {
		tmp = Math.pow((Math.cbrt((2.0 * Math.pow((k / t), -2.0))) / (Math.pow(Math.cbrt(l), -2.0) * (t * Math.cbrt(Math.pow(k, 2.0))))), 3.0);
	} else {
		tmp = 2.0 * ((Math.cos(k) * Math.pow(l, 2.0)) / (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(l * l) <= 5e-323)
		tmp = Float64(cbrt(Float64(2.0 * (Float64(k / t) ^ -2.0))) / Float64((cbrt(l) ^ -2.0) * Float64(t * cbrt((k ^ 2.0))))) ^ 3.0;
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * (l ^ 2.0)) / Float64((k ^ 2.0) * Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e-323], N[Power[N[(N[Power[N[(2.0 * N[Power[N[(k / t), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[(t * N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $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 l l) < 4.94066e-323

   1. Initial program 20.9%

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

      1. *-commutative 20.9%

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

      2. associate-/r* 20.9%

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

   3. Simplified 34.1%

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

      1. add-cube-cbrt 34.1%

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

      2. pow3 34.1%

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

      3. cbrt-prod 34.1%

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

      4. cbrt-div 34.1%

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

      5. rem-cbrt-cube 40.2%

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

      6. cbrt-prod 69.0%

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

      7. pow2 69.0%

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

   6. Applied egg-rr 69.0%

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

      1. add-cube-cbrt 68.9%

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

   8. Applied egg-rr 76.2%

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

      1. unpow2 76.2%

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

      2. unpow3 76.2%

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

      3. associate-*r* 81.9%

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

      4. *-commutative 81.9%

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

   10. Simplified 81.9%

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

   11. Taylor expanded in k around 0 81.9%

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

   if 4.94066e-323 < (*.f64 l l)

   1. Initial program 36.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in t around 0 79.9%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}}{{\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{{k}^{2}}\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq 9.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\frac{{\left(t \cdot \sqrt[3]{\frac{t\_1}{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq 5.4 \cdot 10^{+191}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{+286}:\\ \;\;\;\;\frac{\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{t\_1}}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \log \left({\left(e^{\frac{2}{t}}\right)}^{\left({k}^{-4}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= k 9.5e-101)
     (/ (/ 2.0 (pow (/ k t) 2.0)) (/ (pow (* t (cbrt (/ t_1 l))) 3.0) l))
     (if (<= k 5.4e+191)
       (*
        2.0
        (/ (* (cos k) (pow l 2.0)) (* (pow k 2.0) (* t (pow (sin k) 2.0)))))
       (if (<= k 6e+286)
         (/
          (/ (* 2.0 (pow (/ k t) -2.0)) t_1)
          (pow (* t (pow (cbrt l) -2.0)) 3.0))
         (* (* l l) (log (pow (exp (/ 2.0 t)) (pow k -4.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (k <= 9.5e-101) {
		tmp = (2.0 / pow((k / t), 2.0)) / (pow((t * cbrt((t_1 / l))), 3.0) / l);
	} else if (k <= 5.4e+191) {
		tmp = 2.0 * ((cos(k) * pow(l, 2.0)) / (pow(k, 2.0) * (t * pow(sin(k), 2.0))));
	} else if (k <= 6e+286) {
		tmp = ((2.0 * pow((k / t), -2.0)) / t_1) / pow((t * pow(cbrt(l), -2.0)), 3.0);
	} else {
		tmp = (l * l) * log(pow(exp((2.0 / t)), pow(k, -4.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (k <= 9.5e-101) {
		tmp = (2.0 / Math.pow((k / t), 2.0)) / (Math.pow((t * Math.cbrt((t_1 / l))), 3.0) / l);
	} else if (k <= 5.4e+191) {
		tmp = 2.0 * ((Math.cos(k) * Math.pow(l, 2.0)) / (Math.pow(k, 2.0) * (t * Math.pow(Math.sin(k), 2.0))));
	} else if (k <= 6e+286) {
		tmp = ((2.0 * Math.pow((k / t), -2.0)) / t_1) / Math.pow((t * Math.pow(Math.cbrt(l), -2.0)), 3.0);
	} else {
		tmp = (l * l) * Math.log(Math.pow(Math.exp((2.0 / t)), Math.pow(k, -4.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (k <= 9.5e-101)
		tmp = Float64(Float64(2.0 / (Float64(k / t) ^ 2.0)) / Float64((Float64(t * cbrt(Float64(t_1 / l))) ^ 3.0) / l));
	elseif (k <= 5.4e+191)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * (l ^ 2.0)) / Float64((k ^ 2.0) * Float64(t * (sin(k) ^ 2.0)))));
	elseif (k <= 6e+286)
		tmp = Float64(Float64(Float64(2.0 * (Float64(k / t) ^ -2.0)) / t_1) / (Float64(t * (cbrt(l) ^ -2.0)) ^ 3.0));
	else
		tmp = Float64(Float64(l * l) * log((exp(Float64(2.0 / t)) ^ (k ^ -4.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 9.5e-101], N[(N[(2.0 / N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(t * N[Power[N[(t$95$1 / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.4e+191], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6e+286], N[(N[(N[(2.0 * N[Power[N[(k / t), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[Power[N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[Log[N[Power[N[Exp[N[(2.0 / t), $MachinePrecision]], $MachinePrecision], N[Power[k, -4.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 9.49999999999999994e-101

   1. Initial program 36.3%

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

      1. *-commutative 36.3%

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

      2. associate-/r* 36.3%

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

   3. Simplified 40.9%

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

      1. associate-*l/ 42.0%

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

      2. associate-/r* 52.4%

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

   6. Applied egg-rr 52.4%

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

      1. add-cube-cbrt 52.4%

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

      2. pow3 52.4%

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

      3. associate-/l* 52.9%

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

      4. cbrt-prod 52.9%

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

      5. unpow3 52.9%

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

      6. add-cbrt-cube 69.2%

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

   8. Applied egg-rr 69.2%

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

   if 9.49999999999999994e-101 < k < 5.39999999999999992e191

   1. Initial program 21.9%

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

   3. Simplified 37.0%

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

   5. Taylor expanded in t around 0 78.8%

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

   if 5.39999999999999992e191 < k < 5.9999999999999998e286

   1. Initial program 35.4%

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

      1. *-commutative 35.4%

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

      2. associate-/r* 38.4%

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

   3. Simplified 43.1%

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

      1. add-cube-cbrt 43.1%

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

      2. pow3 43.1%

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

      3. cbrt-prod 43.1%

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

      4. cbrt-div 43.1%

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

      5. rem-cbrt-cube 53.0%

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

      6. cbrt-prod 61.9%

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

      7. pow2 61.9%

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

   6. Applied egg-rr 61.9%

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

      1. *-un-lft-identity 61.9%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{3}}} \]

      2. div-inv 61.9%

         \[\leadsto 1 \cdot \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{3}} \]

      3. +-rgt-identity 61.9%

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

      4. pow-flip 61.9%

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

      5. metadata-eval 61.9%

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

      6. *-commutative 61.9%

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

      7. unpow-prod-down 61.9%

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

      8. pow3 61.9%

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

      9. add-cube-cbrt 61.8%

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

      10. div-inv 61.8%

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

      11. pow-flip 61.9%

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

      12. metadata-eval 61.9%

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

   8. Applied egg-rr 61.9%

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

      1. *-lft-identity 61.9%

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

      2. associate-/r* 62.0%

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

   10. Simplified 62.0%

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

   if 5.9999999999999998e286 < k

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

         \[\leadsto \frac{2}{\color{blue}{t \cdot {k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   7. Simplified 100.0%

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

      1. add-log-exp 100.0%

         \[\leadsto \color{blue}{\log \left(e^{\frac{\frac{2}{t}}{{k}^{4}}}\right)} \cdot \left(\ell \cdot \ell\right) \]

      2. div-inv 100.0%

         \[\leadsto \log \left(e^{\color{blue}{\frac{2}{t} \cdot \frac{1}{{k}^{4}}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      3. exp-prod 100.0%

         \[\leadsto \log \color{blue}{\left({\left(e^{\frac{2}{t}}\right)}^{\left(\frac{1}{{k}^{4}}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]

      4. pow-flip 100.0%

         \[\leadsto \log \left({\left(e^{\frac{2}{t}}\right)}^{\color{blue}{\left({k}^{\left(-4\right)}\right)}}\right) \cdot \left(\ell \cdot \ell\right) \]

      5. metadata-eval 100.0%

         \[\leadsto \log \left({\left(e^{\frac{2}{t}}\right)}^{\left({k}^{\color{blue}{-4}}\right)}\right) \cdot \left(\ell \cdot \ell\right) \]

   9. Applied egg-rr 100.0%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\frac{{\left(t \cdot \sqrt[3]{\frac{\sin k \cdot \tan k}{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq 5.4 \cdot 10^{+191}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{+286}:\\ \;\;\;\;\frac{\frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\sin k \cdot \tan k}}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \log \left({\left(e^{\frac{2}{t}}\right)}^{\left({k}^{-4}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ t_2 := \frac{2}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{if}\;k \leq 9.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{t\_2}{\frac{{\left(t \cdot \sqrt[3]{\frac{t\_1}{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq 8.2 \cdot 10^{+246}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{+286}:\\ \;\;\;\;\frac{t\_2}{t\_1 \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \log \left({\left(e^{\frac{2}{t}}\right)}^{\left({k}^{-4}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))) (t_2 (/ 2.0 (pow (/ k t) 2.0))))
   (if (<= k 9.5e-101)
     (/ t_2 (/ (pow (* t (cbrt (/ t_1 l))) 3.0) l))
     (if (<= k 8.2e+246)
       (*
        2.0
        (/ (* (cos k) (pow l 2.0)) (* (pow k 2.0) (* t (pow (sin k) 2.0)))))
       (if (<= k 1.25e+286)
         (/ t_2 (* t_1 (pow (/ (pow t 1.5) l) 2.0)))
         (* (* l l) (log (pow (exp (/ 2.0 t)) (pow k -4.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double t_2 = 2.0 / pow((k / t), 2.0);
	double tmp;
	if (k <= 9.5e-101) {
		tmp = t_2 / (pow((t * cbrt((t_1 / l))), 3.0) / l);
	} else if (k <= 8.2e+246) {
		tmp = 2.0 * ((cos(k) * pow(l, 2.0)) / (pow(k, 2.0) * (t * pow(sin(k), 2.0))));
	} else if (k <= 1.25e+286) {
		tmp = t_2 / (t_1 * pow((pow(t, 1.5) / l), 2.0));
	} else {
		tmp = (l * l) * log(pow(exp((2.0 / t)), pow(k, -4.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.tan(k);
	double t_2 = 2.0 / Math.pow((k / t), 2.0);
	double tmp;
	if (k <= 9.5e-101) {
		tmp = t_2 / (Math.pow((t * Math.cbrt((t_1 / l))), 3.0) / l);
	} else if (k <= 8.2e+246) {
		tmp = 2.0 * ((Math.cos(k) * Math.pow(l, 2.0)) / (Math.pow(k, 2.0) * (t * Math.pow(Math.sin(k), 2.0))));
	} else if (k <= 1.25e+286) {
		tmp = t_2 / (t_1 * Math.pow((Math.pow(t, 1.5) / l), 2.0));
	} else {
		tmp = (l * l) * Math.log(Math.pow(Math.exp((2.0 / t)), Math.pow(k, -4.0)));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	t_2 = Float64(2.0 / (Float64(k / t) ^ 2.0))
	tmp = 0.0
	if (k <= 9.5e-101)
		tmp = Float64(t_2 / Float64((Float64(t * cbrt(Float64(t_1 / l))) ^ 3.0) / l));
	elseif (k <= 8.2e+246)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * (l ^ 2.0)) / Float64((k ^ 2.0) * Float64(t * (sin(k) ^ 2.0)))));
	elseif (k <= 1.25e+286)
		tmp = Float64(t_2 / Float64(t_1 * (Float64((t ^ 1.5) / l) ^ 2.0)));
	else
		tmp = Float64(Float64(l * l) * log((exp(Float64(2.0 / t)) ^ (k ^ -4.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 9.5e-101], N[(t$95$2 / N[(N[Power[N[(t * N[Power[N[(t$95$1 / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 8.2e+246], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.25e+286], N[(t$95$2 / N[(t$95$1 * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[Log[N[Power[N[Exp[N[(2.0 / t), $MachinePrecision]], $MachinePrecision], N[Power[k, -4.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 9.49999999999999994e-101

   1. Initial program 36.3%

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

      1. *-commutative 36.3%

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

      2. associate-/r* 36.3%

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

   3. Simplified 40.9%

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

      1. associate-*l/ 42.0%

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

      2. associate-/r* 52.4%

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

   6. Applied egg-rr 52.4%

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

      1. add-cube-cbrt 52.4%

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

      2. pow3 52.4%

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

      3. associate-/l* 52.9%

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

      4. cbrt-prod 52.9%

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

      5. unpow3 52.9%

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

      6. add-cbrt-cube 69.2%

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

   8. Applied egg-rr 69.2%

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

   if 9.49999999999999994e-101 < k < 8.19999999999999951e246

   1. Initial program 24.4%

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

   3. Simplified 37.6%

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

   5. Taylor expanded in t around 0 73.9%

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

   if 8.19999999999999951e246 < k < 1.2500000000000001e286

   1. Initial program 41.2%

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

      1. *-commutative 41.2%

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

      2. associate-/r* 41.2%

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

   3. Simplified 41.2%

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

      1. add-sqr-sqrt 40.0%

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

      2. pow2 40.0%

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

      3. sqrt-div 40.0%

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

      4. sqrt-pow1 40.4%

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

      5. metadata-eval 40.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      6. sqrt-prod 20.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

      7. add-sqr-sqrt 59.5%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Applied egg-rr 59.5%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   if 1.2500000000000001e286 < k

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

         \[\leadsto \frac{2}{\color{blue}{t \cdot {k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   7. Simplified 100.0%

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

      1. add-log-exp 100.0%

         \[\leadsto \color{blue}{\log \left(e^{\frac{\frac{2}{t}}{{k}^{4}}}\right)} \cdot \left(\ell \cdot \ell\right) \]

      2. div-inv 100.0%

         \[\leadsto \log \left(e^{\color{blue}{\frac{2}{t} \cdot \frac{1}{{k}^{4}}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      3. exp-prod 100.0%

         \[\leadsto \log \color{blue}{\left({\left(e^{\frac{2}{t}}\right)}^{\left(\frac{1}{{k}^{4}}\right)}\right)} \cdot \left(\ell \cdot \ell\right) \]

      4. pow-flip 100.0%

         \[\leadsto \log \left({\left(e^{\frac{2}{t}}\right)}^{\color{blue}{\left({k}^{\left(-4\right)}\right)}}\right) \cdot \left(\ell \cdot \ell\right) \]

      5. metadata-eval 100.0%

         \[\leadsto \log \left({\left(e^{\frac{2}{t}}\right)}^{\left({k}^{\color{blue}{-4}}\right)}\right) \cdot \left(\ell \cdot \ell\right) \]

   9. Applied egg-rr 100.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\frac{{\left(t \cdot \sqrt[3]{\frac{\sin k \cdot \tan k}{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq 8.2 \cdot 10^{+246}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{+286}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \log \left({\left(e^{\frac{2}{t}}\right)}^{\left({k}^{-4}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\frac{\sin k \cdot \tan k}{\ell} \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e-323)
   (*
    l
    (/ (* 2.0 (pow (/ k t) -2.0)) (* (/ (* (sin k) (tan k)) l) (pow t 3.0))))
   (*
    2.0
    (/ (* (cos k) (pow l 2.0)) (* (pow k 2.0) (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-323) {
		tmp = l * ((2.0 * pow((k / t), -2.0)) / (((sin(k) * tan(k)) / l) * pow(t, 3.0)));
	} else {
		tmp = 2.0 * ((cos(k) * pow(l, 2.0)) / (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 * l) <= 5d-323) then
        tmp = l * ((2.0d0 * ((k / t) ** (-2.0d0))) / (((sin(k) * tan(k)) / l) * (t ** 3.0d0)))
    else
        tmp = 2.0d0 * ((cos(k) * (l ** 2.0d0)) / ((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 * l) <= 5e-323) {
		tmp = l * ((2.0 * Math.pow((k / t), -2.0)) / (((Math.sin(k) * Math.tan(k)) / l) * Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 * ((Math.cos(k) * Math.pow(l, 2.0)) / (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 * l) <= 5e-323:
		tmp = l * ((2.0 * math.pow((k / t), -2.0)) / (((math.sin(k) * math.tan(k)) / l) * math.pow(t, 3.0)))
	else:
		tmp = 2.0 * ((math.cos(k) * math.pow(l, 2.0)) / (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(l * l) <= 5e-323)
		tmp = Float64(l * Float64(Float64(2.0 * (Float64(k / t) ^ -2.0)) / Float64(Float64(Float64(sin(k) * tan(k)) / l) * (t ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * (l ^ 2.0)) / Float64((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 * l) <= 5e-323)
		tmp = l * ((2.0 * ((k / t) ^ -2.0)) / (((sin(k) * tan(k)) / l) * (t ^ 3.0)));
	else
		tmp = 2.0 * ((cos(k) * (l ^ 2.0)) / ((k ^ 2.0) * (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e-323], N[(l * N[(N[(2.0 * N[Power[N[(k / t), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $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 l l) < 4.94066e-323

   1. Initial program 20.9%

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

      1. *-commutative 20.9%

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

      2. associate-/r* 20.9%

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

   3. Simplified 34.1%

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

      1. associate-*l/ 34.0%

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

      2. associate-/r* 59.7%

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

   6. Applied egg-rr 59.7%

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

      1. associate-/r/ 63.2%

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

      2. div-inv 63.2%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell}} \cdot \ell \]

      3. +-rgt-identity 63.2%

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

      4. pow-flip 63.2%

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

      5. metadata-eval 63.2%

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

      6. associate-/l* 64.9%

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

   8. Applied egg-rr 64.9%

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

   if 4.94066e-323 < (*.f64 l l)

   1. Initial program 36.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in t around 0 79.9%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\frac{\sin k \cdot \tan k}{\ell} \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\frac{\sin k \cdot \tan k}{\ell} \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\frac{\cos k}{{k}^{2}}}{t}}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e-323)
   (*
    l
    (/ (* 2.0 (pow (/ k t) -2.0)) (* (/ (* (sin k) (tan k)) l) (pow t 3.0))))
   (* (* 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 * l) <= 5e-323) {
		tmp = l * ((2.0 * pow((k / t), -2.0)) / (((sin(k) * tan(k)) / l) * pow(t, 3.0)));
	} 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 * l) <= 5d-323) then
        tmp = l * ((2.0d0 * ((k / t) ** (-2.0d0))) / (((sin(k) * tan(k)) / l) * (t ** 3.0d0)))
    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 * l) <= 5e-323) {
		tmp = l * ((2.0 * Math.pow((k / t), -2.0)) / (((Math.sin(k) * Math.tan(k)) / l) * Math.pow(t, 3.0)));
	} 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 * l) <= 5e-323:
		tmp = l * ((2.0 * math.pow((k / t), -2.0)) / (((math.sin(k) * math.tan(k)) / l) * math.pow(t, 3.0)))
	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(l * l) <= 5e-323)
		tmp = Float64(l * Float64(Float64(2.0 * (Float64(k / t) ^ -2.0)) / Float64(Float64(Float64(sin(k) * tan(k)) / l) * (t ^ 3.0))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(Float64(cos(k) / (k ^ 2.0)) / t) / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 5e-323)
		tmp = l * ((2.0 * ((k / t) ^ -2.0)) / (((sin(k) * tan(k)) / l) * (t ^ 3.0)));
	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[(l * l), $MachinePrecision], 5e-323], N[(l * N[(N[(2.0 * N[Power[N[(k / t), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.94066e-323

   1. Initial program 20.9%

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

      1. *-commutative 20.9%

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

      2. associate-/r* 20.9%

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

   3. Simplified 34.1%

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

      1. associate-*l/ 34.0%

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

      2. associate-/r* 59.7%

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

   6. Applied egg-rr 59.7%

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

      1. associate-/r/ 63.2%

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

      2. div-inv 63.2%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell}} \cdot \ell \]

      3. +-rgt-identity 63.2%

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

      4. pow-flip 63.2%

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

      5. metadata-eval 63.2%

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

      6. associate-/l* 64.9%

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

   8. Applied egg-rr 64.9%

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

   if 4.94066e-323 < (*.f64 l l)

   1. Initial program 36.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in t around 0 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* 79.6%

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

      2. associate-/r* 79.6%

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

   7. Simplified 79.6%

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

      1. *-un-lft-identity 79.6%

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

      2. associate-/r* 79.6%

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

   9. Applied egg-rr 79.6%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot {\left(\frac{k}{t}\right)}^{-2}}{\frac{\sin k \cdot \tan k}{\ell} \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\frac{\cos k}{{k}^{2}}}{t}}{{\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{{k}^{2} \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\cos k}{t \cdot {k}^{2}}}{{\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e-323)
   (/ (/ 2.0 (pow (/ k t) 2.0)) (* (pow k 2.0) (* (/ (pow t 2.0) l) (/ t l))))
   (* (* 2.0 (/ (/ (cos k) (* t (pow k 2.0))) (pow (sin k) 2.0))) (* l l))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-323) {
		tmp = (2.0 / pow((k / t), 2.0)) / (pow(k, 2.0) * ((pow(t, 2.0) / l) * (t / l)));
	} else {
		tmp = (2.0 * ((cos(k) / (t * pow(k, 2.0))) / pow(sin(k), 2.0))) * (l * 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 ((l * l) <= 5d-323) then
        tmp = (2.0d0 / ((k / t) ** 2.0d0)) / ((k ** 2.0d0) * (((t ** 2.0d0) / l) * (t / l)))
    else
        tmp = (2.0d0 * ((cos(k) / (t * (k ** 2.0d0))) / (sin(k) ** 2.0d0))) * (l * l)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-323) {
		tmp = (2.0 / Math.pow((k / t), 2.0)) / (Math.pow(k, 2.0) * ((Math.pow(t, 2.0) / l) * (t / l)));
	} else {
		tmp = (2.0 * ((Math.cos(k) / (t * Math.pow(k, 2.0))) / Math.pow(Math.sin(k), 2.0))) * (l * l);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 5e-323:
		tmp = (2.0 / math.pow((k / t), 2.0)) / (math.pow(k, 2.0) * ((math.pow(t, 2.0) / l) * (t / l)))
	else:
		tmp = (2.0 * ((math.cos(k) / (t * math.pow(k, 2.0))) / math.pow(math.sin(k), 2.0))) * (l * l)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e-323)
		tmp = Float64(Float64(2.0 / (Float64(k / t) ^ 2.0)) / Float64((k ^ 2.0) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k) / Float64(t * (k ^ 2.0))) / (sin(k) ^ 2.0))) * Float64(l * l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 5e-323)
		tmp = (2.0 / ((k / t) ^ 2.0)) / ((k ^ 2.0) * (((t ^ 2.0) / l) * (t / l)));
	else
		tmp = (2.0 * ((cos(k) / (t * (k ^ 2.0))) / (sin(k) ^ 2.0))) * (l * l);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e-323], N[(N[(2.0 / N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.94066e-323

   1. Initial program 20.9%

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

      1. *-commutative 20.9%

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

      2. associate-/r* 20.9%

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

   3. Simplified 34.1%

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

      1. unpow3 34.1%

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

      2. times-frac 61.2%

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

      3. pow2 61.2%

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

   6. Applied egg-rr 61.2%

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

   7. Taylor expanded in k around 0 61.2%

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

   if 4.94066e-323 < (*.f64 l l)

   1. Initial program 36.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in t around 0 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* 79.6%

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

      2. associate-/r* 79.6%

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

   7. Simplified 79.6%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{{k}^{2} \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\cos k}{t \cdot {k}^{2}}}{{\sin k}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{{k}^{2} \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\frac{\cos k}{{k}^{2}}}{t}}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e-323)
   (/ (/ 2.0 (pow (/ k t) 2.0)) (* (pow k 2.0) (* (/ (pow t 2.0) l) (/ t l))))
   (* (* 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 * l) <= 5e-323) {
		tmp = (2.0 / pow((k / t), 2.0)) / (pow(k, 2.0) * ((pow(t, 2.0) / l) * (t / l)));
	} 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 * l) <= 5d-323) then
        tmp = (2.0d0 / ((k / t) ** 2.0d0)) / ((k ** 2.0d0) * (((t ** 2.0d0) / l) * (t / l)))
    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 * l) <= 5e-323) {
		tmp = (2.0 / Math.pow((k / t), 2.0)) / (Math.pow(k, 2.0) * ((Math.pow(t, 2.0) / l) * (t / l)));
	} 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 * l) <= 5e-323:
		tmp = (2.0 / math.pow((k / t), 2.0)) / (math.pow(k, 2.0) * ((math.pow(t, 2.0) / l) * (t / l)))
	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(l * l) <= 5e-323)
		tmp = Float64(Float64(2.0 / (Float64(k / t) ^ 2.0)) / Float64((k ^ 2.0) * Float64(Float64((t ^ 2.0) / l) * Float64(t / l))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(Float64(cos(k) / (k ^ 2.0)) / t) / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 5e-323)
		tmp = (2.0 / ((k / t) ^ 2.0)) / ((k ^ 2.0) * (((t ^ 2.0) / l) * (t / l)));
	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[(l * l), $MachinePrecision], 5e-323], N[(N[(2.0 / N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $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[(N[Cos[k], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.94066e-323

   1. Initial program 20.9%

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

      1. *-commutative 20.9%

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

      2. associate-/r* 20.9%

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

   3. Simplified 34.1%

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

      1. unpow3 34.1%

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

      2. times-frac 61.2%

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

      3. pow2 61.2%

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

   6. Applied egg-rr 61.2%

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

   7. Taylor expanded in k around 0 61.2%

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

   if 4.94066e-323 < (*.f64 l l)

   1. Initial program 36.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in t around 0 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* 79.6%

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

      2. associate-/r* 79.6%

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

   7. Simplified 79.6%

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

      1. *-un-lft-identity 79.6%

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

      2. associate-/r* 79.6%

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

   9. Applied egg-rr 79.6%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{{k}^{2} \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\frac{\cos k}{{k}^{2}}}{t}}{{\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 68.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-215}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}}\\ \mathbf{elif}\;k \leq 7 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\frac{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{-9}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\frac{1}{{k}^{2}}}{t}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{1}{t \cdot \frac{{k}^{2}}{\cos k}}}{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 9.5e-215)
   (* (* l l) (/ 2.0 (* t (pow k 4.0))))
   (if (<= k 7e-174)
     (/ (/ 2.0 (pow (/ k t) 2.0)) (/ (/ (* (pow k 2.0) (pow t 3.0)) l) l))
     (if (<= k 1.6e-9)
       (* (* l l) (* 2.0 (/ (/ (/ 1.0 (pow k 2.0)) t) (pow (sin k) 2.0))))
       (*
        (* l l)
        (*
         2.0
         (/
          (/ 1.0 (* t (/ (pow k 2.0) (cos k))))
          (- 0.5 (/ (cos (* 2.0 k)) 2.0)))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.5e-215) {
		tmp = (l * l) * (2.0 / (t * pow(k, 4.0)));
	} else if (k <= 7e-174) {
		tmp = (2.0 / pow((k / t), 2.0)) / (((pow(k, 2.0) * pow(t, 3.0)) / l) / l);
	} else if (k <= 1.6e-9) {
		tmp = (l * l) * (2.0 * (((1.0 / pow(k, 2.0)) / t) / pow(sin(k), 2.0)));
	} else {
		tmp = (l * l) * (2.0 * ((1.0 / (t * (pow(k, 2.0) / cos(k)))) / (0.5 - (cos((2.0 * k)) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9.5d-215) then
        tmp = (l * l) * (2.0d0 / (t * (k ** 4.0d0)))
    else if (k <= 7d-174) then
        tmp = (2.0d0 / ((k / t) ** 2.0d0)) / ((((k ** 2.0d0) * (t ** 3.0d0)) / l) / l)
    else if (k <= 1.6d-9) then
        tmp = (l * l) * (2.0d0 * (((1.0d0 / (k ** 2.0d0)) / t) / (sin(k) ** 2.0d0)))
    else
        tmp = (l * l) * (2.0d0 * ((1.0d0 / (t * ((k ** 2.0d0) / cos(k)))) / (0.5d0 - (cos((2.0d0 * k)) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.5e-215) {
		tmp = (l * l) * (2.0 / (t * Math.pow(k, 4.0)));
	} else if (k <= 7e-174) {
		tmp = (2.0 / Math.pow((k / t), 2.0)) / (((Math.pow(k, 2.0) * Math.pow(t, 3.0)) / l) / l);
	} else if (k <= 1.6e-9) {
		tmp = (l * l) * (2.0 * (((1.0 / Math.pow(k, 2.0)) / t) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = (l * l) * (2.0 * ((1.0 / (t * (Math.pow(k, 2.0) / Math.cos(k)))) / (0.5 - (Math.cos((2.0 * k)) / 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 9.5e-215:
		tmp = (l * l) * (2.0 / (t * math.pow(k, 4.0)))
	elif k <= 7e-174:
		tmp = (2.0 / math.pow((k / t), 2.0)) / (((math.pow(k, 2.0) * math.pow(t, 3.0)) / l) / l)
	elif k <= 1.6e-9:
		tmp = (l * l) * (2.0 * (((1.0 / math.pow(k, 2.0)) / t) / math.pow(math.sin(k), 2.0)))
	else:
		tmp = (l * l) * (2.0 * ((1.0 / (t * (math.pow(k, 2.0) / math.cos(k)))) / (0.5 - (math.cos((2.0 * k)) / 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 9.5e-215)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t * (k ^ 4.0))));
	elseif (k <= 7e-174)
		tmp = Float64(Float64(2.0 / (Float64(k / t) ^ 2.0)) / Float64(Float64(Float64((k ^ 2.0) * (t ^ 3.0)) / l) / l));
	elseif (k <= 1.6e-9)
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(Float64(1.0 / (k ^ 2.0)) / t) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(1.0 / Float64(t * Float64((k ^ 2.0) / cos(k)))) / Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 9.5e-215)
		tmp = (l * l) * (2.0 / (t * (k ^ 4.0)));
	elseif (k <= 7e-174)
		tmp = (2.0 / ((k / t) ^ 2.0)) / ((((k ^ 2.0) * (t ^ 3.0)) / l) / l);
	elseif (k <= 1.6e-9)
		tmp = (l * l) * (2.0 * (((1.0 / (k ^ 2.0)) / t) / (sin(k) ^ 2.0)));
	else
		tmp = (l * l) * (2.0 * ((1.0 / (t * ((k ^ 2.0) / cos(k)))) / (0.5 - (cos((2.0 * k)) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 9.5e-215], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 7e-174], N[(N[(2.0 / N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.6e-9], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[(1.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(1.0 / N[(t * N[(N[Power[k, 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if k < 9.5000000000000007e-215

   1. Initial program 33.9%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in k around 0 61.7%

      \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]

   if 9.5000000000000007e-215 < k < 6.99999999999999975e-174

   1. Initial program 33.3%

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

      1. *-commutative 33.3%

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

      2. associate-/r* 33.3%

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

   3. Simplified 33.3%

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

      1. associate-*l/ 33.3%

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

      2. associate-/r* 83.3%

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

   6. Applied egg-rr 83.3%

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

   7. Taylor expanded in k around 0 83.3%

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

   if 6.99999999999999975e-174 < k < 1.60000000000000006e-9

   1. Initial program 38.0%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in t around 0 76.0%

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

      1. associate-*r* 76.0%

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

      2. associate-/r* 76.0%

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

   7. Simplified 76.0%

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

   8. Taylor expanded in k around 0 76.0%

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

      1. associate-/r* 57.2%

         \[\leadsto \left(2 \cdot \frac{\color{blue}{\frac{\frac{1}{{k}^{2}}}{t}}}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

   10. Simplified 76.0%

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

   if 1.60000000000000006e-9 < k

   1. Initial program 29.9%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in t around 0 72.6%

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

      1. associate-*r* 72.7%

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

      2. associate-/r* 72.7%

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

   7. Simplified 72.7%

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

      1. unpow2 72.7%

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

      2. sin-mult 72.7%

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

   9. Applied egg-rr 72.7%

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

       1. div-sub 72.7%

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

       2. +-inverses 72.7%

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

       3. cos-0 72.7%

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

       4. metadata-eval 72.7%

          \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2} \cdot t}}{\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

       5. count-2 72.7%

          \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2} \cdot t}}{0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

       6. *-commutative 72.7%

          \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2} \cdot t}}{0.5 - \frac{\cos \color{blue}{\left(k \cdot 2\right)}}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

   11. Simplified 72.7%

       \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2} \cdot t}}{\color{blue}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]
   12. Step-by-step derivation

       1. clear-num 72.7%

          \[\leadsto \left(2 \cdot \frac{\color{blue}{\frac{1}{\frac{{k}^{2} \cdot t}{\cos k}}}}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

       2. inv-pow 72.7%

          \[\leadsto \left(2 \cdot \frac{\color{blue}{{\left(\frac{{k}^{2} \cdot t}{\cos k}\right)}^{-1}}}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

       3. *-commutative 72.7%

          \[\leadsto \left(2 \cdot \frac{{\left(\frac{\color{blue}{t \cdot {k}^{2}}}{\cos k}\right)}^{-1}}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

   13. Applied egg-rr 72.7%

       \[\leadsto \left(2 \cdot \frac{\color{blue}{{\left(\frac{t \cdot {k}^{2}}{\cos k}\right)}^{-1}}}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]
   14. Step-by-step derivation

       1. unpow-1 72.7%

          \[\leadsto \left(2 \cdot \frac{\color{blue}{\frac{1}{\frac{t \cdot {k}^{2}}{\cos k}}}}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

       2. associate-/l* 72.6%

          \[\leadsto \left(2 \cdot \frac{\frac{1}{\color{blue}{t \cdot \frac{{k}^{2}}{\cos k}}}}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

   15. Simplified 72.6%

       \[\leadsto \left(2 \cdot \frac{\color{blue}{\frac{1}{t \cdot \frac{{k}^{2}}{\cos k}}}}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-215}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}}\\ \mathbf{elif}\;k \leq 7 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\frac{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{-9}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\frac{1}{{k}^{2}}}{t}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{1}{t \cdot \frac{{k}^{2}}{\cos k}}}{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-215}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\frac{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{-9}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\frac{1}{{k}^{2}}}{t}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{t \cdot {k}^{2}}}{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 9.5e-215)
   (* (* l l) (/ 2.0 (* t (pow k 4.0))))
   (if (<= k 3.8e-174)
     (/ (/ 2.0 (pow (/ k t) 2.0)) (/ (/ (* (pow k 2.0) (pow t 3.0)) l) l))
     (if (<= k 1.6e-9)
       (* (* l l) (* 2.0 (/ (/ (/ 1.0 (pow k 2.0)) t) (pow (sin k) 2.0))))
       (*
        (* l l)
        (*
         2.0
         (/
          (/ (cos k) (* t (pow k 2.0)))
          (- 0.5 (/ (cos (* 2.0 k)) 2.0)))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.5e-215) {
		tmp = (l * l) * (2.0 / (t * pow(k, 4.0)));
	} else if (k <= 3.8e-174) {
		tmp = (2.0 / pow((k / t), 2.0)) / (((pow(k, 2.0) * pow(t, 3.0)) / l) / l);
	} else if (k <= 1.6e-9) {
		tmp = (l * l) * (2.0 * (((1.0 / pow(k, 2.0)) / t) / pow(sin(k), 2.0)));
	} else {
		tmp = (l * l) * (2.0 * ((cos(k) / (t * pow(k, 2.0))) / (0.5 - (cos((2.0 * k)) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9.5d-215) then
        tmp = (l * l) * (2.0d0 / (t * (k ** 4.0d0)))
    else if (k <= 3.8d-174) then
        tmp = (2.0d0 / ((k / t) ** 2.0d0)) / ((((k ** 2.0d0) * (t ** 3.0d0)) / l) / l)
    else if (k <= 1.6d-9) then
        tmp = (l * l) * (2.0d0 * (((1.0d0 / (k ** 2.0d0)) / t) / (sin(k) ** 2.0d0)))
    else
        tmp = (l * l) * (2.0d0 * ((cos(k) / (t * (k ** 2.0d0))) / (0.5d0 - (cos((2.0d0 * k)) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.5e-215) {
		tmp = (l * l) * (2.0 / (t * Math.pow(k, 4.0)));
	} else if (k <= 3.8e-174) {
		tmp = (2.0 / Math.pow((k / t), 2.0)) / (((Math.pow(k, 2.0) * Math.pow(t, 3.0)) / l) / l);
	} else if (k <= 1.6e-9) {
		tmp = (l * l) * (2.0 * (((1.0 / Math.pow(k, 2.0)) / t) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / (t * Math.pow(k, 2.0))) / (0.5 - (Math.cos((2.0 * k)) / 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 9.5e-215:
		tmp = (l * l) * (2.0 / (t * math.pow(k, 4.0)))
	elif k <= 3.8e-174:
		tmp = (2.0 / math.pow((k / t), 2.0)) / (((math.pow(k, 2.0) * math.pow(t, 3.0)) / l) / l)
	elif k <= 1.6e-9:
		tmp = (l * l) * (2.0 * (((1.0 / math.pow(k, 2.0)) / t) / math.pow(math.sin(k), 2.0)))
	else:
		tmp = (l * l) * (2.0 * ((math.cos(k) / (t * math.pow(k, 2.0))) / (0.5 - (math.cos((2.0 * k)) / 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 9.5e-215)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t * (k ^ 4.0))));
	elseif (k <= 3.8e-174)
		tmp = Float64(Float64(2.0 / (Float64(k / t) ^ 2.0)) / Float64(Float64(Float64((k ^ 2.0) * (t ^ 3.0)) / l) / l));
	elseif (k <= 1.6e-9)
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(Float64(1.0 / (k ^ 2.0)) / t) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / Float64(t * (k ^ 2.0))) / Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 9.5e-215)
		tmp = (l * l) * (2.0 / (t * (k ^ 4.0)));
	elseif (k <= 3.8e-174)
		tmp = (2.0 / ((k / t) ^ 2.0)) / ((((k ^ 2.0) * (t ^ 3.0)) / l) / l);
	elseif (k <= 1.6e-9)
		tmp = (l * l) * (2.0 * (((1.0 / (k ^ 2.0)) / t) / (sin(k) ^ 2.0)));
	else
		tmp = (l * l) * (2.0 * ((cos(k) / (t * (k ^ 2.0))) / (0.5 - (cos((2.0 * k)) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 9.5e-215], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.8e-174], N[(N[(2.0 / N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.6e-9], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[(1.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if k < 9.5000000000000007e-215

   1. Initial program 33.9%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in k around 0 61.7%

      \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]

   if 9.5000000000000007e-215 < k < 3.80000000000000021e-174

   1. Initial program 33.3%

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

      1. *-commutative 33.3%

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

      2. associate-/r* 33.3%

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

   3. Simplified 33.3%

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

      1. associate-*l/ 33.3%

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

      2. associate-/r* 83.3%

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

   6. Applied egg-rr 83.3%

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

   7. Taylor expanded in k around 0 83.3%

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

   if 3.80000000000000021e-174 < k < 1.60000000000000006e-9

   1. Initial program 38.0%

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

   3. Simplified 38.0%

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

   5. Taylor expanded in t around 0 76.0%

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

      1. associate-*r* 76.0%

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

      2. associate-/r* 76.0%

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

   7. Simplified 76.0%

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

   8. Taylor expanded in k around 0 76.0%

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

      1. associate-/r* 57.2%

         \[\leadsto \left(2 \cdot \frac{\color{blue}{\frac{\frac{1}{{k}^{2}}}{t}}}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

   10. Simplified 76.0%

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

   if 1.60000000000000006e-9 < k

   1. Initial program 29.9%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in t around 0 72.6%

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

      1. associate-*r* 72.7%

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

      2. associate-/r* 72.7%

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

   7. Simplified 72.7%

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

      1. unpow2 72.7%

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

      2. sin-mult 72.7%

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

   9. Applied egg-rr 72.7%

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

       1. div-sub 72.7%

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

       2. +-inverses 72.7%

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

       3. cos-0 72.7%

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

       4. metadata-eval 72.7%

          \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2} \cdot t}}{\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

       5. count-2 72.7%

          \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2} \cdot t}}{0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

       6. *-commutative 72.7%

          \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2} \cdot t}}{0.5 - \frac{\cos \color{blue}{\left(k \cdot 2\right)}}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

   11. Simplified 72.7%

       \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2} \cdot t}}{\color{blue}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-215}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\frac{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{-9}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\frac{1}{{k}^{2}}}{t}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{t \cdot {k}^{2}}}{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 70.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.02 \cdot 10^{-65}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{t \cdot {k}^{2}}}{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}\right)\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t}}}{\frac{\frac{\left(\sin k \cdot \tan k\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\frac{1}{{k}^{2}}}{t}}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1.02e-65)
   (*
    (* l l)
    (* 2.0 (/ (/ (cos k) (* t (pow k 2.0))) (- 0.5 (/ (cos (* 2.0 k)) 2.0)))))
   (if (<= t 1.4e+100)
     (/
      (/ 2.0 (* (/ k t) (/ k t)))
      (/ (/ (* (* (sin k) (tan k)) (pow t 3.0)) l) l))
     (* (* l l) (* 2.0 (/ (/ (/ 1.0 (pow k 2.0)) t) (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.02e-65) {
		tmp = (l * l) * (2.0 * ((cos(k) / (t * pow(k, 2.0))) / (0.5 - (cos((2.0 * k)) / 2.0))));
	} else if (t <= 1.4e+100) {
		tmp = (2.0 / ((k / t) * (k / t))) / ((((sin(k) * tan(k)) * pow(t, 3.0)) / l) / l);
	} else {
		tmp = (l * l) * (2.0 * (((1.0 / 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 (t <= 1.02d-65) then
        tmp = (l * l) * (2.0d0 * ((cos(k) / (t * (k ** 2.0d0))) / (0.5d0 - (cos((2.0d0 * k)) / 2.0d0))))
    else if (t <= 1.4d+100) then
        tmp = (2.0d0 / ((k / t) * (k / t))) / ((((sin(k) * tan(k)) * (t ** 3.0d0)) / l) / l)
    else
        tmp = (l * l) * (2.0d0 * (((1.0d0 / (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 (t <= 1.02e-65) {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / (t * Math.pow(k, 2.0))) / (0.5 - (Math.cos((2.0 * k)) / 2.0))));
	} else if (t <= 1.4e+100) {
		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 * (((1.0 / Math.pow(k, 2.0)) / t) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 1.02e-65:
		tmp = (l * l) * (2.0 * ((math.cos(k) / (t * math.pow(k, 2.0))) / (0.5 - (math.cos((2.0 * k)) / 2.0))))
	elif t <= 1.4e+100:
		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 * (((1.0 / 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 (t <= 1.02e-65)
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / Float64(t * (k ^ 2.0))) / Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0)))));
	elseif (t <= 1.4e+100)
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) * Float64(k / t))) / Float64(Float64(Float64(Float64(sin(k) * tan(k)) * (t ^ 3.0)) / l) / l));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(Float64(1.0 / (k ^ 2.0)) / t) / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1.02e-65)
		tmp = (l * l) * (2.0 * ((cos(k) / (t * (k ^ 2.0))) / (0.5 - (cos((2.0 * k)) / 2.0))));
	elseif (t <= 1.4e+100)
		tmp = (2.0 / ((k / t) * (k / t))) / ((((sin(k) * tan(k)) * (t ^ 3.0)) / l) / l);
	else
		tmp = (l * l) * (2.0 * (((1.0 / (k ^ 2.0)) / t) / (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1.02e-65], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.4e+100], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[(1.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.02000000000000004e-65

   1. Initial program 35.7%

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

   3. Simplified 41.5%

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

   5. Taylor expanded in t around 0 71.7%

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

      1. associate-*r* 71.7%

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

      2. associate-/r* 71.7%

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

   7. Simplified 71.7%

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

      1. unpow2 71.7%

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

      2. sin-mult 67.7%

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

   9. Applied egg-rr 67.7%

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

       1. div-sub 67.7%

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

       2. +-inverses 67.7%

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

       3. cos-0 67.7%

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

       4. metadata-eval 67.7%

          \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2} \cdot t}}{\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

       5. count-2 67.7%

          \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2} \cdot t}}{0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

       6. *-commutative 67.7%

          \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2} \cdot t}}{0.5 - \frac{\cos \color{blue}{\left(k \cdot 2\right)}}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

   11. Simplified 67.7%

       \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2} \cdot t}}{\color{blue}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   if 1.02000000000000004e-65 < t < 1.3999999999999999e100

   1. Initial program 60.4%

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

      1. *-commutative 60.4%

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

      2. associate-/r* 62.3%

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

   3. Simplified 68.3%

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

      1. associate-*l/ 68.2%

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

      2. associate-/r* 76.5%

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

   6. Applied egg-rr 76.5%

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

      1. unpow2 76.5%

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

   8. Applied egg-rr 76.5%

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

   if 1.3999999999999999e100 < t

   1. Initial program 6.3%

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

   3. Simplified 25.2%

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

   5. Taylor expanded in t around 0 81.3%

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

      1. associate-*r* 81.4%

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

      2. associate-/r* 81.4%

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

   7. Simplified 81.4%

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

   8. Taylor expanded in k around 0 73.8%

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

      1. associate-/r* 61.9%

         \[\leadsto \left(2 \cdot \frac{\color{blue}{\frac{\frac{1}{{k}^{2}}}{t}}}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

   10. Simplified 73.8%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.02 \cdot 10^{-65}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{t \cdot {k}^{2}}}{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}\right)\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t}}}{\frac{\frac{\left(\sin k \cdot \tan k\right) \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\frac{1}{{k}^{2}}}{t}}{{\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 64.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\frac{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{t \cdot {k}^{2}}}{{k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e-323)
   (/ (/ 2.0 (pow (/ k t) 2.0)) (/ (* (pow k 2.0) (/ (pow t 3.0) l)) l))
   (* (* l l) (* 2.0 (/ (/ (cos k) (* t (pow k 2.0))) (pow k 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-323) {
		tmp = (2.0 / pow((k / t), 2.0)) / ((pow(k, 2.0) * (pow(t, 3.0) / l)) / l);
	} else {
		tmp = (l * l) * (2.0 * ((cos(k) / (t * pow(k, 2.0))) / pow(k, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 5d-323) then
        tmp = (2.0d0 / ((k / t) ** 2.0d0)) / (((k ** 2.0d0) * ((t ** 3.0d0) / l)) / l)
    else
        tmp = (l * l) * (2.0d0 * ((cos(k) / (t * (k ** 2.0d0))) / (k ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-323) {
		tmp = (2.0 / Math.pow((k / t), 2.0)) / ((Math.pow(k, 2.0) * (Math.pow(t, 3.0) / l)) / l);
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / (t * Math.pow(k, 2.0))) / Math.pow(k, 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 5e-323:
		tmp = (2.0 / math.pow((k / t), 2.0)) / ((math.pow(k, 2.0) * (math.pow(t, 3.0) / l)) / l)
	else:
		tmp = (l * l) * (2.0 * ((math.cos(k) / (t * math.pow(k, 2.0))) / math.pow(k, 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e-323)
		tmp = Float64(Float64(2.0 / (Float64(k / t) ^ 2.0)) / Float64(Float64((k ^ 2.0) * Float64((t ^ 3.0) / l)) / l));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / Float64(t * (k ^ 2.0))) / (k ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 5e-323)
		tmp = (2.0 / ((k / t) ^ 2.0)) / (((k ^ 2.0) * ((t ^ 3.0) / l)) / l);
	else
		tmp = (l * l) * (2.0 * ((cos(k) / (t * (k ^ 2.0))) / (k ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e-323], N[(N[(2.0 / N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.94066e-323

   1. Initial program 20.9%

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

      1. *-commutative 20.9%

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

      2. associate-/r* 20.9%

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

   3. Simplified 34.1%

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

      1. associate-*l/ 34.0%

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

      2. associate-/r* 59.7%

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

   6. Applied egg-rr 59.7%

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

   7. Taylor expanded in k around 0 59.7%

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

      1. associate-/l* 57.8%

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

   9. Simplified 57.8%

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

   if 4.94066e-323 < (*.f64 l l)

   1. Initial program 36.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in t around 0 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* 79.6%

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

      2. associate-/r* 79.6%

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

   7. Simplified 79.6%

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

   8. Taylor expanded in k around 0 70.2%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\frac{{k}^{2} \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{t \cdot {k}^{2}}}{{k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 64.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\frac{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{t \cdot {k}^{2}}}{{k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e-323)
   (/ (/ 2.0 (pow (/ k t) 2.0)) (/ (/ (* (pow k 2.0) (pow t 3.0)) l) l))
   (* (* l l) (* 2.0 (/ (/ (cos k) (* t (pow k 2.0))) (pow k 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-323) {
		tmp = (2.0 / pow((k / t), 2.0)) / (((pow(k, 2.0) * pow(t, 3.0)) / l) / l);
	} else {
		tmp = (l * l) * (2.0 * ((cos(k) / (t * pow(k, 2.0))) / pow(k, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 5d-323) then
        tmp = (2.0d0 / ((k / t) ** 2.0d0)) / ((((k ** 2.0d0) * (t ** 3.0d0)) / l) / l)
    else
        tmp = (l * l) * (2.0d0 * ((cos(k) / (t * (k ** 2.0d0))) / (k ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e-323) {
		tmp = (2.0 / Math.pow((k / t), 2.0)) / (((Math.pow(k, 2.0) * Math.pow(t, 3.0)) / l) / l);
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k) / (t * Math.pow(k, 2.0))) / Math.pow(k, 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (l * l) <= 5e-323:
		tmp = (2.0 / math.pow((k / t), 2.0)) / (((math.pow(k, 2.0) * math.pow(t, 3.0)) / l) / l)
	else:
		tmp = (l * l) * (2.0 * ((math.cos(k) / (t * math.pow(k, 2.0))) / math.pow(k, 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e-323)
		tmp = Float64(Float64(2.0 / (Float64(k / t) ^ 2.0)) / Float64(Float64(Float64((k ^ 2.0) * (t ^ 3.0)) / l) / l));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k) / Float64(t * (k ^ 2.0))) / (k ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 5e-323)
		tmp = (2.0 / ((k / t) ^ 2.0)) / ((((k ^ 2.0) * (t ^ 3.0)) / l) / l);
	else
		tmp = (l * l) * (2.0 * ((cos(k) / (t * (k ^ 2.0))) / (k ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e-323], N[(N[(2.0 / N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.94066e-323

   1. Initial program 20.9%

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

      1. *-commutative 20.9%

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

      2. associate-/r* 20.9%

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

   3. Simplified 34.1%

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

      1. associate-*l/ 34.0%

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

      2. associate-/r* 59.7%

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

   6. Applied egg-rr 59.7%

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

   7. Taylor expanded in k around 0 59.7%

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

   if 4.94066e-323 < (*.f64 l l)

   1. Initial program 36.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in t around 0 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* 79.6%

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

      2. associate-/r* 79.6%

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

   7. Simplified 79.6%

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

   8. Taylor expanded in k around 0 70.2%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-323}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\frac{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{t \cdot {k}^{2}}}{{k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 65.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return (l * l) * (2.0 * ((1.0 / (t * pow(k, 2.0))) / pow(sin(k), 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 * ((1.0d0 / (t * (k ** 2.0d0))) / (sin(k) ** 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 41.2%

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

5. Taylor expanded in t around 0 72.4%

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

   1. associate-*r* 72.5%

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

   2. associate-/r* 72.5%

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

7. Simplified 72.5%

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

8. Taylor expanded in k around 0 63.4%

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

9. Final simplification 63.4%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{1}{t \cdot {k}^{2}}}{{\sin k}^{2}}\right) \]
10. Add Preprocessing

Alternative 20: 66.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 41.2%

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

5. Taylor expanded in t around 0 72.4%

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

   1. associate-*r* 72.5%

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

   2. associate-/r* 72.5%

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

7. Simplified 72.5%

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

8. Taylor expanded in k around 0 65.0%

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

9. Final simplification 65.0%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{t \cdot {k}^{2}}}{{k}^{2}}\right) \]
10. Add Preprocessing

Alternative 21: 55.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.6 \cdot 10^{+74}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{1}{t \cdot {k}^{2}}}{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5.6e+74)
   (*
    (* l l)
    (/
     (+ (* -0.3333333333333333 (/ (pow k 2.0) t)) (* 2.0 (/ 1.0 t)))
     (pow k 4.0)))
   (*
    (* l l)
    (* 2.0 (/ (/ 1.0 (* t (pow k 2.0))) (- 0.5 (/ (cos (* 2.0 k)) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.6e+74) {
		tmp = (l * l) * (((-0.3333333333333333 * (pow(k, 2.0) / t)) + (2.0 * (1.0 / t))) / pow(k, 4.0));
	} else {
		tmp = (l * l) * (2.0 * ((1.0 / (t * pow(k, 2.0))) / (0.5 - (cos((2.0 * k)) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.6d+74) then
        tmp = (l * l) * ((((-0.3333333333333333d0) * ((k ** 2.0d0) / t)) + (2.0d0 * (1.0d0 / t))) / (k ** 4.0d0))
    else
        tmp = (l * l) * (2.0d0 * ((1.0d0 / (t * (k ** 2.0d0))) / (0.5d0 - (cos((2.0d0 * k)) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.6e+74) {
		tmp = (l * l) * (((-0.3333333333333333 * (Math.pow(k, 2.0) / t)) + (2.0 * (1.0 / t))) / Math.pow(k, 4.0));
	} else {
		tmp = (l * l) * (2.0 * ((1.0 / (t * Math.pow(k, 2.0))) / (0.5 - (Math.cos((2.0 * k)) / 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 5.6e+74:
		tmp = (l * l) * (((-0.3333333333333333 * (math.pow(k, 2.0) / t)) + (2.0 * (1.0 / t))) / math.pow(k, 4.0))
	else:
		tmp = (l * l) * (2.0 * ((1.0 / (t * math.pow(k, 2.0))) / (0.5 - (math.cos((2.0 * k)) / 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 5.6e+74)
		tmp = Float64(Float64(l * l) * Float64(Float64(Float64(-0.3333333333333333 * Float64((k ^ 2.0) / t)) + Float64(2.0 * Float64(1.0 / t))) / (k ^ 4.0)));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(1.0 / Float64(t * (k ^ 2.0))) / Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5.6e+74)
		tmp = (l * l) * (((-0.3333333333333333 * ((k ^ 2.0) / t)) + (2.0 * (1.0 / t))) / (k ^ 4.0));
	else
		tmp = (l * l) * (2.0 * ((1.0 / (t * (k ^ 2.0))) / (0.5 - (cos((2.0 * k)) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 5.6e+74], N[(N[(l * l), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(N[Power[k, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(1.0 / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.60000000000000003e74

   1. Initial program 33.6%

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

   3. Simplified 42.1%

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

   5. Taylor expanded in k around 0 56.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   if 5.60000000000000003e74 < k

   1. Initial program 32.0%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in t around 0 59.9%

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

      1. associate-*r* 59.9%

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

      2. associate-/r* 59.9%

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

   7. Simplified 59.9%

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

      1. unpow2 59.9%

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

      2. sin-mult 60.0%

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

   9. Applied egg-rr 60.0%

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

       1. div-sub 60.0%

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

       2. +-inverses 60.0%

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

       3. cos-0 60.0%

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

       4. metadata-eval 60.0%

          \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2} \cdot t}}{\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

       5. count-2 60.0%

          \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2} \cdot t}}{0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

       6. *-commutative 60.0%

          \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2} \cdot t}}{0.5 - \frac{\cos \color{blue}{\left(k \cdot 2\right)}}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

   11. Simplified 60.0%

       \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2} \cdot t}}{\color{blue}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   12. Taylor expanded in k around 0 49.6%

       \[\leadsto \left(2 \cdot \frac{\color{blue}{\frac{1}{{k}^{2} \cdot t}}}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.6 \cdot 10^{+74}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{1}{t \cdot {k}^{2}}}{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 55.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{+74}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\frac{1}{{k}^{2}}}{t}}{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5.5e+74)
   (*
    (* l l)
    (/
     (+ (* -0.3333333333333333 (/ (pow k 2.0) t)) (* 2.0 (/ 1.0 t)))
     (pow k 4.0)))
   (*
    (* l l)
    (* 2.0 (/ (/ (/ 1.0 (pow k 2.0)) t) (- 0.5 (/ (cos (* 2.0 k)) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.5e+74) {
		tmp = (l * l) * (((-0.3333333333333333 * (pow(k, 2.0) / t)) + (2.0 * (1.0 / t))) / pow(k, 4.0));
	} else {
		tmp = (l * l) * (2.0 * (((1.0 / pow(k, 2.0)) / t) / (0.5 - (cos((2.0 * k)) / 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.5d+74) then
        tmp = (l * l) * ((((-0.3333333333333333d0) * ((k ** 2.0d0) / t)) + (2.0d0 * (1.0d0 / t))) / (k ** 4.0d0))
    else
        tmp = (l * l) * (2.0d0 * (((1.0d0 / (k ** 2.0d0)) / t) / (0.5d0 - (cos((2.0d0 * k)) / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.5e+74) {
		tmp = (l * l) * (((-0.3333333333333333 * (Math.pow(k, 2.0) / t)) + (2.0 * (1.0 / t))) / Math.pow(k, 4.0));
	} else {
		tmp = (l * l) * (2.0 * (((1.0 / Math.pow(k, 2.0)) / t) / (0.5 - (Math.cos((2.0 * k)) / 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 5.5e+74:
		tmp = (l * l) * (((-0.3333333333333333 * (math.pow(k, 2.0) / t)) + (2.0 * (1.0 / t))) / math.pow(k, 4.0))
	else:
		tmp = (l * l) * (2.0 * (((1.0 / math.pow(k, 2.0)) / t) / (0.5 - (math.cos((2.0 * k)) / 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 5.5e+74)
		tmp = Float64(Float64(l * l) * Float64(Float64(Float64(-0.3333333333333333 * Float64((k ^ 2.0) / t)) + Float64(2.0 * Float64(1.0 / t))) / (k ^ 4.0)));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(Float64(1.0 / (k ^ 2.0)) / t) / Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5.5e+74)
		tmp = (l * l) * (((-0.3333333333333333 * ((k ^ 2.0) / t)) + (2.0 * (1.0 / t))) / (k ^ 4.0));
	else
		tmp = (l * l) * (2.0 * (((1.0 / (k ^ 2.0)) / t) / (0.5 - (cos((2.0 * k)) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 5.5e+74], N[(N[(l * l), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(N[Power[k, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[(1.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.5000000000000003e74

   1. Initial program 33.6%

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

   3. Simplified 42.1%

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

   5. Taylor expanded in k around 0 56.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   if 5.5000000000000003e74 < k

   1. Initial program 32.0%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in t around 0 59.9%

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

      1. associate-*r* 59.9%

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

      2. associate-/r* 59.9%

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

   7. Simplified 59.9%

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

      1. unpow2 59.9%

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

      2. sin-mult 60.0%

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

   9. Applied egg-rr 60.0%

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

       1. div-sub 60.0%

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

       2. +-inverses 60.0%

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

       3. cos-0 60.0%

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

       4. metadata-eval 60.0%

          \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2} \cdot t}}{\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

       5. count-2 60.0%

          \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2} \cdot t}}{0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

       6. *-commutative 60.0%

          \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2} \cdot t}}{0.5 - \frac{\cos \color{blue}{\left(k \cdot 2\right)}}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

   11. Simplified 60.0%

       \[\leadsto \left(2 \cdot \frac{\frac{\cos k}{{k}^{2} \cdot t}}{\color{blue}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   12. Taylor expanded in k around 0 49.6%

       \[\leadsto \left(2 \cdot \frac{\color{blue}{\frac{1}{{k}^{2} \cdot t}}}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]
   13. Step-by-step derivation

       1. associate-/r* 49.6%

          \[\leadsto \left(2 \cdot \frac{\color{blue}{\frac{\frac{1}{{k}^{2}}}{t}}}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]

   14. Simplified 49.6%

       \[\leadsto \left(2 \cdot \frac{\color{blue}{\frac{\frac{1}{{k}^{2}}}{t}}}{0.5 - \frac{\cos \left(k \cdot 2\right)}{2}}\right) \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{+74}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\frac{1}{{k}^{2}}}{t}}{0.5 - \frac{\cos \left(2 \cdot k\right)}{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 55.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+73}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} \cdot \left({\ell}^{2} \cdot {k}^{-4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 5e+73)
   (*
    (* l l)
    (/
     (+ (* -0.3333333333333333 (/ (pow k 2.0) t)) (* 2.0 (/ 1.0 t)))
     (pow k 4.0)))
   (* (/ 2.0 t) (* (pow l 2.0) (pow k -4.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e+73) {
		tmp = (l * l) * (((-0.3333333333333333 * (pow(k, 2.0) / t)) + (2.0 * (1.0 / t))) / pow(k, 4.0));
	} else {
		tmp = (2.0 / t) * (pow(l, 2.0) * pow(k, -4.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5d+73) then
        tmp = (l * l) * ((((-0.3333333333333333d0) * ((k ** 2.0d0) / t)) + (2.0d0 * (1.0d0 / t))) / (k ** 4.0d0))
    else
        tmp = (2.0d0 / t) * ((l ** 2.0d0) * (k ** (-4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e+73) {
		tmp = (l * l) * (((-0.3333333333333333 * (Math.pow(k, 2.0) / t)) + (2.0 * (1.0 / t))) / Math.pow(k, 4.0));
	} else {
		tmp = (2.0 / t) * (Math.pow(l, 2.0) * Math.pow(k, -4.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 5e+73:
		tmp = (l * l) * (((-0.3333333333333333 * (math.pow(k, 2.0) / t)) + (2.0 * (1.0 / t))) / math.pow(k, 4.0))
	else:
		tmp = (2.0 / t) * (math.pow(l, 2.0) * math.pow(k, -4.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 5e+73)
		tmp = Float64(Float64(l * l) * Float64(Float64(Float64(-0.3333333333333333 * Float64((k ^ 2.0) / t)) + Float64(2.0 * Float64(1.0 / t))) / (k ^ 4.0)));
	else
		tmp = Float64(Float64(2.0 / t) * Float64((l ^ 2.0) * (k ^ -4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5e+73)
		tmp = (l * l) * (((-0.3333333333333333 * ((k ^ 2.0) / t)) + (2.0 * (1.0 / t))) / (k ^ 4.0));
	else
		tmp = (2.0 / t) * ((l ^ 2.0) * (k ^ -4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 5e+73], N[(N[(l * l), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(N[Power[k, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.99999999999999976e73

   1. Initial program 33.6%

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

   3. Simplified 42.1%

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

   5. Taylor expanded in k around 0 56.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   if 4.99999999999999976e73 < k

   1. Initial program 32.0%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in k around 0 45.7%

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

      1. *-commutative 45.7%

         \[\leadsto \frac{2}{\color{blue}{t \cdot {k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

      2. associate-/r* 45.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   7. Simplified 45.7%

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

      1. pow1 45.7%

         \[\leadsto \color{blue}{{\left(\frac{\frac{2}{t}}{{k}^{4}} \cdot \left(\ell \cdot \ell\right)\right)}^{1}} \]

      2. div-inv 45.7%

         \[\leadsto {\left(\color{blue}{\left(\frac{2}{t} \cdot \frac{1}{{k}^{4}}\right)} \cdot \left(\ell \cdot \ell\right)\right)}^{1} \]

      3. pow-flip 45.7%

         \[\leadsto {\left(\left(\frac{2}{t} \cdot \color{blue}{{k}^{\left(-4\right)}}\right) \cdot \left(\ell \cdot \ell\right)\right)}^{1} \]

      4. metadata-eval 45.7%

         \[\leadsto {\left(\left(\frac{2}{t} \cdot {k}^{\color{blue}{-4}}\right) \cdot \left(\ell \cdot \ell\right)\right)}^{1} \]

      5. pow2 45.7%

         \[\leadsto {\left(\left(\frac{2}{t} \cdot {k}^{-4}\right) \cdot \color{blue}{{\ell}^{2}}\right)}^{1} \]

   9. Applied egg-rr 45.7%

      \[\leadsto \color{blue}{{\left(\left(\frac{2}{t} \cdot {k}^{-4}\right) \cdot {\ell}^{2}\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 45.7%

          \[\leadsto \color{blue}{\left(\frac{2}{t} \cdot {k}^{-4}\right) \cdot {\ell}^{2}} \]

       2. associate-*l* 45.8%

          \[\leadsto \color{blue}{\frac{2}{t} \cdot \left({k}^{-4} \cdot {\ell}^{2}\right)} \]

   11. Simplified 45.8%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+73}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} \cdot \left({\ell}^{2} \cdot {k}^{-4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 63.2% 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 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 41.2%

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

5. Taylor expanded in k around 0 60.5%

   \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \left(t + 0.16666666666666666 \cdot \left({k}^{2} \cdot t\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]

6. Final simplification 60.5%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{4} \cdot \left(t + \left(t \cdot {k}^{2}\right) \cdot 0.16666666666666666\right)} \]
7. Add Preprocessing

Alternative 25: 63.2% 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 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 41.2%

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

5. Taylor expanded in k around 0 60.5%

   \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot t}} \cdot \left(\ell \cdot \ell\right) \]

6. Final simplification 60.5%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(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))))