Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.1% → 93.1%

Time: 22.1s

Alternatives: 19

Speedup: 3.8×

• Could not converge on a ground truth

  1. reg0 = (bf #e6.1508834540591852402258155052550137043)
  2. reg1 = (bf 48626664558662180977678813559231017385980)
  3. reg2 = (bf "3.456494347097202893240306012236189754129e230")

• 40.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.1% 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: 93.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 35.0%

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

3. Simplified 35.0%

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

6. Applied egg-rr 81.5%

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

   1. associate-*l/ 77.6%

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

   2. associate-*r/ 71.0%

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

   3. unpow2 71.0%

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

   4. associate-/r/ 71.0%

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

8. Simplified 71.0%

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

   1. add-cube-cbrt 70.9%

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

   2. pow3 70.9%

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

10. Applied egg-rr 88.6%

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

    1. cbrt-div 94.1%

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

12. Applied egg-rr 94.1%

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

Alternative 2: 88.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 35.0%

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

3. Simplified 35.0%

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

6. Applied egg-rr 81.5%

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

   1. associate-*l/ 77.6%

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

   2. associate-*r/ 71.0%

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

   3. unpow2 71.0%

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

   4. associate-/r/ 71.0%

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

8. Simplified 71.0%

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

   1. add-cube-cbrt 70.9%

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

   2. pow3 70.9%

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

10. Applied egg-rr 88.6%

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

    1. *-commutative 88.6%

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

    2. cbrt-prod 89.3%

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

12. Applied egg-rr 89.3%

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

13. Final simplification 89.3%

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

Alternative 3: 87.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 35.0%

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

3. Simplified 35.0%

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

6. Applied egg-rr 81.5%

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

   1. associate-*l/ 77.6%

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

   2. associate-*r/ 71.0%

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

   3. unpow2 71.0%

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

   4. associate-/r/ 71.0%

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

8. Simplified 71.0%

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

   1. add-cube-cbrt 70.9%

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

   2. pow3 70.9%

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

10. Applied egg-rr 88.6%

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

Alternative 4: 82.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ t_2 := t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-310}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\frac{\sqrt{2} \cdot t}{k}}\right)}^{2}}{t\_2 \cdot \sqrt[3]{{k}^{2}}}\right)}^{3}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+295}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2 \cdot {\left(\frac{t}{k}\right)}^{2}}}{\sqrt[3]{t\_1} \cdot t\_2}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))) (t_2 (* t (pow (cbrt l) -2.0))))
   (if (<= (* l l) 1e-310)
     (pow
      (/ (pow (cbrt (/ (* (sqrt 2.0) t) k)) 2.0) (* t_2 (cbrt (pow k 2.0))))
      3.0)
     (if (<= (* l l) 2e+295)
       (* (* l l) (/ (* 2.0 (pow k -2.0)) (* t t_1)))
       (pow (/ (cbrt (* 2.0 (pow (/ t k) 2.0))) (* (cbrt t_1) t_2)) 3.0)))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double t_2 = t * pow(cbrt(l), -2.0);
	double tmp;
	if ((l * l) <= 1e-310) {
		tmp = pow((pow(cbrt(((sqrt(2.0) * t) / k)), 2.0) / (t_2 * cbrt(pow(k, 2.0)))), 3.0);
	} else if ((l * l) <= 2e+295) {
		tmp = (l * l) * ((2.0 * pow(k, -2.0)) / (t * t_1));
	} else {
		tmp = pow((cbrt((2.0 * pow((t / k), 2.0))) / (cbrt(t_1) * t_2)), 3.0);
	}
	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 = t * Math.pow(Math.cbrt(l), -2.0);
	double tmp;
	if ((l * l) <= 1e-310) {
		tmp = Math.pow((Math.pow(Math.cbrt(((Math.sqrt(2.0) * t) / k)), 2.0) / (t_2 * Math.cbrt(Math.pow(k, 2.0)))), 3.0);
	} else if ((l * l) <= 2e+295) {
		tmp = (l * l) * ((2.0 * Math.pow(k, -2.0)) / (t * t_1));
	} else {
		tmp = Math.pow((Math.cbrt((2.0 * Math.pow((t / k), 2.0))) / (Math.cbrt(t_1) * t_2)), 3.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	t_2 = Float64(t * (cbrt(l) ^ -2.0))
	tmp = 0.0
	if (Float64(l * l) <= 1e-310)
		tmp = Float64((cbrt(Float64(Float64(sqrt(2.0) * t) / k)) ^ 2.0) / Float64(t_2 * cbrt((k ^ 2.0)))) ^ 3.0;
	elseif (Float64(l * l) <= 2e+295)
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 * (k ^ -2.0)) / Float64(t * t_1)));
	else
		tmp = Float64(cbrt(Float64(2.0 * (Float64(t / k) ^ 2.0))) / Float64(cbrt(t_1) * t_2)) ^ 3.0;
	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[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 1e-310], N[Power[N[(N[Power[N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / k), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$2 * N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+295], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[(2.0 * N[Power[N[(t / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(N[Power[t$95$1, 1/3], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.999999999999969e-311

   1. Initial program 26.1%

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

   3. Simplified 26.1%

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

   6. Applied egg-rr 86.4%

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

      1. associate-*l/ 84.9%

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

      2. associate-*r/ 81.2%

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

      3. unpow2 81.2%

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

      4. associate-/r/ 81.2%

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

   8. Simplified 81.2%

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

      1. add-cube-cbrt 81.2%

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

      2. pow3 81.2%

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

   10. Applied egg-rr 90.9%

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

   11. Taylor expanded in k around 0 89.5%

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

   if 9.999999999999969e-311 < (*.f64 l l) < 2e295

   1. Initial program 43.3%

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

   3. Simplified 55.2%

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

      1. add-log-exp 37.0%

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

      2. exp-prod 39.5%

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

   6. Applied egg-rr 39.5%

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

   7. Taylor expanded in k around inf 87.4%

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

      1. associate-/r* 87.5%

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

   9. Simplified 87.5%

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

       1. *-un-lft-identity 87.5%

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

       2. div-inv 87.5%

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

       3. pow-flip 88.3%

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

       4. metadata-eval 88.3%

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

   11. Applied egg-rr 88.3%

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

       1. *-lft-identity 88.3%

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

   13. Simplified 88.3%

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

   if 2e295 < (*.f64 l l)

   1. Initial program 28.6%

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

   3. Simplified 28.6%

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

   6. Applied egg-rr 83.9%

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

      1. associate-*l/ 78.0%

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

      2. associate-*r/ 63.0%

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

      3. unpow2 63.0%

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

      4. associate-/r/ 62.9%

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

   8. Simplified 62.9%

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

      1. add-cube-cbrt 62.9%

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

      2. pow3 62.9%

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

   10. Applied egg-rr 90.7%

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

       1. div-inv 90.6%

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

       2. unpow2 90.6%

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

       3. cbrt-unprod 64.7%

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

       4. pow2 64.7%

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

       5. associate-/l* 64.7%

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

   12. Applied egg-rr 64.7%

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

       1. associate-*r/ 64.7%

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

       2. *-rgt-identity 64.7%

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

       3. unpow2 64.7%

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

       4. swap-sqr 64.8%

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

       5. rem-square-sqrt 64.8%

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

       6. unpow2 64.8%

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

       7. *-commutative 64.8%

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

   14. Simplified 64.8%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-310}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\frac{\sqrt{2} \cdot t}{k}}\right)}^{2}}{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{{k}^{2}}}\right)}^{3}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+295}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2 \cdot {\left(\frac{t}{k}\right)}^{2}}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.7 \cdot 10^{-155}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\frac{\sqrt{2} \cdot t}{k}}\right)}^{2}}{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{{k}^{2}}}\right)}^{3}\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{+149}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 2.7e-155)
   (pow
    (/
     (pow (cbrt (/ (* (sqrt 2.0) t) k)) 2.0)
     (* (* t (pow (cbrt l) -2.0)) (cbrt (pow k 2.0))))
    3.0)
   (if (<= l 1.85e+149)
     (* (* l l) (/ (* 2.0 (pow k -2.0)) (* t (* (sin k) (tan k)))))
     (/
      2.0
      (pow
       (*
        (/ t (pow (cbrt l) 2.0))
        (cbrt (* (sin k) (* (tan k) (pow (/ k t) 2.0)))))
       3.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 2.7e-155) {
		tmp = pow((pow(cbrt(((sqrt(2.0) * t) / k)), 2.0) / ((t * pow(cbrt(l), -2.0)) * cbrt(pow(k, 2.0)))), 3.0);
	} else if (l <= 1.85e+149) {
		tmp = (l * l) * ((2.0 * pow(k, -2.0)) / (t * (sin(k) * tan(k))));
	} else {
		tmp = 2.0 / pow(((t / pow(cbrt(l), 2.0)) * cbrt((sin(k) * (tan(k) * pow((k / t), 2.0))))), 3.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 2.7e-155) {
		tmp = Math.pow((Math.pow(Math.cbrt(((Math.sqrt(2.0) * t) / k)), 2.0) / ((t * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt(Math.pow(k, 2.0)))), 3.0);
	} else if (l <= 1.85e+149) {
		tmp = (l * l) * ((2.0 * Math.pow(k, -2.0)) / (t * (Math.sin(k) * Math.tan(k))));
	} else {
		tmp = 2.0 / Math.pow(((t / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((Math.sin(k) * (Math.tan(k) * Math.pow((k / t), 2.0))))), 3.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 2.7e-155)
		tmp = Float64((cbrt(Float64(Float64(sqrt(2.0) * t) / k)) ^ 2.0) / Float64(Float64(t * (cbrt(l) ^ -2.0)) * cbrt((k ^ 2.0)))) ^ 3.0;
	elseif (l <= 1.85e+149)
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 * (k ^ -2.0)) / Float64(t * Float64(sin(k) * tan(k)))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t / (cbrt(l) ^ 2.0)) * cbrt(Float64(sin(k) * Float64(tan(k) * (Float64(k / t) ^ 2.0))))) ^ 3.0));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 2.7e-155], N[Power[N[(N[Power[N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / k), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[l, 1.85e+149], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] / N[(t * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.69999999999999981e-155

   1. Initial program 32.6%

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

   3. Simplified 32.6%

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

   6. Applied egg-rr 84.2%

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

      1. associate-*l/ 81.9%

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

      2. associate-*r/ 76.0%

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

      3. unpow2 76.0%

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

      4. associate-/r/ 76.0%

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

   8. Simplified 76.0%

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

      1. add-cube-cbrt 76.0%

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

      2. pow3 76.0%

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

   10. Applied egg-rr 91.5%

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

   11. Taylor expanded in k around 0 70.8%

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

   if 2.69999999999999981e-155 < l < 1.84999999999999989e149

   1. Initial program 41.9%

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

   3. Simplified 52.9%

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

      1. add-log-exp 35.7%

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

      2. exp-prod 36.9%

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

   6. Applied egg-rr 36.9%

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

   7. Taylor expanded in k around inf 85.3%

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

      1. associate-/r* 85.4%

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

   9. Simplified 85.4%

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

       1. *-un-lft-identity 85.4%

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

       2. div-inv 85.4%

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

       3. pow-flip 85.5%

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

       4. metadata-eval 85.5%

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

   11. Applied egg-rr 85.5%

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

       1. *-lft-identity 85.5%

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

   13. Simplified 85.5%

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

   if 1.84999999999999989e149 < l

   1. Initial program 31.3%

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

   3. Simplified 31.3%

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

      1. add-cube-cbrt 31.3%

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

      2. pow3 31.3%

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

   6. Applied egg-rr 62.7%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.7 \cdot 10^{-155}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\frac{\sqrt{2} \cdot t}{k}}\right)}^{2}}{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{{k}^{2}}}\right)}^{3}\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{+149}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq 1.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{{\left(\sqrt[3]{t\_1} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{+157}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{t\_1 \cdot {\left(\frac{t}{k}\right)}^{-2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= k 1.2e-108)
     (/
      (/ 2.0 (pow (/ k t) 2.0))
      (pow (* (cbrt t_1) (/ t (pow (cbrt l) 2.0))) 3.0))
     (if (<= k 4.1e+157)
       (* (* l l) (/ (* 2.0 (pow k -2.0)) (* t t_1)))
       (/
        2.0
        (pow
         (* (* t (pow (cbrt l) -2.0)) (cbrt (* t_1 (pow (/ t k) -2.0))))
         3.0))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (k <= 1.2e-108) {
		tmp = (2.0 / pow((k / t), 2.0)) / pow((cbrt(t_1) * (t / pow(cbrt(l), 2.0))), 3.0);
	} else if (k <= 4.1e+157) {
		tmp = (l * l) * ((2.0 * pow(k, -2.0)) / (t * t_1));
	} else {
		tmp = 2.0 / pow(((t * pow(cbrt(l), -2.0)) * cbrt((t_1 * pow((t / k), -2.0)))), 3.0);
	}
	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 <= 1.2e-108) {
		tmp = (2.0 / Math.pow((k / t), 2.0)) / Math.pow((Math.cbrt(t_1) * (t / Math.pow(Math.cbrt(l), 2.0))), 3.0);
	} else if (k <= 4.1e+157) {
		tmp = (l * l) * ((2.0 * Math.pow(k, -2.0)) / (t * t_1));
	} else {
		tmp = 2.0 / Math.pow(((t * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt((t_1 * Math.pow((t / k), -2.0)))), 3.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (k <= 1.2e-108)
		tmp = Float64(Float64(2.0 / (Float64(k / t) ^ 2.0)) / (Float64(cbrt(t_1) * Float64(t / (cbrt(l) ^ 2.0))) ^ 3.0));
	elseif (k <= 4.1e+157)
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 * (k ^ -2.0)) / Float64(t * t_1)));
	else
		tmp = Float64(2.0 / (Float64(Float64(t * (cbrt(l) ^ -2.0)) * cbrt(Float64(t_1 * (Float64(t / k) ^ -2.0)))) ^ 3.0));
	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, 1.2e-108], N[(N[(2.0 / N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[Power[t$95$1, 1/3], $MachinePrecision] * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.1e+157], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$1 * N[Power[N[(t / k), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.20000000000000009e-108

   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 41.5%

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

      1. add-cube-cbrt 41.5%

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

      2. pow3 41.5%

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

      3. cbrt-prod 41.5%

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

      4. cbrt-div 41.5%

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

      5. rem-cbrt-cube 59.6%

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

      6. cbrt-prod 71.2%

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

      7. pow2 71.2%

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

   6. Applied egg-rr 71.2%

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

   if 1.20000000000000009e-108 < k < 4.10000000000000016e157

   1. Initial program 25.2%

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

   3. Simplified 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. Step-by-step derivation

      1. add-log-exp 19.8%

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

      2. exp-prod 23.5%

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

   6. Applied egg-rr 23.5%

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

   7. Taylor expanded in k around inf 77.0%

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

      1. associate-/r* 77.2%

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

   9. Simplified 77.2%

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

       1. *-un-lft-identity 77.2%

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

       2. div-inv 77.2%

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

       3. pow-flip 78.7%

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

       4. metadata-eval 78.7%

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

   11. Applied egg-rr 78.7%

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

       1. *-lft-identity 78.7%

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

   13. Simplified 78.7%

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

   if 4.10000000000000016e157 < k

   1. Initial program 47.1%

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

   3. Simplified 47.1%

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

      1. +-commutative 47.1%

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

      2. associate-+l- 50.0%

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

      3. metadata-eval 50.0%

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

      4. --rgt-identity 50.0%

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

      5. unpow2 50.0%

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

      6. clear-num 50.0%

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

      7. clear-num 50.0%

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

      8. frac-times 50.0%

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

      9. metadata-eval 50.0%

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

   6. Applied egg-rr 50.0%

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

      1. add-cube-cbrt 50.0%

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

      2. pow3 50.0%

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

   8. Applied egg-rr 72.8%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{+157}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{t}{k}\right)}^{-2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq 9.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{t\_1} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \frac{1}{\frac{t}{k} \cdot \frac{t}{k}}}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{+157}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{t\_1 \cdot {\left(\frac{t}{k}\right)}^{-2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= k 9.5e-109)
     (/
      2.0
      (*
       (pow (* (cbrt t_1) (/ t (pow (cbrt l) 2.0))) 3.0)
       (/ 1.0 (* (/ t k) (/ t k)))))
     (if (<= k 4.1e+157)
       (* (* l l) (/ (* 2.0 (pow k -2.0)) (* t t_1)))
       (/
        2.0
        (pow
         (* (* t (pow (cbrt l) -2.0)) (cbrt (* t_1 (pow (/ t k) -2.0))))
         3.0))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (k <= 9.5e-109) {
		tmp = 2.0 / (pow((cbrt(t_1) * (t / pow(cbrt(l), 2.0))), 3.0) * (1.0 / ((t / k) * (t / k))));
	} else if (k <= 4.1e+157) {
		tmp = (l * l) * ((2.0 * pow(k, -2.0)) / (t * t_1));
	} else {
		tmp = 2.0 / pow(((t * pow(cbrt(l), -2.0)) * cbrt((t_1 * pow((t / k), -2.0)))), 3.0);
	}
	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-109) {
		tmp = 2.0 / (Math.pow((Math.cbrt(t_1) * (t / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (1.0 / ((t / k) * (t / k))));
	} else if (k <= 4.1e+157) {
		tmp = (l * l) * ((2.0 * Math.pow(k, -2.0)) / (t * t_1));
	} else {
		tmp = 2.0 / Math.pow(((t * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt((t_1 * Math.pow((t / k), -2.0)))), 3.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (k <= 9.5e-109)
		tmp = Float64(2.0 / Float64((Float64(cbrt(t_1) * Float64(t / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(1.0 / Float64(Float64(t / k) * Float64(t / k)))));
	elseif (k <= 4.1e+157)
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 * (k ^ -2.0)) / Float64(t * t_1)));
	else
		tmp = Float64(2.0 / (Float64(Float64(t * (cbrt(l) ^ -2.0)) * cbrt(Float64(t_1 * (Float64(t / k) ^ -2.0)))) ^ 3.0));
	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-109], N[(2.0 / N[(N[Power[N[(N[Power[t$95$1, 1/3], $MachinePrecision] * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(1.0 / N[(N[(t / k), $MachinePrecision] * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.1e+157], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$1 * N[Power[N[(t / k), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 9.49999999999999933e-109

   1. Initial program 36.3%

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

   3. Simplified 36.3%

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

      1. +-commutative 36.3%

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

      2. associate-+l- 41.5%

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

      3. metadata-eval 41.5%

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

      4. --rgt-identity 41.5%

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

      5. unpow2 41.5%

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

      6. clear-num 41.5%

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

      7. clear-num 41.5%

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

      8. frac-times 41.5%

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

      9. metadata-eval 41.5%

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

   6. Applied egg-rr 41.5%

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

      1. add-cube-cbrt 41.5%

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

      2. pow3 41.5%

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

      3. cbrt-prod 41.5%

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

      4. cbrt-div 41.5%

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

      5. rem-cbrt-cube 59.6%

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

      6. cbrt-prod 71.2%

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

      7. pow2 71.2%

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

   8. Applied egg-rr 71.2%

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

   if 9.49999999999999933e-109 < k < 4.10000000000000016e157

   1. Initial program 25.2%

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

   3. Simplified 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. Step-by-step derivation

      1. add-log-exp 19.8%

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

      2. exp-prod 23.5%

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

   6. Applied egg-rr 23.5%

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

   7. Taylor expanded in k around inf 77.0%

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

      1. associate-/r* 77.2%

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

   9. Simplified 77.2%

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

       1. *-un-lft-identity 77.2%

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

       2. div-inv 77.2%

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

       3. pow-flip 78.7%

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

       4. metadata-eval 78.7%

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

   11. Applied egg-rr 78.7%

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

       1. *-lft-identity 78.7%

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

   13. Simplified 78.7%

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

   if 4.10000000000000016e157 < k

   1. Initial program 47.1%

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

   3. Simplified 47.1%

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

      1. +-commutative 47.1%

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

      2. associate-+l- 50.0%

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

      3. metadata-eval 50.0%

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

      4. --rgt-identity 50.0%

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

      5. unpow2 50.0%

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

      6. clear-num 50.0%

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

      7. clear-num 50.0%

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

      8. frac-times 50.0%

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

      9. metadata-eval 50.0%

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

   6. Applied egg-rr 50.0%

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

      1. add-cube-cbrt 50.0%

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

      2. pow3 50.0%

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

   8. Applied egg-rr 72.8%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \frac{1}{\frac{t}{k} \cdot \frac{t}{k}}}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{+157}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{t}{k}\right)}^{-2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq 1.3 \cdot 10^{-108}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{t\_1} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \frac{1}{\frac{t}{k} \cdot \frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= k 1.3e-108)
     (/
      2.0
      (*
       (pow (* (cbrt t_1) (/ t (pow (cbrt l) 2.0))) 3.0)
       (/ 1.0 (* (/ t k) (/ t k)))))
     (* (* l l) (/ (* 2.0 (pow k -2.0)) (* t t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (k <= 1.3e-108) {
		tmp = 2.0 / (pow((cbrt(t_1) * (t / pow(cbrt(l), 2.0))), 3.0) * (1.0 / ((t / k) * (t / k))));
	} else {
		tmp = (l * l) * ((2.0 * pow(k, -2.0)) / (t * t_1));
	}
	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 <= 1.3e-108) {
		tmp = 2.0 / (Math.pow((Math.cbrt(t_1) * (t / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (1.0 / ((t / k) * (t / k))));
	} else {
		tmp = (l * l) * ((2.0 * Math.pow(k, -2.0)) / (t * t_1));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (k <= 1.3e-108)
		tmp = Float64(2.0 / Float64((Float64(cbrt(t_1) * Float64(t / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(1.0 / Float64(Float64(t / k) * Float64(t / k)))));
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 * (k ^ -2.0)) / Float64(t * t_1)));
	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, 1.3e-108], N[(2.0 / N[(N[Power[N[(N[Power[t$95$1, 1/3], $MachinePrecision] * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(1.0 / N[(N[(t / k), $MachinePrecision] * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.29999999999999992e-108

   1. Initial program 36.3%

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

   3. Simplified 36.3%

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

      1. +-commutative 36.3%

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

      2. associate-+l- 41.5%

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

      3. metadata-eval 41.5%

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

      4. --rgt-identity 41.5%

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

      5. unpow2 41.5%

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

      6. clear-num 41.5%

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

      7. clear-num 41.5%

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

      8. frac-times 41.5%

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

      9. metadata-eval 41.5%

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

   6. Applied egg-rr 41.5%

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

      1. add-cube-cbrt 41.5%

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

      2. pow3 41.5%

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

      3. cbrt-prod 41.5%

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

      4. cbrt-div 41.5%

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

      5. rem-cbrt-cube 59.6%

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

      6. cbrt-prod 71.2%

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

      7. pow2 71.2%

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

   8. Applied egg-rr 71.2%

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

   if 1.29999999999999992e-108 < k

   1. Initial program 32.8%

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

   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. Step-by-step derivation

      1. add-log-exp 30.9%

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

      2. exp-prod 28.8%

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

   6. Applied egg-rr 28.8%

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

   7. Taylor expanded in k around inf 72.4%

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

      1. associate-/r* 72.5%

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

   9. Simplified 72.5%

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

       1. *-un-lft-identity 72.5%

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

       2. div-inv 72.5%

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

       3. pow-flip 73.5%

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

       4. metadata-eval 73.5%

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

   11. Applied egg-rr 73.5%

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

       1. *-lft-identity 73.5%

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

   13. Simplified 73.5%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-108}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \frac{1}{\frac{t}{k} \cdot \frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;\ell \leq 5.1 \cdot 10^{-204}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{t\_1 \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= l 5.1e-204)
     (/ (/ 2.0 (pow (/ k t) 2.0)) (* t_1 (pow (/ t (pow (cbrt l) 2.0)) 3.0)))
     (* (* l l) (/ (* 2.0 (pow k -2.0)) (* t t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (l <= 5.1e-204) {
		tmp = (2.0 / pow((k / t), 2.0)) / (t_1 * pow((t / pow(cbrt(l), 2.0)), 3.0));
	} else {
		tmp = (l * l) * ((2.0 * pow(k, -2.0)) / (t * t_1));
	}
	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 (l <= 5.1e-204) {
		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) * ((2.0 * Math.pow(k, -2.0)) / (t * t_1));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (l <= 5.1e-204)
		tmp = Float64(Float64(2.0 / (Float64(k / t) ^ 2.0)) / Float64(t_1 * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0)));
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 * (k ^ -2.0)) / Float64(t * t_1)));
	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[l, 5.1e-204], N[(N[(2.0 / N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 5.10000000000000027e-204

   1. Initial program 32.0%

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

      1. *-commutative 32.0%

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

      2. associate-/r* 32.0%

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

   3. Simplified 39.3%

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

      1. add-cube-cbrt 39.2%

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

      2. associate-*l* 39.2%

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

      3. cbrt-div 39.3%

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

      4. rem-cbrt-cube 39.3%

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

      5. cbrt-prod 39.2%

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

      6. pow2 39.2%

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

      7. pow2 39.2%

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

      8. cbrt-div 39.2%

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

      9. rem-cbrt-cube 52.9%

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

      10. cbrt-prod 62.9%

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

      11. pow2 62.9%

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

   6. Applied egg-rr 62.9%

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

      1. unpow2 62.9%

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

      2. cube-mult 62.9%

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

   8. Simplified 62.9%

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

   if 5.10000000000000027e-204 < l

   1. Initial program 38.8%

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

   3. Simplified 45.6%

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

      1. add-log-exp 30.4%

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

      2. exp-prod 33.9%

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

   6. Applied egg-rr 33.9%

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

   7. Taylor expanded in k around inf 76.7%

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

      1. associate-/r* 76.8%

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

   9. Simplified 76.8%

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

       1. *-un-lft-identity 76.8%

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

       2. div-inv 76.8%

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

       3. pow-flip 76.8%

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

       4. metadata-eval 76.8%

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

   11. Applied egg-rr 76.8%

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

       1. *-lft-identity 76.8%

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

   13. Simplified 76.8%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.1 \cdot 10^{-204}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;\ell \leq 8.2 \cdot 10^{-204}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}} \cdot \left(t\_1 \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= l 8.2e-204)
     (/
      2.0
      (*
       (/ 1.0 (* (/ t k) (/ t k)))
       (* t_1 (pow (/ t (pow (cbrt l) 2.0)) 3.0))))
     (* (* l l) (/ (* 2.0 (pow k -2.0)) (* t t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (l <= 8.2e-204) {
		tmp = 2.0 / ((1.0 / ((t / k) * (t / k))) * (t_1 * pow((t / pow(cbrt(l), 2.0)), 3.0)));
	} else {
		tmp = (l * l) * ((2.0 * pow(k, -2.0)) / (t * t_1));
	}
	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 (l <= 8.2e-204) {
		tmp = 2.0 / ((1.0 / ((t / k) * (t / k))) * (t_1 * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0)));
	} else {
		tmp = (l * l) * ((2.0 * Math.pow(k, -2.0)) / (t * t_1));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (l <= 8.2e-204)
		tmp = Float64(2.0 / Float64(Float64(1.0 / Float64(Float64(t / k) * Float64(t / k))) * Float64(t_1 * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0))));
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 * (k ^ -2.0)) / Float64(t * t_1)));
	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[l, 8.2e-204], N[(2.0 / N[(N[(1.0 / N[(N[(t / k), $MachinePrecision] * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 8.2000000000000002e-204

   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 32.0%

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

      1. +-commutative 32.0%

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

      2. associate-+l- 39.4%

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

      3. metadata-eval 39.4%

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

      4. --rgt-identity 39.4%

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

      5. unpow2 39.4%

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

      6. clear-num 39.4%

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

      7. clear-num 39.4%

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

      8. frac-times 39.3%

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

      9. metadata-eval 39.3%

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

   6. Applied egg-rr 39.3%

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

      1. add-cube-cbrt 39.2%

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

      2. associate-*l* 39.2%

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

      3. cbrt-div 39.3%

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

      4. rem-cbrt-cube 39.3%

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

      5. cbrt-prod 39.2%

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

      6. pow2 39.2%

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

      7. pow2 39.2%

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

      8. cbrt-div 39.2%

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

      9. rem-cbrt-cube 52.9%

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

      10. cbrt-prod 62.9%

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

      11. pow2 62.9%

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

   8. Applied egg-rr 62.9%

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

      1. unpow2 62.9%

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

      2. cube-mult 62.9%

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

   10. Simplified 62.9%

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

   if 8.2000000000000002e-204 < l

   1. Initial program 38.8%

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

   3. Simplified 45.6%

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

      1. add-log-exp 30.4%

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

      2. exp-prod 33.9%

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

   6. Applied egg-rr 33.9%

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

   7. Taylor expanded in k around inf 76.7%

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

      1. associate-/r* 76.8%

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

   9. Simplified 76.8%

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

       1. *-un-lft-identity 76.8%

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

       2. div-inv 76.8%

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

       3. pow-flip 76.8%

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

       4. metadata-eval 76.8%

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

   11. Applied egg-rr 76.8%

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

       1. *-lft-identity 76.8%

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

   13. Simplified 76.8%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.2 \cdot 10^{-204}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;\ell \leq 1.4 \cdot 10^{-204}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}} \cdot \left(t\_1 \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= l 1.4e-204)
     (/
      2.0
      (* (/ 1.0 (* (/ t k) (/ t k))) (* t_1 (pow (/ (pow t 1.5) l) 2.0))))
     (* (* l l) (/ (* 2.0 (pow k -2.0)) (* t t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (l <= 1.4e-204) {
		tmp = 2.0 / ((1.0 / ((t / k) * (t / k))) * (t_1 * pow((pow(t, 1.5) / l), 2.0)));
	} else {
		tmp = (l * l) * ((2.0 * pow(k, -2.0)) / (t * t_1));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) * tan(k)
    if (l <= 1.4d-204) then
        tmp = 2.0d0 / ((1.0d0 / ((t / k) * (t / k))) * (t_1 * (((t ** 1.5d0) / l) ** 2.0d0)))
    else
        tmp = (l * l) * ((2.0d0 * (k ** (-2.0d0))) / (t * t_1))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (l <= 1.4e-204) {
		tmp = 2.0 / ((1.0 / ((t / k) * (t / k))) * (t_1 * Math.pow((Math.pow(t, 1.5) / l), 2.0)));
	} else {
		tmp = (l * l) * ((2.0 * Math.pow(k, -2.0)) / (t * t_1));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(k) * math.tan(k)
	tmp = 0
	if l <= 1.4e-204:
		tmp = 2.0 / ((1.0 / ((t / k) * (t / k))) * (t_1 * math.pow((math.pow(t, 1.5) / l), 2.0)))
	else:
		tmp = (l * l) * ((2.0 * math.pow(k, -2.0)) / (t * t_1))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (l <= 1.4e-204)
		tmp = Float64(2.0 / Float64(Float64(1.0 / Float64(Float64(t / k) * Float64(t / k))) * Float64(t_1 * (Float64((t ^ 1.5) / l) ^ 2.0))));
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 * (k ^ -2.0)) / Float64(t * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) * tan(k);
	tmp = 0.0;
	if (l <= 1.4e-204)
		tmp = 2.0 / ((1.0 / ((t / k) * (t / k))) * (t_1 * (((t ^ 1.5) / l) ^ 2.0)));
	else
		tmp = (l * l) * ((2.0 * (k ^ -2.0)) / (t * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 1.4e-204], N[(2.0 / N[(N[(1.0 / N[(N[(t / k), $MachinePrecision] * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.4e-204

   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 32.0%

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

      1. +-commutative 32.0%

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

      2. associate-+l- 39.4%

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

      3. metadata-eval 39.4%

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

      4. --rgt-identity 39.4%

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

      5. unpow2 39.4%

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

      6. clear-num 39.4%

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

      7. clear-num 39.4%

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

      8. frac-times 39.3%

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

      9. metadata-eval 39.3%

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

   6. Applied egg-rr 39.3%

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

      1. add-sqr-sqrt 19.7%

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

      2. pow2 19.7%

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

      3. sqrt-div 19.7%

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

      4. sqrt-pow1 25.4%

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

      5. metadata-eval 25.4%

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

      6. sqrt-prod 5.6%

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

      7. add-sqr-sqrt 32.3%

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

   8. Applied egg-rr 32.3%

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

   if 1.4e-204 < l

   1. Initial program 38.8%

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

   3. Simplified 45.6%

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

      1. add-log-exp 30.4%

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

      2. exp-prod 33.9%

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

   6. Applied egg-rr 33.9%

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

   7. Taylor expanded in k around inf 76.7%

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

      1. associate-/r* 76.8%

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

   9. Simplified 76.8%

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

       1. *-un-lft-identity 76.8%

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

       2. div-inv 76.8%

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

       3. pow-flip 76.8%

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

       4. metadata-eval 76.8%

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

   11. Applied egg-rr 76.8%

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

       1. *-lft-identity 76.8%

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

   13. Simplified 76.8%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.4 \cdot 10^{-204}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9.8 \cdot 10^{-205}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{{k}^{2} \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 9.8e-205)
   (/ (/ 2.0 (pow (/ k t) 2.0)) (* (pow k 2.0) (pow (/ (pow t 1.5) l) 2.0)))
   (* (* l l) (/ (* 2.0 (pow k -2.0)) (* t (* (sin k) (tan k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 9.8e-205) {
		tmp = (2.0 / pow((k / t), 2.0)) / (pow(k, 2.0) * pow((pow(t, 1.5) / l), 2.0));
	} else {
		tmp = (l * l) * ((2.0 * pow(k, -2.0)) / (t * (sin(k) * tan(k))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 9.8d-205) then
        tmp = (2.0d0 / ((k / t) ** 2.0d0)) / ((k ** 2.0d0) * (((t ** 1.5d0) / l) ** 2.0d0))
    else
        tmp = (l * l) * ((2.0d0 * (k ** (-2.0d0))) / (t * (sin(k) * tan(k))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 9.8e-205) {
		tmp = (2.0 / Math.pow((k / t), 2.0)) / (Math.pow(k, 2.0) * Math.pow((Math.pow(t, 1.5) / l), 2.0));
	} else {
		tmp = (l * l) * ((2.0 * Math.pow(k, -2.0)) / (t * (Math.sin(k) * Math.tan(k))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 9.8e-205:
		tmp = (2.0 / math.pow((k / t), 2.0)) / (math.pow(k, 2.0) * math.pow((math.pow(t, 1.5) / l), 2.0))
	else:
		tmp = (l * l) * ((2.0 * math.pow(k, -2.0)) / (t * (math.sin(k) * math.tan(k))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 9.8e-205)
		tmp = Float64(Float64(2.0 / (Float64(k / t) ^ 2.0)) / Float64((k ^ 2.0) * (Float64((t ^ 1.5) / l) ^ 2.0)));
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 * (k ^ -2.0)) / Float64(t * Float64(sin(k) * tan(k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 9.8e-205)
		tmp = (2.0 / ((k / t) ^ 2.0)) / ((k ^ 2.0) * (((t ^ 1.5) / l) ^ 2.0));
	else
		tmp = (l * l) * ((2.0 * (k ^ -2.0)) / (t * (sin(k) * tan(k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 9.8e-205], N[(N[(2.0 / N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] / N[(t * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.7999999999999995e-205

   1. Initial program 32.0%

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

      1. *-commutative 32.0%

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

      2. associate-/r* 32.0%

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

   3. Simplified 39.3%

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

      1. add-sqr-sqrt 19.7%

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

      2. pow2 19.7%

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

      3. sqrt-div 19.7%

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

      4. sqrt-pow1 25.4%

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

      5. metadata-eval 25.4%

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

      6. sqrt-prod 5.6%

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

      7. add-sqr-sqrt 32.3%

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

   6. Applied egg-rr 32.2%

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

   7. Taylor expanded in k around 0 26.8%

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

   if 9.7999999999999995e-205 < l

   1. Initial program 38.8%

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

   3. Simplified 45.6%

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

      1. add-log-exp 30.4%

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

      2. exp-prod 33.9%

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

   6. Applied egg-rr 33.9%

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

   7. Taylor expanded in k around inf 76.7%

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

      1. associate-/r* 76.8%

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

   9. Simplified 76.8%

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

       1. *-un-lft-identity 76.8%

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

       2. div-inv 76.8%

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

       3. pow-flip 76.8%

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

       4. metadata-eval 76.8%

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

   11. Applied egg-rr 76.8%

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

       1. *-lft-identity 76.8%

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

   13. Simplified 76.8%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.8 \cdot 10^{-205}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{{k}^{2} \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;\ell \leq 5 \cdot 10^{-205}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}} \cdot \left(t\_1 \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= l 5e-205)
     (/
      2.0
      (* (/ 1.0 (* (/ t k) (/ t k))) (* t_1 (* (/ (pow t 2.0) l) (/ t l)))))
     (* (* l l) (/ (* 2.0 (pow k -2.0)) (* t t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (l <= 5e-205) {
		tmp = 2.0 / ((1.0 / ((t / k) * (t / k))) * (t_1 * ((pow(t, 2.0) / l) * (t / l))));
	} else {
		tmp = (l * l) * ((2.0 * pow(k, -2.0)) / (t * t_1));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) * tan(k)
    if (l <= 5d-205) then
        tmp = 2.0d0 / ((1.0d0 / ((t / k) * (t / k))) * (t_1 * (((t ** 2.0d0) / l) * (t / l))))
    else
        tmp = (l * l) * ((2.0d0 * (k ** (-2.0d0))) / (t * t_1))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (l <= 5e-205) {
		tmp = 2.0 / ((1.0 / ((t / k) * (t / k))) * (t_1 * ((Math.pow(t, 2.0) / l) * (t / l))));
	} else {
		tmp = (l * l) * ((2.0 * Math.pow(k, -2.0)) / (t * t_1));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(k) * math.tan(k)
	tmp = 0
	if l <= 5e-205:
		tmp = 2.0 / ((1.0 / ((t / k) * (t / k))) * (t_1 * ((math.pow(t, 2.0) / l) * (t / l))))
	else:
		tmp = (l * l) * ((2.0 * math.pow(k, -2.0)) / (t * t_1))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (l <= 5e-205)
		tmp = Float64(2.0 / Float64(Float64(1.0 / Float64(Float64(t / k) * Float64(t / k))) * Float64(t_1 * Float64(Float64((t ^ 2.0) / l) * Float64(t / l)))));
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 * (k ^ -2.0)) / Float64(t * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) * tan(k);
	tmp = 0.0;
	if (l <= 5e-205)
		tmp = 2.0 / ((1.0 / ((t / k) * (t / k))) * (t_1 * (((t ^ 2.0) / l) * (t / l))));
	else
		tmp = (l * l) * ((2.0 * (k ^ -2.0)) / (t * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 5e-205], N[(2.0 / N[(N[(1.0 / N[(N[(t / k), $MachinePrecision] * N[(t / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[(N[Power[t, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 5.00000000000000001e-205

   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 32.0%

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

      1. +-commutative 32.0%

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

      2. associate-+l- 39.4%

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

      3. metadata-eval 39.4%

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

      4. --rgt-identity 39.4%

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

      5. unpow2 39.4%

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

      6. clear-num 39.4%

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

      7. clear-num 39.4%

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

      8. frac-times 39.3%

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

      9. metadata-eval 39.3%

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

   6. Applied egg-rr 39.3%

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

      1. unpow3 39.3%

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

      2. times-frac 55.3%

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

      3. pow2 55.3%

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

   8. Applied egg-rr 55.3%

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

   if 5.00000000000000001e-205 < l

   1. Initial program 38.8%

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

   3. Simplified 45.6%

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

      1. add-log-exp 30.4%

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

      2. exp-prod 33.9%

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

   6. Applied egg-rr 33.9%

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

   7. Taylor expanded in k around inf 76.7%

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

      1. associate-/r* 76.8%

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

   9. Simplified 76.8%

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

       1. *-un-lft-identity 76.8%

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

       2. div-inv 76.8%

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

       3. pow-flip 76.8%

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

       4. metadata-eval 76.8%

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

   11. Applied egg-rr 76.8%

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

       1. *-lft-identity 76.8%

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

   13. Simplified 76.8%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-205}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-2}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.0%

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

3. Simplified 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. Step-by-step derivation

   1. add-log-exp 27.9%

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

   2. exp-prod 30.8%

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

6. Applied egg-rr 30.8%

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

7. Taylor expanded in k around inf 71.0%

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

   1. associate-/r* 71.1%

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

9. Simplified 71.1%

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

    1. *-un-lft-identity 71.1%

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

    2. div-inv 71.1%

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

    3. pow-flip 71.5%

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

    4. metadata-eval 71.5%

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

11. Applied egg-rr 71.5%

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

    1. *-lft-identity 71.5%

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

13. Simplified 71.5%

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

14. Final simplification 71.5%

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

Alternative 15: 73.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.0%

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

3. Simplified 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. Step-by-step derivation

   1. add-log-exp 27.9%

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

   2. exp-prod 30.8%

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

6. Applied egg-rr 30.8%

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

7. Taylor expanded in k around inf 71.0%

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

   1. associate-/r* 71.1%

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

9. Simplified 71.1%

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

    1. unpow2 71.1%

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

11. Applied egg-rr 71.1%

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

12. Final simplification 71.1%

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

Alternative 16: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.0%

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

3. Simplified 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. Step-by-step derivation

   1. add-log-exp 27.9%

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

   2. exp-prod 30.8%

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

6. Applied egg-rr 30.8%

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

7. Taylor expanded in k around inf 71.0%

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

   1. associate-/r* 71.1%

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

9. Simplified 71.1%

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

10. Taylor expanded in k around 0 61.7%

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

11. Final simplification 61.7%

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

Alternative 17: 29.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 6.1e+38) (/ (* 2.0 (pow l 2.0)) 0.0) 0.0))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.1e+38) {
		tmp = (2.0 * pow(l, 2.0)) / 0.0;
	} else {
		tmp = 0.0;
	}
	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 <= 6.1d+38) then
        tmp = (2.0d0 * (l ** 2.0d0)) / 0.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.1e+38) {
		tmp = (2.0 * Math.pow(l, 2.0)) / 0.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 6.1e+38:
		tmp = (2.0 * math.pow(l, 2.0)) / 0.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 6.1e+38)
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) / 0.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6.1e+38)
		tmp = (2.0 * (l ^ 2.0)) / 0.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 6.1e+38], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / 0.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if k < 6.0999999999999999e38

   1. Initial program 34.8%

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

   3. Simplified 40.9%

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

      1. add-log-exp 23.6%

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

      2. exp-prod 30.4%

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

   6. Applied egg-rr 30.4%

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

   7. Taylor expanded in t around 0 22.6%

      \[\leadsto \frac{2}{\log \color{blue}{1}} \cdot \left(\ell \cdot \ell\right) \]
   8. Step-by-step derivation

      1. associate-*l/ 22.6%

         \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\log 1}} \]

      2. pow2 22.6%

         \[\leadsto \frac{2 \cdot \color{blue}{{\ell}^{2}}}{\log 1} \]

      3. metadata-eval 22.6%

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{0}} \]

   9. Applied egg-rr 22.6%

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{0}} \]

   if 6.0999999999999999e38 < k

   1. Initial program 35.3%

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

   3. Simplified 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. Step-by-step derivation

      1. add-log-exp 39.8%

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

      2. exp-prod 32.1%

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

   6. Applied egg-rr 32.1%

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

   7. Taylor expanded in t around 0 4.2%

      \[\leadsto \frac{2}{\log \color{blue}{1}} \cdot \left(\ell \cdot \ell\right) \]
   8. Step-by-step derivation

      1. add-cube-cbrt 4.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{2}{\log 1}} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \sqrt[3]{\frac{2}{\log 1}}\right)} \cdot \left(\ell \cdot \ell\right) \]

      2. pow2 4.2%

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

      3. clear-num 4.2%

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

      4. metadata-eval 4.2%

         \[\leadsto \left({\left(\sqrt[3]{\frac{1}{\frac{\color{blue}{0}}{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      5. metadata-eval 4.2%

         \[\leadsto \left({\left(\sqrt[3]{\frac{1}{\color{blue}{0}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      6. metadata-eval 4.2%

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

      7. cbrt-div 4.2%

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

      8. metadata-eval 4.2%

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

      9. metadata-eval 4.2%

         \[\leadsto \left({\left(\frac{1}{\sqrt[3]{\color{blue}{0}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      10. clear-num 4.2%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{1}{\frac{\log 1}{2}}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      11. metadata-eval 4.2%

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

      12. metadata-eval 4.2%

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

      13. metadata-eval 4.2%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\color{blue}{\log 1}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      14. cbrt-div 4.2%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\log 1}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      15. metadata-eval 4.2%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{\color{blue}{1}}{\sqrt[3]{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      16. metadata-eval 4.2%

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

   9. Applied egg-rr 4.2%

      \[\leadsto \color{blue}{\left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{1}{\sqrt[3]{0}}\right)} \cdot \left(\ell \cdot \ell\right) \]
   10. Step-by-step derivation

       1. pow-plus 4.2%

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

       2. metadata-eval 4.2%

          \[\leadsto {\left(\frac{1}{\sqrt[3]{0}}\right)}^{\color{blue}{3}} \cdot \left(\ell \cdot \ell\right) \]

       3. cube-div 4.2%

          \[\leadsto \color{blue}{\frac{{1}^{3}}{{\left(\sqrt[3]{0}\right)}^{3}}} \cdot \left(\ell \cdot \ell\right) \]

       4. metadata-eval 4.2%

          \[\leadsto \frac{\color{blue}{1}}{{\left(\sqrt[3]{0}\right)}^{3}} \cdot \left(\ell \cdot \ell\right) \]

       5. rem-cube-cbrt 4.2%

          \[\leadsto \frac{1}{\color{blue}{0}} \cdot \left(\ell \cdot \ell\right) \]

       6. unpow-1 4.2%

          \[\leadsto \color{blue}{{0}^{-1}} \cdot \left(\ell \cdot \ell\right) \]

       7. pow-base-0 50.6%

          \[\leadsto \color{blue}{0} \cdot \left(\ell \cdot \ell\right) \]

   11. Simplified 50.6%

       \[\leadsto \color{blue}{0} \cdot \left(\ell \cdot \ell\right) \]

   12. Taylor expanded in l around 0 51.4%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 62.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (* l l) (/ 2.0 (* t (pow k 4.0)))))

C

double code(double t, double l, double k) {
	return (l * l) * (2.0 / (t * pow(k, 4.0)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l * l) * (2.0d0 / (t * (k ** 4.0d0)))
end function

Java

public static double code(double t, double l, double k) {
	return (l * l) * (2.0 / (t * Math.pow(k, 4.0)));
}

Python

def code(t, l, k):
	return (l * l) * (2.0 / (t * math.pow(k, 4.0)))

Julia

function code(t, l, k)
	return Float64(Float64(l * l) * Float64(2.0 / Float64(t * (k ^ 4.0))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l * l) * (2.0 / (t * (k ^ 4.0)));
end

Wolfram

code[t_, l_, k_] := N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.0%

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

3. Simplified 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 60.8%

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

6. Final simplification 60.8%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}} \]
7. Add Preprocessing

Alternative 19: 28.6% accurate, 421.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 0.0)

C

double code(double t, double l, double k) {
	return 0.0;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 0.0d0
end function

Java

public static double code(double t, double l, double k) {
	return 0.0;
}

Python

def code(t, l, k):
	return 0.0

Julia

function code(t, l, k)
	return 0.0
end

MATLAB

function tmp = code(t, l, k)
	tmp = 0.0;
end

Wolfram

code[t_, l_, k_] := 0.0

Derivation

1. Initial program 35.0%

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

3. Simplified 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. Step-by-step derivation

   1. add-log-exp 27.9%

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

   2. exp-prod 30.8%

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

6. Applied egg-rr 30.8%

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

7. Taylor expanded in t around 0 17.7%

   \[\leadsto \frac{2}{\log \color{blue}{1}} \cdot \left(\ell \cdot \ell\right) \]
8. Step-by-step derivation

   1. add-cube-cbrt 17.7%

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{2}{\log 1}} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \sqrt[3]{\frac{2}{\log 1}}\right)} \cdot \left(\ell \cdot \ell\right) \]

   2. pow2 17.7%

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

   3. clear-num 17.7%

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

   4. metadata-eval 17.7%

      \[\leadsto \left({\left(\sqrt[3]{\frac{1}{\frac{\color{blue}{0}}{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

   5. metadata-eval 17.7%

      \[\leadsto \left({\left(\sqrt[3]{\frac{1}{\color{blue}{0}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

   6. metadata-eval 17.7%

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

   7. cbrt-div 17.7%

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

   8. metadata-eval 17.7%

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

   9. metadata-eval 17.7%

      \[\leadsto \left({\left(\frac{1}{\sqrt[3]{\color{blue}{0}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

   10. clear-num 17.7%

       \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{1}{\frac{\log 1}{2}}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   11. metadata-eval 17.7%

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

   12. metadata-eval 17.7%

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

   13. metadata-eval 17.7%

       \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\color{blue}{\log 1}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   14. cbrt-div 17.7%

       \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\log 1}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   15. metadata-eval 17.7%

       \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{\color{blue}{1}}{\sqrt[3]{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

   16. metadata-eval 17.7%

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

9. Applied egg-rr 17.7%

   \[\leadsto \color{blue}{\left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{1}{\sqrt[3]{0}}\right)} \cdot \left(\ell \cdot \ell\right) \]
10. Step-by-step derivation

    1. pow-plus 17.7%

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

    2. metadata-eval 17.7%

       \[\leadsto {\left(\frac{1}{\sqrt[3]{0}}\right)}^{\color{blue}{3}} \cdot \left(\ell \cdot \ell\right) \]

    3. cube-div 17.7%

       \[\leadsto \color{blue}{\frac{{1}^{3}}{{\left(\sqrt[3]{0}\right)}^{3}}} \cdot \left(\ell \cdot \ell\right) \]

    4. metadata-eval 17.7%

       \[\leadsto \frac{\color{blue}{1}}{{\left(\sqrt[3]{0}\right)}^{3}} \cdot \left(\ell \cdot \ell\right) \]

    5. rem-cube-cbrt 17.7%

       \[\leadsto \frac{1}{\color{blue}{0}} \cdot \left(\ell \cdot \ell\right) \]

    6. unpow-1 17.7%

       \[\leadsto \color{blue}{{0}^{-1}} \cdot \left(\ell \cdot \ell\right) \]

    7. pow-base-0 28.1%

       \[\leadsto \color{blue}{0} \cdot \left(\ell \cdot \ell\right) \]

11. Simplified 28.1%

    \[\leadsto \color{blue}{0} \cdot \left(\ell \cdot \ell\right) \]

12. Taylor expanded in l around 0 29.1%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024141 
(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))))