Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.9% → 94.6%

Time: 19.4s

Alternatives: 20

Speedup: 3.8×

• Could not converge on a ground truth

  1. t = 1.4562009836986044e-98
  2. l = 8.451138165835098e+226
  3. k = 5.912613404075968e+53

• 40.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.6% accurate, 0.4× speedup

Math

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

FPCore

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

Java

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

Julia

function code(t, l, k)
	return Float64((Float64(cbrt(Float64(sqrt(2.0) * t)) / cbrt(k)) ^ 2.0) / Float64(t * Float64(cbrt(Float64(sin(k) * tan(k))) * (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[(t * N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 42.9%

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

3. Simplified 42.9%

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

6. Applied egg-rr 85.5%

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

   1. add-cube-cbrt 85.5%

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

   2. pow3 85.5%

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

8. Applied egg-rr 90.2%

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

   1. associate-*r/ 90.2%

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

   2. cbrt-div 94.3%

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

10. Applied egg-rr 94.3%

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

Alternative 2: 89.4% accurate, 0.4× speedup

Math

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

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(sqrt(2.0) * Float64(t / k))) ^ 2.0) / Float64(t * Float64((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[Sqrt[2.0], $MachinePrecision] * N[(t / k), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(t * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 42.9%

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

3. Simplified 42.9%

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

6. Applied egg-rr 85.5%

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

   1. add-cube-cbrt 85.5%

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

   2. pow3 85.5%

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

8. Applied egg-rr 90.2%

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

   1. *-commutative 90.2%

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

   2. cbrt-prod 90.6%

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

10. Applied egg-rr 90.6%

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

11. Final simplification 90.6%

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

Alternative 3: 88.9% accurate, 0.5× speedup

Math

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

FPCore

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.9%

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

3. Simplified 42.9%

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

6. Applied egg-rr 85.5%

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

   1. add-cube-cbrt 85.5%

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

   2. pow3 85.5%

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

8. Applied egg-rr 90.2%

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

Alternative 4: 88.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.9%

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

3. Simplified 42.9%

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

6. Applied egg-rr 85.5%

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

   1. add-cube-cbrt 85.5%

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

   2. pow3 85.5%

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

8. Applied egg-rr 90.2%

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

   1. cube-mult 90.2%

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

10. Applied egg-rr 89.7%

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

    1. unpow2 89.7%

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

    2. cube-unmult 89.7%

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

    3. associate-/l* 89.7%

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

12. Simplified 90.2%

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

Alternative 5: 77.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ t_2 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ \mathbf{if}\;\ell \leq 2.2 \cdot 10^{-161}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\sqrt{2} \cdot \frac{t}{k}}\right)}^{2}}{t \cdot \left(t\_2 \cdot \sqrt[3]{{k}^{2}}\right)}\right)}^{3}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+164}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2 \cdot {\left(\frac{t}{k}\right)}^{2}}}{t \cdot \left(\sqrt[3]{t\_1} \cdot t\_2\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))) (t_2 (pow (cbrt l) -2.0)))
   (if (<= l 2.2e-161)
     (pow
      (/
       (pow (cbrt (* (sqrt 2.0) (/ t k))) 2.0)
       (* t (* t_2 (cbrt (pow k 2.0)))))
      3.0)
     (if (<= l 1.55e+164)
       (* (/ 2.0 (pow k 2.0)) (/ (pow l 2.0) (* t t_1)))
       (pow
        (/ (cbrt (* 2.0 (pow (/ t k) 2.0))) (* t (* (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 = pow(cbrt(l), -2.0);
	double tmp;
	if (l <= 2.2e-161) {
		tmp = pow((pow(cbrt((sqrt(2.0) * (t / k))), 2.0) / (t * (t_2 * cbrt(pow(k, 2.0))))), 3.0);
	} else if (l <= 1.55e+164) {
		tmp = (2.0 / pow(k, 2.0)) * (pow(l, 2.0) / (t * t_1));
	} else {
		tmp = pow((cbrt((2.0 * pow((t / k), 2.0))) / (t * (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 = Math.pow(Math.cbrt(l), -2.0);
	double tmp;
	if (l <= 2.2e-161) {
		tmp = Math.pow((Math.pow(Math.cbrt((Math.sqrt(2.0) * (t / k))), 2.0) / (t * (t_2 * Math.cbrt(Math.pow(k, 2.0))))), 3.0);
	} else if (l <= 1.55e+164) {
		tmp = (2.0 / Math.pow(k, 2.0)) * (Math.pow(l, 2.0) / (t * t_1));
	} else {
		tmp = Math.pow((Math.cbrt((2.0 * Math.pow((t / k), 2.0))) / (t * (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 = cbrt(l) ^ -2.0
	tmp = 0.0
	if (l <= 2.2e-161)
		tmp = Float64((cbrt(Float64(sqrt(2.0) * Float64(t / k))) ^ 2.0) / Float64(t * Float64(t_2 * cbrt((k ^ 2.0))))) ^ 3.0;
	elseif (l <= 1.55e+164)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64((l ^ 2.0) / Float64(t * t_1)));
	else
		tmp = Float64(cbrt(Float64(2.0 * (Float64(t / k) ^ 2.0))) / Float64(t * 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[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, If[LessEqual[l, 2.2e-161], N[Power[N[(N[Power[N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / k), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(t * N[(t$95$2 * N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[l, 1.55e+164], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $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[(t * N[(N[Power[t$95$1, 1/3], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.20000000000000002e-161

   1. Initial program 40.9%

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

   3. Simplified 41.0%

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

   6. Applied egg-rr 83.2%

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

      1. add-cube-cbrt 83.1%

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

      2. pow3 83.2%

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

   8. Applied egg-rr 87.8%

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

   9. Taylor expanded in k around 0 77.2%

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

   if 2.20000000000000002e-161 < l < 1.5500000000000001e164

   1. Initial program 54.0%

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

   3. Simplified 62.9%

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

      1. add-log-exp 41.1%

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

      2. exp-prod 39.9%

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

   6. Applied egg-rr 39.9%

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

   7. Taylor expanded in k around inf 90.0%

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

      1. associate-*l/ 90.0%

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

      2. pow2 90.0%

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

      3. associate-*r* 90.0%

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

   9. Applied egg-rr 90.0%

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

       1. times-frac 93.9%

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

       2. associate-*l* 94.0%

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

   11. Simplified 94.0%

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

   if 1.5500000000000001e164 < l

   1. Initial program 29.2%

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

   3. Simplified 29.2%

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

   6. Applied egg-rr 94.9%

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

      1. associate-/r/ 95.0%

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

      2. associate-/r* 95.0%

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

      3. associate-/r/ 95.1%

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

   8. Simplified 95.1%

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

      1. associate-*l/ 90.0%

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

      2. associate-*l/ 90.0%

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

      3. associate-/l/ 89.9%

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

      4. associate-*l/ 89.9%

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

      5. *-commutative 89.9%

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

      6. div-inv 89.9%

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

      7. pow-flip 89.8%

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

      8. metadata-eval 89.8%

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

   10. Applied egg-rr 89.9%

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

       1. associate-*r/ 83.6%

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

       2. associate-/l* 83.6%

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

       3. associate-/l* 83.7%

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

       4. swap-sqr 83.6%

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

       5. rem-square-sqrt 83.7%

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

       6. associate-*l* 83.8%

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

       7. associate-*l* 83.8%

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

   12. Simplified 83.8%

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

       1. add-cube-cbrt 83.5%

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

       2. pow3 83.5%

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

   14. Applied egg-rr 83.8%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{-161}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\sqrt{2} \cdot \frac{t}{k}}\right)}^{2}}{t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{{k}^{2}}\right)}\right)}^{3}\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+164}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2 \cdot {\left(\frac{t}{k}\right)}^{2}}}{t \cdot \left(\sqrt[3]{\sin k \cdot \tan k} \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-161}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\sqrt{2} \cdot \frac{t}{k}}\right)}^{2}}{t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{{k}^{2}}\right)}\right)}^{3}\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{+169}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 1e-161)
   (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 1.06e+169)
     (* (/ 2.0 (pow k 2.0)) (/ (pow l 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 <= 1e-161) {
		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 <= 1.06e+169) {
		tmp = (2.0 / pow(k, 2.0)) * (pow(l, 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 <= 1e-161) {
		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 <= 1.06e+169) {
		tmp = (2.0 / Math.pow(k, 2.0)) * (Math.pow(l, 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 <= 1e-161)
		tmp = Float64((cbrt(Float64(sqrt(2.0) * Float64(t / k))) ^ 2.0) / Float64(t * Float64((cbrt(l) ^ -2.0) * cbrt((k ^ 2.0))))) ^ 3.0;
	elseif (l <= 1.06e+169)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64((l ^ 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, 1e-161], N[Power[N[(N[Power[N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / k), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(t * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[l, 1.06e+169], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $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 < 1.00000000000000003e-161

   1. Initial program 40.9%

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

   3. Simplified 41.0%

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

   6. Applied egg-rr 83.2%

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

      1. add-cube-cbrt 83.1%

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

      2. pow3 83.2%

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

   8. Applied egg-rr 87.8%

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

   9. Taylor expanded in k around 0 77.2%

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

   if 1.00000000000000003e-161 < l < 1.05999999999999995e169

   1. Initial program 54.0%

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

   3. Simplified 62.9%

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

      1. add-log-exp 41.1%

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

      2. exp-prod 39.9%

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

   6. Applied egg-rr 39.9%

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

   7. Taylor expanded in k around inf 90.0%

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

      1. associate-*l/ 90.0%

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

      2. pow2 90.0%

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

      3. associate-*r* 90.0%

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

   9. Applied egg-rr 90.0%

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

       1. times-frac 93.9%

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

       2. associate-*l* 94.0%

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

   11. Simplified 94.0%

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

   if 1.05999999999999995e169 < l

   1. Initial program 29.2%

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

   3. Simplified 29.2%

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

      1. add-cube-cbrt 29.2%

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

      2. pow3 29.2%

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

   6. Applied egg-rr 81.4%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-161}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\sqrt{2} \cdot \frac{t}{k}}\right)}^{2}}{t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{{k}^{2}}\right)}\right)}^{3}\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{+169}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-134}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{t}{k}\right)}^{2}}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{+82}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 8e-134)
   (*
    2.0
    (/
     (pow (/ t k) 2.0)
     (pow (* (pow (cbrt l) -2.0) (* t (cbrt (* (sin k) (tan k))))) 3.0)))
   (if (<= k 1.15e+82)
     (/
      2.0
      (* (pow k 2.0) (* (/ t (pow l 2.0)) (/ (pow (sin k) 2.0) (cos 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 (k <= 8e-134) {
		tmp = 2.0 * (pow((t / k), 2.0) / pow((pow(cbrt(l), -2.0) * (t * cbrt((sin(k) * tan(k))))), 3.0));
	} else if (k <= 1.15e+82) {
		tmp = 2.0 / (pow(k, 2.0) * ((t / pow(l, 2.0)) * (pow(sin(k), 2.0) / cos(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 (k <= 8e-134) {
		tmp = 2.0 * (Math.pow((t / k), 2.0) / Math.pow((Math.pow(Math.cbrt(l), -2.0) * (t * Math.cbrt((Math.sin(k) * Math.tan(k))))), 3.0));
	} else if (k <= 1.15e+82) {
		tmp = 2.0 / (Math.pow(k, 2.0) * ((t / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(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 (k <= 8e-134)
		tmp = Float64(2.0 * Float64((Float64(t / k) ^ 2.0) / (Float64((cbrt(l) ^ -2.0) * Float64(t * cbrt(Float64(sin(k) * tan(k))))) ^ 3.0)));
	elseif (k <= 1.15e+82)
		tmp = Float64(2.0 / Float64((k ^ 2.0) * Float64(Float64(t / (l ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(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[k, 8e-134], N[(2.0 * N[(N[Power[N[(t / k), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[(t * N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.15e+82], N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(t / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[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 k < 8.00000000000000032e-134

   1. Initial program 45.4%

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

   3. Simplified 45.4%

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

   6. Applied egg-rr 87.0%

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

      1. frac-times 76.2%

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

      2. pow2 76.2%

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

      3. div-inv 76.2%

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

      4. clear-num 76.2%

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

      5. unpow2 76.2%

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

      6. pow3 76.2%

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

   8. Applied egg-rr 77.6%

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

      1. unpow2 77.6%

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

      2. swap-sqr 77.6%

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

      3. rem-square-sqrt 77.6%

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

      4. unpow2 77.6%

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

      5. associate-/l* 77.6%

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

      6. associate-*r* 77.6%

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

   10. Simplified 77.6%

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

   if 8.00000000000000032e-134 < k < 1.14999999999999994e82

   1. Initial program 42.1%

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

   3. Simplified 42.1%

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

   5. Taylor expanded in t around 0 89.9%

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

      1. associate-/l* 91.8%

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

   7. Simplified 91.8%

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

      1. pow2 91.8%

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

      2. times-frac 92.0%

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

      3. pow2 92.0%

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

   9. Applied egg-rr 92.0%

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

   if 1.14999999999999994e82 < k

   1. Initial program 35.6%

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

   3. Simplified 35.7%

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

      1. add-cube-cbrt 35.7%

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

      2. pow3 35.7%

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

   6. Applied egg-rr 79.1%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-134}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{t}{k}\right)}^{2}}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{+82}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(\frac{t}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ t_2 := \sqrt[3]{t\_1}\\ \mathbf{if}\;k \leq 6.7 \cdot 10^{-134}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{t}{k}\right)}^{2}}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot t\_2\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{+134}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t}}}{{\left(t\_2 \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))) (t_2 (cbrt t_1)))
   (if (<= k 6.7e-134)
     (* 2.0 (/ (pow (/ t k) 2.0) (pow (* (pow (cbrt l) -2.0) (* t t_2)) 3.0)))
     (if (<= k 1.2e+134)
       (* (/ 2.0 (pow k 2.0)) (/ (pow l 2.0) (* t t_1)))
       (/
        (/ 2.0 (* (/ k t) (/ k t)))
        (pow (* t_2 (/ t (pow (cbrt l) 2.0))) 3.0))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double t_2 = cbrt(t_1);
	double tmp;
	if (k <= 6.7e-134) {
		tmp = 2.0 * (pow((t / k), 2.0) / pow((pow(cbrt(l), -2.0) * (t * t_2)), 3.0));
	} else if (k <= 1.2e+134) {
		tmp = (2.0 / pow(k, 2.0)) * (pow(l, 2.0) / (t * t_1));
	} else {
		tmp = (2.0 / ((k / t) * (k / t))) / pow((t_2 * (t / pow(cbrt(l), 2.0))), 3.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.tan(k);
	double t_2 = Math.cbrt(t_1);
	double tmp;
	if (k <= 6.7e-134) {
		tmp = 2.0 * (Math.pow((t / k), 2.0) / Math.pow((Math.pow(Math.cbrt(l), -2.0) * (t * t_2)), 3.0));
	} else if (k <= 1.2e+134) {
		tmp = (2.0 / Math.pow(k, 2.0)) * (Math.pow(l, 2.0) / (t * t_1));
	} else {
		tmp = (2.0 / ((k / t) * (k / t))) / Math.pow((t_2 * (t / Math.pow(Math.cbrt(l), 2.0))), 3.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	t_2 = cbrt(t_1)
	tmp = 0.0
	if (k <= 6.7e-134)
		tmp = Float64(2.0 * Float64((Float64(t / k) ^ 2.0) / (Float64((cbrt(l) ^ -2.0) * Float64(t * t_2)) ^ 3.0)));
	elseif (k <= 1.2e+134)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64((l ^ 2.0) / Float64(t * t_1)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) * Float64(k / t))) / (Float64(t_2 * Float64(t / (cbrt(l) ^ 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]}, Block[{t$95$2 = N[Power[t$95$1, 1/3], $MachinePrecision]}, If[LessEqual[k, 6.7e-134], N[(2.0 * N[(N[Power[N[(t / k), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[(t * t$95$2), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.2e+134], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(t$95$2 * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 6.69999999999999996e-134

   1. Initial program 45.4%

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

   3. Simplified 45.4%

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

   6. Applied egg-rr 87.0%

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

      1. frac-times 76.2%

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

      2. pow2 76.2%

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

      3. div-inv 76.2%

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

      4. clear-num 76.2%

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

      5. unpow2 76.2%

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

      6. pow3 76.2%

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

   8. Applied egg-rr 77.6%

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

      1. unpow2 77.6%

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

      2. swap-sqr 77.6%

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

      3. rem-square-sqrt 77.6%

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

      4. unpow2 77.6%

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

      5. associate-/l* 77.6%

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

      6. associate-*r* 77.6%

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

   10. Simplified 77.6%

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

   if 6.69999999999999996e-134 < k < 1.20000000000000003e134

   1. Initial program 36.2%

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

   3. Simplified 49.5%

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

      1. add-log-exp 22.0%

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

      2. exp-prod 27.5%

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

   6. Applied egg-rr 27.5%

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

   7. Taylor expanded in k around inf 83.0%

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

      1. associate-*l/ 83.0%

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

      2. pow2 83.0%

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

      3. associate-*r* 83.0%

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

   9. Applied egg-rr 83.0%

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

       1. times-frac 86.9%

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

       2. associate-*l* 86.9%

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

   11. Simplified 86.9%

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

   if 1.20000000000000003e134 < k

   1. Initial program 43.4%

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

      1. *-commutative 43.4%

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

      2. associate-/r* 43.4%

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

   3. Simplified 54.2%

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

      1. add-cube-cbrt 54.2%

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

      2. pow3 54.2%

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

      3. cbrt-prod 54.2%

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

      4. cbrt-div 54.2%

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

      5. rem-cbrt-cube 67.9%

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

      6. cbrt-prod 76.6%

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

      7. pow2 76.6%

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

   6. Applied egg-rr 76.6%

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

      1. +-rgt-identity 76.6%

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

      2. unpow2 76.6%

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

   8. Applied egg-rr 76.6%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.7 \cdot 10^{-134}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{t}{k}\right)}^{2}}{{\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{+134}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t}}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_2 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq 6.8 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{{\left(\sqrt[3]{{k}^{2}} \cdot t\_1\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{+134}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t}}}{{\left(\sqrt[3]{t\_2} \cdot t\_1\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (pow (cbrt l) 2.0))) (t_2 (* (sin k) (tan k))))
   (if (<= k 6.8e-134)
     (/ (/ 2.0 (pow (/ k t) 2.0)) (pow (* (cbrt (pow k 2.0)) t_1) 3.0))
     (if (<= k 1.2e+134)
       (* (/ 2.0 (pow k 2.0)) (/ (pow l 2.0) (* t t_2)))
       (/ (/ 2.0 (* (/ k t) (/ k t))) (pow (* (cbrt t_2) t_1) 3.0))))))

C

double code(double t, double l, double k) {
	double t_1 = t / pow(cbrt(l), 2.0);
	double t_2 = sin(k) * tan(k);
	double tmp;
	if (k <= 6.8e-134) {
		tmp = (2.0 / pow((k / t), 2.0)) / pow((cbrt(pow(k, 2.0)) * t_1), 3.0);
	} else if (k <= 1.2e+134) {
		tmp = (2.0 / pow(k, 2.0)) * (pow(l, 2.0) / (t * t_2));
	} else {
		tmp = (2.0 / ((k / t) * (k / t))) / pow((cbrt(t_2) * t_1), 3.0);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = t / Math.pow(Math.cbrt(l), 2.0);
	double t_2 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (k <= 6.8e-134) {
		tmp = (2.0 / Math.pow((k / t), 2.0)) / Math.pow((Math.cbrt(Math.pow(k, 2.0)) * t_1), 3.0);
	} else if (k <= 1.2e+134) {
		tmp = (2.0 / Math.pow(k, 2.0)) * (Math.pow(l, 2.0) / (t * t_2));
	} else {
		tmp = (2.0 / ((k / t) * (k / t))) / Math.pow((Math.cbrt(t_2) * t_1), 3.0);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(t / (cbrt(l) ^ 2.0))
	t_2 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (k <= 6.8e-134)
		tmp = Float64(Float64(2.0 / (Float64(k / t) ^ 2.0)) / (Float64(cbrt((k ^ 2.0)) * t_1) ^ 3.0));
	elseif (k <= 1.2e+134)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64((l ^ 2.0) / Float64(t * t_2)));
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(k / t) * Float64(k / t))) / (Float64(cbrt(t_2) * t_1) ^ 3.0));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 6.8e-134], N[(N[(2.0 / N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision] * t$95$1), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.2e+134], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[(k / t), $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[Power[t$95$2, 1/3], $MachinePrecision] * t$95$1), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 6.79999999999999954e-134

   1. Initial program 45.4%

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

      1. *-commutative 45.4%

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

      2. associate-/r* 45.4%

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

   3. Simplified 49.2%

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

      1. add-cube-cbrt 49.1%

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

      2. pow3 49.2%

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

      3. cbrt-prod 49.2%

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

      4. cbrt-div 50.3%

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

      5. rem-cbrt-cube 65.8%

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

      6. cbrt-prod 75.6%

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

      7. pow2 75.6%

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

   6. Applied egg-rr 75.6%

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

   7. Taylor expanded in k around 0 70.5%

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

   if 6.79999999999999954e-134 < k < 1.20000000000000003e134

   1. Initial program 36.2%

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

   3. Simplified 49.5%

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

      1. add-log-exp 22.0%

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

      2. exp-prod 27.5%

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

   6. Applied egg-rr 27.5%

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

   7. Taylor expanded in k around inf 83.0%

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

      1. associate-*l/ 83.0%

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

      2. pow2 83.0%

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

      3. associate-*r* 83.0%

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

   9. Applied egg-rr 83.0%

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

       1. times-frac 86.9%

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

       2. associate-*l* 86.9%

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

   11. Simplified 86.9%

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

   if 1.20000000000000003e134 < k

   1. Initial program 43.4%

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

      1. *-commutative 43.4%

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

      2. associate-/r* 43.4%

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

   3. Simplified 54.2%

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

      1. add-cube-cbrt 54.2%

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

      2. pow3 54.2%

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

      3. cbrt-prod 54.2%

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

      4. cbrt-div 54.2%

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

      5. rem-cbrt-cube 67.9%

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

      6. cbrt-prod 76.6%

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

      7. pow2 76.6%

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

   6. Applied egg-rr 76.6%

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

      1. +-rgt-identity 76.6%

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

      2. unpow2 76.6%

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

   8. Applied egg-rr 76.6%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{{\left(\sqrt[3]{{k}^{2}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{+134}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t}}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ t_3 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq 6.7 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{2}{t\_2}}{{\left(\sqrt[3]{{k}^{2}} \cdot t\_1\right)}^{3}}\\ \mathbf{elif}\;k \leq 9.5 \cdot 10^{+133}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_3 \cdot t\_2\right) \cdot {t\_1}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ t (pow (cbrt l) 2.0)))
        (t_2 (pow (/ k t) 2.0))
        (t_3 (* (sin k) (tan k))))
   (if (<= k 6.7e-134)
     (/ (/ 2.0 t_2) (pow (* (cbrt (pow k 2.0)) t_1) 3.0))
     (if (<= k 9.5e+133)
       (* (/ 2.0 (pow k 2.0)) (/ (pow l 2.0) (* t t_3)))
       (/ 2.0 (* (* t_3 t_2) (pow t_1 3.0)))))))

C

double code(double t, double l, double k) {
	double t_1 = t / pow(cbrt(l), 2.0);
	double t_2 = pow((k / t), 2.0);
	double t_3 = sin(k) * tan(k);
	double tmp;
	if (k <= 6.7e-134) {
		tmp = (2.0 / t_2) / pow((cbrt(pow(k, 2.0)) * t_1), 3.0);
	} else if (k <= 9.5e+133) {
		tmp = (2.0 / pow(k, 2.0)) * (pow(l, 2.0) / (t * t_3));
	} else {
		tmp = 2.0 / ((t_3 * t_2) * pow(t_1, 3.0));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = t / Math.pow(Math.cbrt(l), 2.0);
	double t_2 = Math.pow((k / t), 2.0);
	double t_3 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (k <= 6.7e-134) {
		tmp = (2.0 / t_2) / Math.pow((Math.cbrt(Math.pow(k, 2.0)) * t_1), 3.0);
	} else if (k <= 9.5e+133) {
		tmp = (2.0 / Math.pow(k, 2.0)) * (Math.pow(l, 2.0) / (t * t_3));
	} else {
		tmp = 2.0 / ((t_3 * t_2) * Math.pow(t_1, 3.0));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(t / (cbrt(l) ^ 2.0))
	t_2 = Float64(k / t) ^ 2.0
	t_3 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (k <= 6.7e-134)
		tmp = Float64(Float64(2.0 / t_2) / (Float64(cbrt((k ^ 2.0)) * t_1) ^ 3.0));
	elseif (k <= 9.5e+133)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64((l ^ 2.0) / Float64(t * t_3)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_3 * t_2) * (t_1 ^ 3.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 6.7e-134], N[(N[(2.0 / t$95$2), $MachinePrecision] / N[Power[N[(N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision] * t$95$1), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 9.5e+133], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$3 * t$95$2), $MachinePrecision] * N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 6.69999999999999996e-134

   1. Initial program 45.4%

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

      1. *-commutative 45.4%

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

      2. associate-/r* 45.4%

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

   3. Simplified 49.2%

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

      1. add-cube-cbrt 49.1%

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

      2. pow3 49.2%

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

      3. cbrt-prod 49.2%

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

      4. cbrt-div 50.3%

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

      5. rem-cbrt-cube 65.8%

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

      6. cbrt-prod 75.6%

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

      7. pow2 75.6%

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

   6. Applied egg-rr 75.6%

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

   7. Taylor expanded in k around 0 70.5%

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

   if 6.69999999999999996e-134 < k < 9.49999999999999996e133

   1. Initial program 36.2%

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

   3. Simplified 49.5%

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

      1. add-log-exp 22.0%

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

      2. exp-prod 27.5%

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

   6. Applied egg-rr 27.5%

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

   7. Taylor expanded in k around inf 83.0%

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

      1. associate-*l/ 83.0%

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

      2. pow2 83.0%

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

      3. associate-*r* 83.0%

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

   9. Applied egg-rr 83.0%

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

       1. times-frac 86.9%

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

       2. associate-*l* 86.9%

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

   11. Simplified 86.9%

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

   if 9.49999999999999996e133 < k

   1. Initial program 43.4%

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

   3. Simplified 43.4%

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

      1. add-cube-cbrt 43.4%

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

      2. pow3 43.4%

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

   6. Applied egg-rr 79.7%

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

      1. *-commutative 79.7%

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

      2. cube-prod 76.6%

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

      3. rem-cube-cbrt 76.6%

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

      4. associate-*r* 76.6%

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

   8. Simplified 76.6%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.7 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{{\left(\sqrt[3]{{k}^{2}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 9.5 \cdot 10^{+133}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq 1.2 \cdot 10^{+134}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_1 \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= k 1.2e+134)
     (* (/ 2.0 (pow k 2.0)) (/ (pow l 2.0) (* t t_1)))
     (/
      2.0
      (* (* t_1 (pow (/ k t) 2.0)) (pow (/ t (pow (cbrt l) 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+134) {
		tmp = (2.0 / pow(k, 2.0)) * (pow(l, 2.0) / (t * t_1));
	} else {
		tmp = 2.0 / ((t_1 * pow((k / t), 2.0)) * pow((t / pow(cbrt(l), 2.0)), 3.0));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (k <= 1.2e+134) {
		tmp = (2.0 / Math.pow(k, 2.0)) * (Math.pow(l, 2.0) / (t * t_1));
	} else {
		tmp = 2.0 / ((t_1 * Math.pow((k / t), 2.0)) * Math.pow((t / Math.pow(Math.cbrt(l), 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+134)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64((l ^ 2.0) / Float64(t * t_1)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_1 * (Float64(k / t) ^ 2.0)) * (Float64(t / (cbrt(l) ^ 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+134], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$1 * N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.20000000000000003e134

   1. Initial program 42.8%

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

   3. Simplified 49.7%

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

      1. add-log-exp 28.6%

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

      2. exp-prod 32.5%

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

   6. Applied egg-rr 32.5%

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

   7. Taylor expanded in k around inf 76.4%

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

      1. associate-*l/ 76.4%

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

      2. pow2 76.4%

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

      3. associate-*r* 76.5%

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

   9. Applied egg-rr 76.5%

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

       1. times-frac 78.7%

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

       2. associate-*l* 78.7%

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

   11. Simplified 78.7%

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

   if 1.20000000000000003e134 < k

   1. Initial program 43.4%

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

   3. Simplified 43.4%

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

      1. add-cube-cbrt 43.4%

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

      2. pow3 43.4%

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

   6. Applied egg-rr 79.7%

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

      1. *-commutative 79.7%

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

      2. cube-prod 76.6%

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

      3. rem-cube-cbrt 76.6%

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

      4. associate-*r* 76.6%

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

   8. Simplified 76.6%

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

Alternative 12: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq 1.7 \cdot 10^{+133}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot t\_1}\\ \mathbf{elif}\;k \leq 5.2 \cdot 10^{+221}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{t\_1 \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{2}{t \cdot {k}^{4}}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= k 1.7e+133)
     (* (/ 2.0 (pow k 2.0)) (/ (pow l 2.0) (* t t_1)))
     (if (<= k 5.2e+221)
       (/ (/ 2.0 (pow (/ k t) 2.0)) (* t_1 (pow (/ (pow t 1.5) l) 2.0)))
       (log (pow (exp (pow l 2.0)) (/ 2.0 (* t (pow k 4.0)))))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (k <= 1.7e+133) {
		tmp = (2.0 / pow(k, 2.0)) * (pow(l, 2.0) / (t * t_1));
	} else if (k <= 5.2e+221) {
		tmp = (2.0 / pow((k / t), 2.0)) / (t_1 * pow((pow(t, 1.5) / l), 2.0));
	} else {
		tmp = log(pow(exp(pow(l, 2.0)), (2.0 / (t * pow(k, 4.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) * tan(k)
    if (k <= 1.7d+133) then
        tmp = (2.0d0 / (k ** 2.0d0)) * ((l ** 2.0d0) / (t * t_1))
    else if (k <= 5.2d+221) then
        tmp = (2.0d0 / ((k / t) ** 2.0d0)) / (t_1 * (((t ** 1.5d0) / l) ** 2.0d0))
    else
        tmp = log((exp((l ** 2.0d0)) ** (2.0d0 / (t * (k ** 4.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (k <= 1.7e+133) {
		tmp = (2.0 / Math.pow(k, 2.0)) * (Math.pow(l, 2.0) / (t * t_1));
	} else if (k <= 5.2e+221) {
		tmp = (2.0 / Math.pow((k / t), 2.0)) / (t_1 * Math.pow((Math.pow(t, 1.5) / l), 2.0));
	} else {
		tmp = Math.log(Math.pow(Math.exp(Math.pow(l, 2.0)), (2.0 / (t * Math.pow(k, 4.0)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(k) * math.tan(k)
	tmp = 0
	if k <= 1.7e+133:
		tmp = (2.0 / math.pow(k, 2.0)) * (math.pow(l, 2.0) / (t * t_1))
	elif k <= 5.2e+221:
		tmp = (2.0 / math.pow((k / t), 2.0)) / (t_1 * math.pow((math.pow(t, 1.5) / l), 2.0))
	else:
		tmp = math.log(math.pow(math.exp(math.pow(l, 2.0)), (2.0 / (t * math.pow(k, 4.0)))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (k <= 1.7e+133)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64((l ^ 2.0) / Float64(t * t_1)));
	elseif (k <= 5.2e+221)
		tmp = Float64(Float64(2.0 / (Float64(k / t) ^ 2.0)) / Float64(t_1 * (Float64((t ^ 1.5) / l) ^ 2.0)));
	else
		tmp = log((exp((l ^ 2.0)) ^ Float64(2.0 / Float64(t * (k ^ 4.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) * tan(k);
	tmp = 0.0;
	if (k <= 1.7e+133)
		tmp = (2.0 / (k ^ 2.0)) * ((l ^ 2.0) / (t * t_1));
	elseif (k <= 5.2e+221)
		tmp = (2.0 / ((k / t) ^ 2.0)) / (t_1 * (((t ^ 1.5) / l) ^ 2.0));
	else
		tmp = log((exp((l ^ 2.0)) ^ (2.0 / (t * (k ^ 4.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.7e+133], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.2e+221], N[(N[(2.0 / N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[Power[N[Exp[N[Power[l, 2.0], $MachinePrecision]], $MachinePrecision], N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.69999999999999994e133

   1. Initial program 42.8%

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

   3. Simplified 49.7%

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

      1. add-log-exp 28.6%

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

      2. exp-prod 32.5%

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

   6. Applied egg-rr 32.5%

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

   7. Taylor expanded in k around inf 76.4%

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

      1. associate-*l/ 76.4%

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

      2. pow2 76.4%

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

      3. associate-*r* 76.5%

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

   9. Applied egg-rr 76.5%

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

       1. times-frac 78.7%

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

       2. associate-*l* 78.7%

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

   11. Simplified 78.7%

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

   if 1.69999999999999994e133 < k < 5.20000000000000008e221

   1. Initial program 47.7%

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

      1. *-commutative 47.7%

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

      2. associate-/r* 47.7%

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

   3. Simplified 58.3%

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

      1. add-sqr-sqrt 10.6%

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

      2. pow2 10.6%

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

      3. sqrt-div 10.5%

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

      4. sqrt-pow1 15.9%

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

      5. metadata-eval 15.9%

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

      6. sqrt-prod 5.3%

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

      7. add-sqr-sqrt 21.0%

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

   6. Applied egg-rr 21.0%

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

   if 5.20000000000000008e221 < k

   1. Initial program 38.9%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in k around 0 67.3%

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

      1. add-log-exp 67.3%

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

      2. *-commutative 67.3%

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

      3. exp-prod 73.2%

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

      4. pow2 73.2%

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

      5. *-commutative 73.2%

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

   7. Applied egg-rr 73.2%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{+133}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{elif}\;k \leq 5.2 \cdot 10^{+221}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{2}{t \cdot {k}^{4}}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{+134}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{{\ell}^{2}}\right)}^{\left(\frac{2}{t \cdot {k}^{4}}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.2e+134)
   (* (/ 2.0 (pow k 2.0)) (/ (pow l 2.0) (* t (* (sin k) (tan k)))))
   (log (pow (exp (pow l 2.0)) (/ 2.0 (* t (pow k 4.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.2e+134) {
		tmp = (2.0 / pow(k, 2.0)) * (pow(l, 2.0) / (t * (sin(k) * tan(k))));
	} else {
		tmp = log(pow(exp(pow(l, 2.0)), (2.0 / (t * pow(k, 4.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.2d+134) then
        tmp = (2.0d0 / (k ** 2.0d0)) * ((l ** 2.0d0) / (t * (sin(k) * tan(k))))
    else
        tmp = log((exp((l ** 2.0d0)) ** (2.0d0 / (t * (k ** 4.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.2e+134) {
		tmp = (2.0 / Math.pow(k, 2.0)) * (Math.pow(l, 2.0) / (t * (Math.sin(k) * Math.tan(k))));
	} else {
		tmp = Math.log(Math.pow(Math.exp(Math.pow(l, 2.0)), (2.0 / (t * Math.pow(k, 4.0)))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.2e+134:
		tmp = (2.0 / math.pow(k, 2.0)) * (math.pow(l, 2.0) / (t * (math.sin(k) * math.tan(k))))
	else:
		tmp = math.log(math.pow(math.exp(math.pow(l, 2.0)), (2.0 / (t * math.pow(k, 4.0)))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.2e+134)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64((l ^ 2.0) / Float64(t * Float64(sin(k) * tan(k)))));
	else
		tmp = log((exp((l ^ 2.0)) ^ Float64(2.0 / Float64(t * (k ^ 4.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.2e+134)
		tmp = (2.0 / (k ^ 2.0)) * ((l ^ 2.0) / (t * (sin(k) * tan(k))));
	else
		tmp = log((exp((l ^ 2.0)) ^ (2.0 / (t * (k ^ 4.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.2e+134], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[Power[N[Exp[N[Power[l, 2.0], $MachinePrecision]], $MachinePrecision], N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.20000000000000003e134

   1. Initial program 42.8%

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

   3. Simplified 49.7%

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

      1. add-log-exp 28.6%

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

      2. exp-prod 32.5%

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

   6. Applied egg-rr 32.5%

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

   7. Taylor expanded in k around inf 76.4%

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

      1. associate-*l/ 76.4%

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

      2. pow2 76.4%

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

      3. associate-*r* 76.5%

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

   9. Applied egg-rr 76.5%

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

       1. times-frac 78.7%

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

       2. associate-*l* 78.7%

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

   11. Simplified 78.7%

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

   if 1.20000000000000003e134 < k

   1. Initial program 43.4%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in k around 0 68.0%

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

      1. add-log-exp 68.0%

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

      2. *-commutative 68.0%

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

      3. exp-prod 72.3%

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

      4. pow2 72.3%

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

      5. *-commutative 72.3%

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

   7. Applied egg-rr 72.3%

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

Alternative 14: 74.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.9%

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

3. Simplified 50.3%

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

   1. add-log-exp 32.3%

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

   2. exp-prod 34.9%

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

6. Applied egg-rr 34.9%

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

7. Taylor expanded in k around inf 75.2%

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

   1. associate-*l/ 75.2%

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

   2. pow2 75.2%

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

   3. associate-*r* 75.2%

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

9. Applied egg-rr 75.2%

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

    1. times-frac 77.2%

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

    2. associate-*l* 77.2%

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

11. Simplified 77.2%

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

Alternative 15: 74.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.9%

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

3. Simplified 50.3%

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

   1. add-log-exp 32.3%

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

   2. exp-prod 34.9%

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

6. Applied egg-rr 34.9%

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

7. Taylor expanded in k around inf 75.2%

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

   1. *-un-lft-identity 75.2%

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

   2. associate-/r* 75.3%

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

   3. associate-*r* 75.3%

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

9. Applied egg-rr 75.3%

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

10. Final simplification 75.3%

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

Alternative 16: 74.1% accurate, 1.3× speedup

Math

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

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 42.9%

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

3. Simplified 50.3%

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

   1. add-log-exp 32.3%

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

   2. exp-prod 34.9%

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

6. Applied egg-rr 34.9%

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

7. Taylor expanded in k around inf 75.2%

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

   1. div-inv 75.2%

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

   2. associate-*r* 75.2%

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

9. Applied egg-rr 75.2%

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

    1. associate-*r/ 75.2%

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

    2. metadata-eval 75.2%

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

    3. associate-/r* 75.3%

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

    4. associate-*l* 75.3%

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

11. Simplified 75.3%

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

12. Final simplification 75.3%

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

Alternative 17: 74.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.9%

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

3. Simplified 50.3%

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

   1. add-log-exp 32.3%

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

   2. exp-prod 34.9%

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

6. Applied egg-rr 34.9%

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

7. Taylor expanded in k around inf 75.2%

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

   1. unpow2 75.2%

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

9. Applied egg-rr 75.2%

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

10. Final simplification 75.2%

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

Alternative 18: 64.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return (l * l) * (2.0 * (cos(k) / (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 * (cos(k) / (t * (k ** 4.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.9%

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

3. Simplified 50.3%

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

5. Taylor expanded in t around 0 75.2%

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

6. Taylor expanded in k around 0 68.3%

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

7. Final simplification 68.3%

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

Alternative 19: 63.0% 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 42.9%

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

3. Simplified 50.3%

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

5. Taylor expanded in k around 0 66.7%

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

6. Final simplification 66.7%

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

Alternative 20: 63.0% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.9%

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

3. Simplified 50.3%

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

5. Taylor expanded in k around 0 66.7%

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

   1. *-commutative 66.7%

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

   2. associate-/r* 66.7%

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

7. Simplified 66.7%

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

   1. div-inv 66.7%

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

   2. pow-flip 66.7%

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

   3. metadata-eval 66.7%

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

9. Applied egg-rr 66.7%

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

    1. associate-*l/ 66.7%

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

    2. associate-/l* 66.7%

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

11. Simplified 66.7%

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

12. Final simplification 66.7%

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

Reproduce

herbie shell --seed 2024182 
(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))))