Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.9% → 91.1%

Time: 17.9s

Alternatives: 15

Speedup: 28.1×

• 27.4% 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: 54.9% 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: 91.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 0.0235:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k} \cdot \sqrt[3]{k \cdot 2}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 0.0235)
   (/ 2.0 (pow (/ (* (cbrt k) (cbrt (* k 2.0))) (/ (pow (cbrt l) 2.0) t)) 3.0))
   (/ 2.0 (* (* (/ k l) (/ t (/ l k))) (* (sin k) (tan k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.0235) {
		tmp = 2.0 / pow(((cbrt(k) * cbrt((k * 2.0))) / (pow(cbrt(l), 2.0) / t)), 3.0);
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (sin(k) * tan(k)));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.0235) {
		tmp = 2.0 / Math.pow(((Math.cbrt(k) * Math.cbrt((k * 2.0))) / (Math.pow(Math.cbrt(l), 2.0) / t)), 3.0);
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 0.0235)
		tmp = Float64(2.0 / (Float64(Float64(cbrt(k) * cbrt(Float64(k * 2.0))) / Float64((cbrt(l) ^ 2.0) / t)) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t / Float64(l / k))) * Float64(sin(k) * tan(k))));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 0.0235], N[(2.0 / N[Power[N[(N[(N[Power[k, 1/3], $MachinePrecision] * N[Power[N[(k * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0235

   1. Initial program 51.0%

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

      1. associate-*l* 51.0%

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

      2. +-commutative 51.0%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in k around 0 51.4%

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

   5. Taylor expanded in k around 0 53.8%

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

      1. associate-/l* 52.4%

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

      2. unpow2 52.4%

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

      3. associate-*r/ 59.7%

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

   7. Simplified 59.7%

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

      1. add-cube-cbrt 59.6%

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

      2. pow3 59.6%

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

      3. associate-*l/ 54.5%

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

      4. cbrt-div 54.5%

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

      5. *-commutative 54.5%

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

      6. associate-*r/ 47.7%

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

      7. cbrt-div 48.7%

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

      8. cbrt-unprod 56.0%

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

      9. pow2 56.0%

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

      10. rem-cbrt-cube 65.8%

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

   9. Applied egg-rr 65.8%

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

       1. cbrt-prod 75.6%

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

   11. Applied egg-rr 75.6%

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

   if 0.0235 < k

   1. Initial program 41.8%

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

      1. *-commutative 41.8%

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

      2. associate-*l* 41.8%

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

      3. associate-*r* 41.8%

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

      4. +-commutative 41.8%

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

      5. associate-+r+ 41.8%

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

      6. metadata-eval 41.8%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in k around inf 63.7%

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

      1. *-commutative 63.7%

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

      2. unpow2 63.7%

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

      3. times-frac 68.7%

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

      4. unpow2 68.7%

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

   6. Simplified 68.7%

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

      1. associate-*l/ 70.2%

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

      2. associate-/l* 83.6%

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

   8. Applied egg-rr 83.6%

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

      1. associate-/l* 84.8%

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

      2. associate-/r/ 84.8%

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

   10. Simplified 84.8%

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

       1. *-un-lft-identity 84.8%

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

       2. add-sqr-sqrt 84.6%

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

       3. times-frac 84.6%

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

       4. *-un-lft-identity 84.6%

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

       5. frac-times 84.6%

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

       6. clear-num 84.7%

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

       7. sqrt-unprod 60.2%

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

       8. add-sqr-sqrt 66.9%

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

       9. clear-num 66.8%

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

       10. remove-double-div 66.9%

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

       11. *-un-lft-identity 66.9%

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

       12. frac-times 69.8%

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

       13. clear-num 69.8%

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

       14. sqrt-unprod 63.1%

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

       15. add-sqr-sqrt 91.7%

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

   12. Applied egg-rr 91.7%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0235:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{k} \cdot \sqrt[3]{k \cdot 2}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 2: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-169}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left(\sqrt[3]{\frac{k}{\ell} \cdot \frac{k}{\ell}} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 0.00126:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{k \cdot \left(k \cdot 2\right)} \cdot \frac{1}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 5.5e-169)
   (/ 2.0 (pow (* t (* (cbrt (* (/ k l) (/ k l))) (cbrt 2.0))) 3.0))
   (if (<= k 0.00126)
     (/
      2.0
      (pow (* (cbrt (* k (* k 2.0))) (/ 1.0 (/ (pow (cbrt l) 2.0) t))) 3.0))
     (/ 2.0 (* (* (/ k l) (/ t (/ l k))) (* (sin k) (tan k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.5e-169) {
		tmp = 2.0 / pow((t * (cbrt(((k / l) * (k / l))) * cbrt(2.0))), 3.0);
	} else if (k <= 0.00126) {
		tmp = 2.0 / pow((cbrt((k * (k * 2.0))) * (1.0 / (pow(cbrt(l), 2.0) / t))), 3.0);
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (sin(k) * tan(k)));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.5e-169) {
		tmp = 2.0 / Math.pow((t * (Math.cbrt(((k / l) * (k / l))) * Math.cbrt(2.0))), 3.0);
	} else if (k <= 0.00126) {
		tmp = 2.0 / Math.pow((Math.cbrt((k * (k * 2.0))) * (1.0 / (Math.pow(Math.cbrt(l), 2.0) / t))), 3.0);
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 5.5e-169)
		tmp = Float64(2.0 / (Float64(t * Float64(cbrt(Float64(Float64(k / l) * Float64(k / l))) * cbrt(2.0))) ^ 3.0));
	elseif (k <= 0.00126)
		tmp = Float64(2.0 / (Float64(cbrt(Float64(k * Float64(k * 2.0))) * Float64(1.0 / Float64((cbrt(l) ^ 2.0) / t))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t / Float64(l / k))) * Float64(sin(k) * tan(k))));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 5.5e-169], N[(2.0 / N[Power[N[(t * N[(N[Power[N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.00126], N[(2.0 / N[Power[N[(N[Power[N[(k * N[(k * 2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(1.0 / N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.4999999999999994e-169

   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. associate-*l* 45.4%

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

      2. +-commutative 45.4%

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

   3. Simplified 45.4%

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

   4. Taylor expanded in k around 0 44.5%

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

   5. Taylor expanded in k around 0 47.1%

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

      1. associate-/l* 45.8%

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

      2. unpow2 45.8%

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

      3. associate-*r/ 54.8%

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

   7. Simplified 54.8%

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

      1. add-cube-cbrt 54.7%

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

      2. pow3 54.7%

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

      3. associate-*l/ 47.8%

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

      4. cbrt-div 47.8%

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

      5. *-commutative 47.8%

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

      6. associate-*r/ 39.6%

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

      7. cbrt-div 39.6%

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

      8. cbrt-unprod 48.7%

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

      9. pow2 48.7%

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

      10. rem-cbrt-cube 59.2%

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

   9. Applied egg-rr 59.2%

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

   10. Taylor expanded in t around 0 46.0%

       \[\leadsto \frac{2}{{\color{blue}{\left({\left(\frac{{k}^{2} \cdot 1}{{\ell}^{2}}\right)}^{0.3333333333333333} \cdot \left(\sqrt[3]{2} \cdot t\right)\right)}}^{3}} \]
   11. Step-by-step derivation

       1. associate-*r* 46.0%

          \[\leadsto \frac{2}{{\color{blue}{\left(\left({\left(\frac{{k}^{2} \cdot 1}{{\ell}^{2}}\right)}^{0.3333333333333333} \cdot \sqrt[3]{2}\right) \cdot t\right)}}^{3}} \]

       2. *-commutative 46.0%

          \[\leadsto \frac{2}{{\color{blue}{\left(t \cdot \left({\left(\frac{{k}^{2} \cdot 1}{{\ell}^{2}}\right)}^{0.3333333333333333} \cdot \sqrt[3]{2}\right)\right)}}^{3}} \]

       3. unpow1/3 46.1%

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

       4. unpow2 46.1%

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

       5. *-rgt-identity 46.1%

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

       6. unpow2 46.1%

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

       7. times-frac 65.8%

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

   12. Simplified 65.8%

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

   if 5.4999999999999994e-169 < k < 0.00126000000000000005

   1. Initial program 66.9%

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

      1. associate-*l* 66.9%

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

      2. +-commutative 66.9%

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

   3. Simplified 66.9%

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

   4. Taylor expanded in k around 0 71.0%

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

   5. Taylor expanded in k around 0 72.5%

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

      1. associate-/l* 71.0%

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

      2. unpow2 71.0%

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

      3. associate-*r/ 73.5%

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

   7. Simplified 73.5%

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

      1. add-cube-cbrt 73.4%

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

      2. pow3 73.4%

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

      3. associate-*l/ 73.4%

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

      4. cbrt-div 73.3%

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

      5. *-commutative 73.3%

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

      6. associate-*r/ 70.8%

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

      7. cbrt-div 74.4%

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

      8. cbrt-unprod 76.6%

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

      9. pow2 76.6%

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

      10. rem-cbrt-cube 84.2%

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

   9. Applied egg-rr 84.2%

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

       1. div-inv 84.2%

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

   11. Applied egg-rr 84.2%

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

   if 0.00126000000000000005 < k

   1. Initial program 41.8%

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

      1. *-commutative 41.8%

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

      2. associate-*l* 41.8%

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

      3. associate-*r* 41.8%

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

      4. +-commutative 41.8%

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

      5. associate-+r+ 41.8%

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

      6. metadata-eval 41.8%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in k around inf 63.7%

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

      1. *-commutative 63.7%

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

      2. unpow2 63.7%

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

      3. times-frac 68.7%

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

      4. unpow2 68.7%

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

   6. Simplified 68.7%

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

      1. associate-*l/ 70.2%

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

      2. associate-/l* 83.6%

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

   8. Applied egg-rr 83.6%

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

      1. associate-/l* 84.8%

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

      2. associate-/r/ 84.8%

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

   10. Simplified 84.8%

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

       1. *-un-lft-identity 84.8%

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

       2. add-sqr-sqrt 84.6%

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

       3. times-frac 84.6%

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

       4. *-un-lft-identity 84.6%

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

       5. frac-times 84.6%

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

       6. clear-num 84.7%

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

       7. sqrt-unprod 60.2%

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

       8. add-sqr-sqrt 66.9%

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

       9. clear-num 66.8%

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

       10. remove-double-div 66.9%

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

       11. *-un-lft-identity 66.9%

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

       12. frac-times 69.8%

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

       13. clear-num 69.8%

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

       14. sqrt-unprod 63.1%

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

       15. add-sqr-sqrt 91.7%

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

   12. Applied egg-rr 91.7%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-169}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left(\sqrt[3]{\frac{k}{\ell} \cdot \frac{k}{\ell}} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 0.00126:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{k \cdot \left(k \cdot 2\right)} \cdot \frac{1}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 3: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-169}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left(\sqrt[3]{\frac{k}{\ell} \cdot \frac{k}{\ell}} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 0.008:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{k \cdot \left(k \cdot 2\right)}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 5.5e-169)
   (/ 2.0 (pow (* t (* (cbrt (* (/ k l) (/ k l))) (cbrt 2.0))) 3.0))
   (if (<= k 0.008)
     (/ 2.0 (pow (* t (/ (cbrt (* k (* k 2.0))) (pow (cbrt l) 2.0))) 3.0))
     (/ 2.0 (* (* (/ k l) (/ t (/ l k))) (* (sin k) (tan k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.5e-169) {
		tmp = 2.0 / pow((t * (cbrt(((k / l) * (k / l))) * cbrt(2.0))), 3.0);
	} else if (k <= 0.008) {
		tmp = 2.0 / pow((t * (cbrt((k * (k * 2.0))) / pow(cbrt(l), 2.0))), 3.0);
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (sin(k) * tan(k)));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.5e-169) {
		tmp = 2.0 / Math.pow((t * (Math.cbrt(((k / l) * (k / l))) * Math.cbrt(2.0))), 3.0);
	} else if (k <= 0.008) {
		tmp = 2.0 / Math.pow((t * (Math.cbrt((k * (k * 2.0))) / Math.pow(Math.cbrt(l), 2.0))), 3.0);
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 5.5e-169)
		tmp = Float64(2.0 / (Float64(t * Float64(cbrt(Float64(Float64(k / l) * Float64(k / l))) * cbrt(2.0))) ^ 3.0));
	elseif (k <= 0.008)
		tmp = Float64(2.0 / (Float64(t * Float64(cbrt(Float64(k * Float64(k * 2.0))) / (cbrt(l) ^ 2.0))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t / Float64(l / k))) * Float64(sin(k) * tan(k))));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 5.5e-169], N[(2.0 / N[Power[N[(t * N[(N[Power[N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.008], N[(2.0 / N[Power[N[(t * N[(N[Power[N[(k * N[(k * 2.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.4999999999999994e-169

   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. associate-*l* 45.4%

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

      2. +-commutative 45.4%

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

   3. Simplified 45.4%

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

   4. Taylor expanded in k around 0 44.5%

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

   5. Taylor expanded in k around 0 47.1%

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

      1. associate-/l* 45.8%

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

      2. unpow2 45.8%

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

      3. associate-*r/ 54.8%

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

   7. Simplified 54.8%

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

      1. add-cube-cbrt 54.7%

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

      2. pow3 54.7%

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

      3. associate-*l/ 47.8%

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

      4. cbrt-div 47.8%

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

      5. *-commutative 47.8%

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

      6. associate-*r/ 39.6%

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

      7. cbrt-div 39.6%

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

      8. cbrt-unprod 48.7%

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

      9. pow2 48.7%

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

      10. rem-cbrt-cube 59.2%

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

   9. Applied egg-rr 59.2%

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

   10. Taylor expanded in t around 0 46.0%

       \[\leadsto \frac{2}{{\color{blue}{\left({\left(\frac{{k}^{2} \cdot 1}{{\ell}^{2}}\right)}^{0.3333333333333333} \cdot \left(\sqrt[3]{2} \cdot t\right)\right)}}^{3}} \]
   11. Step-by-step derivation

       1. associate-*r* 46.0%

          \[\leadsto \frac{2}{{\color{blue}{\left(\left({\left(\frac{{k}^{2} \cdot 1}{{\ell}^{2}}\right)}^{0.3333333333333333} \cdot \sqrt[3]{2}\right) \cdot t\right)}}^{3}} \]

       2. *-commutative 46.0%

          \[\leadsto \frac{2}{{\color{blue}{\left(t \cdot \left({\left(\frac{{k}^{2} \cdot 1}{{\ell}^{2}}\right)}^{0.3333333333333333} \cdot \sqrt[3]{2}\right)\right)}}^{3}} \]

       3. unpow1/3 46.1%

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

       4. unpow2 46.1%

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

       5. *-rgt-identity 46.1%

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

       6. unpow2 46.1%

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

       7. times-frac 65.8%

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

   12. Simplified 65.8%

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

   if 5.4999999999999994e-169 < k < 0.0080000000000000002

   1. Initial program 66.9%

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

      1. associate-*l* 66.9%

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

      2. +-commutative 66.9%

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

   3. Simplified 66.9%

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

   4. Taylor expanded in k around 0 71.0%

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

   5. Taylor expanded in k around 0 72.5%

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

      1. associate-/l* 71.0%

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

      2. unpow2 71.0%

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

      3. associate-*r/ 73.5%

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

   7. Simplified 73.5%

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

      1. add-cube-cbrt 73.4%

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

      2. pow3 73.4%

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

      3. associate-*l/ 73.4%

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

      4. cbrt-div 73.3%

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

      5. *-commutative 73.3%

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

      6. associate-*r/ 70.8%

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

      7. cbrt-div 74.4%

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

      8. cbrt-unprod 76.6%

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

      9. pow2 76.6%

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

      10. rem-cbrt-cube 84.2%

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

   9. Applied egg-rr 84.2%

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

       1. associate-/r/ 84.2%

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

   11. Applied egg-rr 84.2%

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

   if 0.0080000000000000002 < k

   1. Initial program 41.8%

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

      1. *-commutative 41.8%

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

      2. associate-*l* 41.8%

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

      3. associate-*r* 41.8%

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

      4. +-commutative 41.8%

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

      5. associate-+r+ 41.8%

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

      6. metadata-eval 41.8%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in k around inf 63.7%

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

      1. *-commutative 63.7%

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

      2. unpow2 63.7%

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

      3. times-frac 68.7%

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

      4. unpow2 68.7%

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

   6. Simplified 68.7%

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

      1. associate-*l/ 70.2%

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

      2. associate-/l* 83.6%

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

   8. Applied egg-rr 83.6%

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

      1. associate-/l* 84.8%

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

      2. associate-/r/ 84.8%

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

   10. Simplified 84.8%

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

       1. *-un-lft-identity 84.8%

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

       2. add-sqr-sqrt 84.6%

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

       3. times-frac 84.6%

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

       4. *-un-lft-identity 84.6%

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

       5. frac-times 84.6%

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

       6. clear-num 84.7%

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

       7. sqrt-unprod 60.2%

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

       8. add-sqr-sqrt 66.9%

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

       9. clear-num 66.8%

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

       10. remove-double-div 66.9%

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

       11. *-un-lft-identity 66.9%

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

       12. frac-times 69.8%

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

       13. clear-num 69.8%

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

       14. sqrt-unprod 63.1%

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

       15. add-sqr-sqrt 91.7%

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

   12. Applied egg-rr 91.7%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-169}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left(\sqrt[3]{\frac{k}{\ell} \cdot \frac{k}{\ell}} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 0.008:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{k \cdot \left(k \cdot 2\right)}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 4: 86.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 0.054:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{\frac{k}{\ell}}}{\frac{\sqrt[3]{\ell}}{t}}\right)}^{3} \cdot \left(k \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 0.054)
   (/ 2.0 (* (pow (/ (cbrt (/ k l)) (/ (cbrt l) t)) 3.0) (* k 2.0)))
   (/ 2.0 (* (* (/ k l) (/ t (/ l k))) (* (sin k) (tan k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.054) {
		tmp = 2.0 / (pow((cbrt((k / l)) / (cbrt(l) / t)), 3.0) * (k * 2.0));
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (sin(k) * tan(k)));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.054) {
		tmp = 2.0 / (Math.pow((Math.cbrt((k / l)) / (Math.cbrt(l) / t)), 3.0) * (k * 2.0));
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 0.054)
		tmp = Float64(2.0 / Float64((Float64(cbrt(Float64(k / l)) / Float64(cbrt(l) / t)) ^ 3.0) * Float64(k * 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t / Float64(l / k))) * Float64(sin(k) * tan(k))));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 0.054], N[(2.0 / N[(N[Power[N[(N[Power[N[(k / l), $MachinePrecision], 1/3], $MachinePrecision] / N[(N[Power[l, 1/3], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(k * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0539999999999999994

   1. Initial program 51.0%

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

      1. associate-*l* 51.0%

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

      2. +-commutative 51.0%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in k around 0 51.4%

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

   5. Taylor expanded in k around 0 53.8%

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

      1. associate-/l* 52.4%

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

      2. unpow2 52.4%

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

      3. associate-*r/ 59.7%

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

   7. Simplified 59.7%

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

      1. add-cube-cbrt 59.6%

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

      2. pow3 59.6%

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

      3. associate-/r* 62.2%

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

      4. cbrt-div 62.1%

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

      5. cbrt-div 62.3%

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

      6. rem-cbrt-cube 68.6%

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

   9. Applied egg-rr 68.6%

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

   if 0.0539999999999999994 < k

   1. Initial program 41.8%

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

      1. *-commutative 41.8%

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

      2. associate-*l* 41.8%

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

      3. associate-*r* 41.8%

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

      4. +-commutative 41.8%

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

      5. associate-+r+ 41.8%

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

      6. metadata-eval 41.8%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in k around inf 63.7%

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

      1. *-commutative 63.7%

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

      2. unpow2 63.7%

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

      3. times-frac 68.7%

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

      4. unpow2 68.7%

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

   6. Simplified 68.7%

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

      1. associate-*l/ 70.2%

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

      2. associate-/l* 83.6%

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

   8. Applied egg-rr 83.6%

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

      1. associate-/l* 84.8%

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

      2. associate-/r/ 84.8%

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

   10. Simplified 84.8%

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

       1. *-un-lft-identity 84.8%

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

       2. add-sqr-sqrt 84.6%

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

       3. times-frac 84.6%

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

       4. *-un-lft-identity 84.6%

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

       5. frac-times 84.6%

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

       6. clear-num 84.7%

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

       7. sqrt-unprod 60.2%

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

       8. add-sqr-sqrt 66.9%

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

       9. clear-num 66.8%

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

       10. remove-double-div 66.9%

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

       11. *-un-lft-identity 66.9%

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

       12. frac-times 69.8%

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

       13. clear-num 69.8%

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

       14. sqrt-unprod 63.1%

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

       15. add-sqr-sqrt 91.7%

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

   12. Applied egg-rr 91.7%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.054:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{\frac{k}{\ell}}}{\frac{\sqrt[3]{\ell}}{t}}\right)}^{3} \cdot \left(k \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 5: 84.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 0.122:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \frac{k}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 0.122)
   (/ 2.0 (* (* k 2.0) (/ k (pow (/ (pow (cbrt l) 2.0) t) 3.0))))
   (/ 2.0 (* (* (/ k l) (/ t (/ l k))) (* (sin k) (tan k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.122) {
		tmp = 2.0 / ((k * 2.0) * (k / pow((pow(cbrt(l), 2.0) / t), 3.0)));
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (sin(k) * tan(k)));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.122) {
		tmp = 2.0 / ((k * 2.0) * (k / Math.pow((Math.pow(Math.cbrt(l), 2.0) / t), 3.0)));
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 0.122)
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(k / (Float64((cbrt(l) ^ 2.0) / t) ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t / Float64(l / k))) * Float64(sin(k) * tan(k))));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 0.122], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(k / N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.122

   1. Initial program 51.0%

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

      1. associate-*l* 51.0%

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

      2. +-commutative 51.0%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in k around 0 51.4%

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

   5. Taylor expanded in k around 0 53.8%

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

      1. associate-/l* 52.4%

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

      2. unpow2 52.4%

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

      3. associate-*r/ 59.7%

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

   7. Simplified 59.7%

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

      1. add-cube-cbrt 59.6%

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

      2. pow3 59.6%

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

      3. associate-*r/ 52.3%

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

      4. cbrt-div 52.4%

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

      5. cbrt-unprod 59.6%

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

      6. pow2 59.6%

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

      7. rem-cbrt-cube 65.7%

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

   9. Applied egg-rr 65.7%

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

   if 0.122 < k

   1. Initial program 41.8%

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

      1. *-commutative 41.8%

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

      2. associate-*l* 41.8%

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

      3. associate-*r* 41.8%

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

      4. +-commutative 41.8%

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

      5. associate-+r+ 41.8%

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

      6. metadata-eval 41.8%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in k around inf 63.7%

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

      1. *-commutative 63.7%

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

      2. unpow2 63.7%

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

      3. times-frac 68.7%

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

      4. unpow2 68.7%

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

   6. Simplified 68.7%

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

      1. associate-*l/ 70.2%

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

      2. associate-/l* 83.6%

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

   8. Applied egg-rr 83.6%

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

      1. associate-/l* 84.8%

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

      2. associate-/r/ 84.8%

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

   10. Simplified 84.8%

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

       1. *-un-lft-identity 84.8%

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

       2. add-sqr-sqrt 84.6%

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

       3. times-frac 84.6%

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

       4. *-un-lft-identity 84.6%

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

       5. frac-times 84.6%

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

       6. clear-num 84.7%

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

       7. sqrt-unprod 60.2%

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

       8. add-sqr-sqrt 66.9%

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

       9. clear-num 66.8%

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

       10. remove-double-div 66.9%

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

       11. *-un-lft-identity 66.9%

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

       12. frac-times 69.8%

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

       13. clear-num 69.8%

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

       14. sqrt-unprod 63.1%

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

       15. add-sqr-sqrt 91.7%

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

   12. Applied egg-rr 91.7%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.122:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \frac{k}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 6: 69.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.95 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{2}{{t}^{-3}}\right)}\\ \mathbf{elif}\;t \leq 1.36 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \frac{k}{{\left(\frac{\ell}{{t}^{1.5}}\right)}^{2}}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t -1.95e-99)
   (/ 2.0 (* (/ k l) (* (/ k l) (/ 2.0 (pow t -3.0)))))
   (if (<= t 1.36e-95)
     (/ 2.0 (* (* k k) (/ (* t (/ k (/ l k))) l)))
     (/ 2.0 (* (* k 2.0) (/ k (pow (/ l (pow t 1.5)) 2.0)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.95e-99) {
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / pow(t, -3.0))));
	} else if (t <= 1.36e-95) {
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	} else {
		tmp = 2.0 / ((k * 2.0) * (k / pow((l / pow(t, 1.5)), 2.0)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-1.95d-99)) then
        tmp = 2.0d0 / ((k / l) * ((k / l) * (2.0d0 / (t ** (-3.0d0)))))
    else if (t <= 1.36d-95) then
        tmp = 2.0d0 / ((k * k) * ((t * (k / (l / k))) / l))
    else
        tmp = 2.0d0 / ((k * 2.0d0) * (k / ((l / (t ** 1.5d0)) ** 2.0d0)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.95e-99) {
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / Math.pow(t, -3.0))));
	} else if (t <= 1.36e-95) {
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	} else {
		tmp = 2.0 / ((k * 2.0) * (k / Math.pow((l / Math.pow(t, 1.5)), 2.0)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if t <= -1.95e-99:
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / math.pow(t, -3.0))))
	elif t <= 1.36e-95:
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l))
	else:
		tmp = 2.0 / ((k * 2.0) * (k / math.pow((l / math.pow(t, 1.5)), 2.0)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (t <= -1.95e-99)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(k / l) * Float64(2.0 / (t ^ -3.0)))));
	elseif (t <= 1.36e-95)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t * Float64(k / Float64(l / k))) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(k * 2.0) * Float64(k / (Float64(l / (t ^ 1.5)) ^ 2.0))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -1.95e-99)
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / (t ^ -3.0))));
	elseif (t <= 1.36e-95)
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	else
		tmp = 2.0 / ((k * 2.0) * (k / ((l / (t ^ 1.5)) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, -1.95e-99], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(2.0 / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.36e-95], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * 2.0), $MachinePrecision] * N[(k / N[Power[N[(l / N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.94999999999999994e-99

   1. Initial program 62.2%

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

      1. associate-*l* 62.2%

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

      2. +-commutative 62.2%

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

   3. Simplified 62.2%

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

   4. Taylor expanded in k around 0 59.1%

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

   5. Taylor expanded in k around 0 61.2%

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

      1. associate-/l* 59.1%

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

      2. unpow2 59.1%

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

      3. associate-*r/ 61.9%

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

   7. Simplified 61.9%

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

      1. expm1-log1p-u 17.9%

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

      2. expm1-udef 1.3%

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

      3. associate-*l/ 1.2%

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

      4. times-frac 1.7%

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

      5. *-commutative 1.7%

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

      6. div-inv 1.7%

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

      7. pow-flip 1.7%

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

      8. metadata-eval 1.7%

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

   9. Applied egg-rr 1.7%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{k}{\ell} \cdot \frac{k \cdot 2}{\ell \cdot {t}^{-3}}\right)} - 1}} \]
   10. Step-by-step derivation

       1. expm1-def 19.8%

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

       2. expm1-log1p 66.7%

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

       3. times-frac 67.0%

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

   11. Simplified 67.0%

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

   if -1.94999999999999994e-99 < t < 1.36e-95

   1. Initial program 32.5%

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

      1. *-commutative 32.5%

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

      2. associate-*l* 32.5%

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

      3. associate-*r* 32.5%

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

      4. +-commutative 32.5%

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

      5. associate-+r+ 32.5%

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

      6. metadata-eval 32.5%

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

   3. Simplified 32.5%

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

   4. Taylor expanded in k around inf 65.2%

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

      1. *-commutative 65.2%

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

      2. unpow2 65.2%

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

      3. times-frac 83.2%

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

      4. unpow2 83.2%

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

   6. Simplified 83.2%

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

   7. Taylor expanded in k around 0 72.8%

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

      1. unpow2 73.0%

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

   9. Simplified 72.8%

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

       1. associate-*l/ 84.4%

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

       2. associate-/l* 88.7%

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

   11. Applied egg-rr 73.0%

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

   if 1.36e-95 < t

   1. Initial program 52.5%

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

      1. associate-*l* 52.5%

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

      2. +-commutative 52.5%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in k around 0 47.4%

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

   5. Taylor expanded in k around 0 51.3%

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

      1. associate-/l* 50.1%

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

      2. unpow2 50.1%

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

      3. associate-*r/ 54.7%

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

   7. Simplified 54.7%

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

      1. add-sqr-sqrt 54.7%

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

      2. pow2 54.7%

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

      3. associate-*r/ 50.1%

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

      4. sqrt-div 50.1%

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

      5. sqrt-unprod 34.1%

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

      6. add-sqr-sqrt 54.7%

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

      7. sqrt-pow1 65.4%

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

      8. metadata-eval 65.4%

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

   9. Applied egg-rr 65.4%

      \[\leadsto \frac{2}{\frac{k}{\color{blue}{{\left(\frac{\ell}{{t}^{1.5}}\right)}^{2}}} \cdot \left(2 \cdot k\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.95 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{2}{{t}^{-3}}\right)}\\ \mathbf{elif}\;t \leq 1.36 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \frac{k}{{\left(\frac{\ell}{{t}^{1.5}}\right)}^{2}}}\\ \end{array} \]

Alternative 7: 73.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 0.025:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{2}{{t}^{-3}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 0.025)
   (/ 2.0 (* (/ k l) (* (/ k l) (/ 2.0 (pow t -3.0)))))
   (* (* l l) (/ 2.0 (* (tan k) (* (* k k) (* t (sin k))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.025) {
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / pow(t, -3.0))));
	} else {
		tmp = (l * l) * (2.0 / (tan(k) * ((k * k) * (t * sin(k)))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 <= 0.025d0) then
        tmp = 2.0d0 / ((k / l) * ((k / l) * (2.0d0 / (t ** (-3.0d0)))))
    else
        tmp = (l * l) * (2.0d0 / (tan(k) * ((k * k) * (t * sin(k)))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 0.025) {
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / Math.pow(t, -3.0))));
	} else {
		tmp = (l * l) * (2.0 / (Math.tan(k) * ((k * k) * (t * Math.sin(k)))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 0.025:
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / math.pow(t, -3.0))))
	else:
		tmp = (l * l) * (2.0 / (math.tan(k) * ((k * k) * (t * math.sin(k)))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 0.025)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(k / l) * Float64(2.0 / (t ^ -3.0)))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * Float64(Float64(k * k) * Float64(t * sin(k))))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 0.025)
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / (t ^ -3.0))));
	else
		tmp = (l * l) * (2.0 / (tan(k) * ((k * k) * (t * sin(k)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 0.025], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(2.0 / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.025000000000000001

   1. Initial program 51.0%

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

      1. associate-*l* 51.0%

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

      2. +-commutative 51.0%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in k around 0 51.4%

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

   5. Taylor expanded in k around 0 53.8%

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

      1. associate-/l* 52.4%

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

      2. unpow2 52.4%

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

      3. associate-*r/ 59.7%

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

   7. Simplified 59.7%

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

      1. expm1-log1p-u 45.6%

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

      2. expm1-udef 27.2%

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

      3. associate-*l/ 24.3%

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

      4. times-frac 28.6%

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

      5. *-commutative 28.6%

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

      6. div-inv 28.6%

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

      7. pow-flip 28.6%

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

      8. metadata-eval 28.6%

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

   9. Applied egg-rr 28.6%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{k}{\ell} \cdot \frac{k \cdot 2}{\ell \cdot {t}^{-3}}\right)} - 1}} \]
   10. Step-by-step derivation

       1. expm1-def 47.7%

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

       2. expm1-log1p 63.2%

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

       3. times-frac 63.8%

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

   11. Simplified 63.8%

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

   if 0.025000000000000001 < k

   1. Initial program 41.8%

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

      1. associate-/l/ 41.8%

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

      2. associate-*l/ 41.8%

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

      3. associate-*l/ 41.8%

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

      4. associate-/r/ 41.4%

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

      5. *-commutative 41.4%

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

      6. associate-/l/ 40.3%

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

      7. associate-*r* 40.3%

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

      8. *-commutative 40.3%

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

      9. associate-*r* 40.3%

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

      10. *-commutative 40.3%

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

   3. Simplified 40.3%

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

   4. Taylor expanded in k around inf 63.6%

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

      1. unpow2 63.6%

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

      2. *-commutative 63.6%

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

   6. Simplified 63.6%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.025:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{2}{{t}^{-3}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\\ \end{array} \]

Alternative 8: 79.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 4.9 \cdot 10^{-42}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{2}{{t}^{-3}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(k \cdot \left(\frac{k}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 4.9e-42)
   (/ 2.0 (* (/ k l) (* (/ k l) (/ 2.0 (pow t -3.0)))))
   (/ 2.0 (* (* (sin k) (tan k)) (* k (* (/ k l) (/ t l)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.9e-42) {
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / pow(t, -3.0))));
	} else {
		tmp = 2.0 / ((sin(k) * tan(k)) * (k * ((k / l) * (t / l))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 <= 4.9d-42) then
        tmp = 2.0d0 / ((k / l) * ((k / l) * (2.0d0 / (t ** (-3.0d0)))))
    else
        tmp = 2.0d0 / ((sin(k) * tan(k)) * (k * ((k / l) * (t / l))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.9e-42) {
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / Math.pow(t, -3.0))));
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.tan(k)) * (k * ((k / l) * (t / l))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 4.9e-42:
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / math.pow(t, -3.0))))
	else:
		tmp = 2.0 / ((math.sin(k) * math.tan(k)) * (k * ((k / l) * (t / l))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 4.9e-42)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(k / l) * Float64(2.0 / (t ^ -3.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * tan(k)) * Float64(k * Float64(Float64(k / l) * Float64(t / l)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4.9e-42)
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / (t ^ -3.0))));
	else
		tmp = 2.0 / ((sin(k) * tan(k)) * (k * ((k / l) * (t / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 4.9e-42], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(2.0 / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[(k / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.9e-42

   1. Initial program 49.4%

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

      1. associate-*l* 49.4%

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

      2. +-commutative 49.4%

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

   3. Simplified 49.4%

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

   4. Taylor expanded in k around 0 49.2%

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

   5. Taylor expanded in k around 0 51.7%

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

      1. associate-/l* 50.3%

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

      2. unpow2 50.3%

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

      3. associate-*r/ 58.1%

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

   7. Simplified 58.1%

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

      1. expm1-log1p-u 44.7%

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

      2. expm1-udef 27.9%

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

      3. associate-*l/ 24.8%

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

      4. times-frac 29.4%

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

      5. *-commutative 29.4%

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

      6. div-inv 29.4%

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

      7. pow-flip 29.4%

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

      8. metadata-eval 29.4%

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

   9. Applied egg-rr 29.4%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{k}{\ell} \cdot \frac{k \cdot 2}{\ell \cdot {t}^{-3}}\right)} - 1}} \]
   10. Step-by-step derivation

       1. expm1-def 46.9%

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

       2. expm1-log1p 61.8%

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

       3. times-frac 62.5%

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

   11. Simplified 62.5%

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

   if 4.9e-42 < k

   1. Initial program 47.0%

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

      1. *-commutative 47.0%

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

      2. associate-*l* 47.0%

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

      3. associate-*r* 47.0%

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

      4. +-commutative 47.0%

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

      5. associate-+r+ 47.0%

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

      6. metadata-eval 47.0%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in k around inf 65.5%

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

      1. *-commutative 65.5%

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

      2. unpow2 65.5%

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

      3. times-frac 69.8%

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

      4. unpow2 69.8%

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

   6. Simplified 69.8%

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

      1. associate-*l/ 71.2%

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

      2. associate-/l* 82.4%

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

   8. Applied egg-rr 82.4%

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

      1. associate-/l* 83.5%

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

      2. associate-/r/ 83.5%

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

   10. Simplified 83.5%

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

       1. associate-/r/ 81.8%

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

       2. associate-*r* 83.2%

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

   12. Applied egg-rr 83.2%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.9 \cdot 10^{-42}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{2}{{t}^{-3}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(k \cdot \left(\frac{k}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \]

Alternative 9: 82.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.12 \cdot 10^{-42}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{2}{{t}^{-3}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.12e-42)
   (/ 2.0 (* (/ k l) (* (/ k l) (/ 2.0 (pow t -3.0)))))
   (/ 2.0 (* (* (/ k l) (/ t (/ l k))) (* (sin k) (tan k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.12e-42) {
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / pow(t, -3.0))));
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (sin(k) * tan(k)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 <= 1.12d-42) then
        tmp = 2.0d0 / ((k / l) * ((k / l) * (2.0d0 / (t ** (-3.0d0)))))
    else
        tmp = 2.0d0 / (((k / l) * (t / (l / k))) * (sin(k) * tan(k)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.12e-42) {
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / Math.pow(t, -3.0))));
	} else {
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (Math.sin(k) * Math.tan(k)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.12e-42:
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / math.pow(t, -3.0))))
	else:
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (math.sin(k) * math.tan(k)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.12e-42)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(k / l) * Float64(2.0 / (t ^ -3.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(t / Float64(l / k))) * Float64(sin(k) * tan(k))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.12e-42)
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / (t ^ -3.0))));
	else
		tmp = 2.0 / (((k / l) * (t / (l / k))) * (sin(k) * tan(k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.12e-42], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(2.0 / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.1199999999999999e-42

   1. Initial program 49.4%

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

      1. associate-*l* 49.4%

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

      2. +-commutative 49.4%

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

   3. Simplified 49.4%

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

   4. Taylor expanded in k around 0 49.2%

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

   5. Taylor expanded in k around 0 51.7%

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

      1. associate-/l* 50.3%

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

      2. unpow2 50.3%

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

      3. associate-*r/ 58.1%

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

   7. Simplified 58.1%

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

      1. expm1-log1p-u 44.7%

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

      2. expm1-udef 27.9%

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

      3. associate-*l/ 24.8%

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

      4. times-frac 29.4%

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

      5. *-commutative 29.4%

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

      6. div-inv 29.4%

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

      7. pow-flip 29.4%

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

      8. metadata-eval 29.4%

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

   9. Applied egg-rr 29.4%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{k}{\ell} \cdot \frac{k \cdot 2}{\ell \cdot {t}^{-3}}\right)} - 1}} \]
   10. Step-by-step derivation

       1. expm1-def 46.9%

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

       2. expm1-log1p 61.8%

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

       3. times-frac 62.5%

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

   11. Simplified 62.5%

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

   if 1.1199999999999999e-42 < k

   1. Initial program 47.0%

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

      1. *-commutative 47.0%

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

      2. associate-*l* 47.0%

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

      3. associate-*r* 47.0%

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

      4. +-commutative 47.0%

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

      5. associate-+r+ 47.0%

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

      6. metadata-eval 47.0%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in k around inf 65.5%

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

      1. *-commutative 65.5%

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

      2. unpow2 65.5%

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

      3. times-frac 69.8%

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

      4. unpow2 69.8%

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

   6. Simplified 69.8%

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

      1. associate-*l/ 71.2%

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

      2. associate-/l* 82.4%

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

   8. Applied egg-rr 82.4%

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

      1. associate-/l* 83.5%

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

      2. associate-/r/ 83.5%

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

   10. Simplified 83.5%

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

       1. *-un-lft-identity 83.5%

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

       2. add-sqr-sqrt 83.3%

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

       3. times-frac 83.3%

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

       4. *-un-lft-identity 83.3%

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

       5. frac-times 83.3%

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

       6. clear-num 83.3%

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

       7. sqrt-unprod 56.0%

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

       8. add-sqr-sqrt 64.3%

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

       9. clear-num 64.3%

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

       10. remove-double-div 64.4%

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

       11. *-un-lft-identity 64.4%

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

       12. frac-times 66.8%

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

       13. clear-num 66.8%

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

       14. sqrt-unprod 58.4%

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

       15. add-sqr-sqrt 89.3%

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

   12. Applied egg-rr 89.3%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.12 \cdot 10^{-42}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{2}{{t}^{-3}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]

Alternative 10: 68.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.22 \cdot 10^{-99} \lor \neg \left(t \leq 5.6 \cdot 10^{-95}\right):\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{2}{{t}^{-3}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.22e-99) (not (<= t 5.6e-95)))
   (/ 2.0 (* (/ k l) (* (/ k l) (/ 2.0 (pow t -3.0)))))
   (/ 2.0 (* (* k k) (/ (* t (/ k (/ l k))) l)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.22e-99) || !(t <= 5.6e-95)) {
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / pow(t, -3.0))));
	} else {
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-1.22d-99)) .or. (.not. (t <= 5.6d-95))) then
        tmp = 2.0d0 / ((k / l) * ((k / l) * (2.0d0 / (t ** (-3.0d0)))))
    else
        tmp = 2.0d0 / ((k * k) * ((t * (k / (l / k))) / l))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.22e-99) || !(t <= 5.6e-95)) {
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / Math.pow(t, -3.0))));
	} else {
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if (t <= -1.22e-99) or not (t <= 5.6e-95):
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / math.pow(t, -3.0))))
	else:
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.22e-99) || !(t <= 5.6e-95))
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(k / l) * Float64(2.0 / (t ^ -3.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t * Float64(k / Float64(l / k))) / l)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.22e-99) || ~((t <= 5.6e-95)))
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / (t ^ -3.0))));
	else
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -1.22e-99], N[Not[LessEqual[t, 5.6e-95]], $MachinePrecision]], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(2.0 / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.22e-99 or 5.5999999999999998e-95 < t

   1. Initial program 57.9%

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

      1. associate-*l* 57.9%

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

      2. +-commutative 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in k around 0 53.9%

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

   5. Taylor expanded in k around 0 56.8%

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

      1. associate-/l* 55.1%

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

      2. unpow2 55.1%

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

      3. associate-*r/ 58.7%

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

   7. Simplified 58.7%

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

      1. expm1-log1p-u 34.2%

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

      2. expm1-udef 23.2%

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

      3. associate-*l/ 21.6%

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

      4. times-frac 24.8%

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

      5. *-commutative 24.8%

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

      6. div-inv 24.8%

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

      7. pow-flip 24.8%

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

      8. metadata-eval 24.8%

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

   9. Applied egg-rr 24.8%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{k}{\ell} \cdot \frac{k \cdot 2}{\ell \cdot {t}^{-3}}\right)} - 1}} \]
   10. Step-by-step derivation

       1. expm1-def 36.6%

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

       2. expm1-log1p 62.9%

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

       3. times-frac 63.7%

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

   11. Simplified 63.7%

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

   if -1.22e-99 < t < 5.5999999999999998e-95

   1. Initial program 32.5%

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

      1. *-commutative 32.5%

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

      2. associate-*l* 32.5%

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

      3. associate-*r* 32.5%

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

      4. +-commutative 32.5%

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

      5. associate-+r+ 32.5%

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

      6. metadata-eval 32.5%

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

   3. Simplified 32.5%

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

   4. Taylor expanded in k around inf 65.2%

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

      1. *-commutative 65.2%

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

      2. unpow2 65.2%

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

      3. times-frac 83.2%

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

      4. unpow2 83.2%

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

   6. Simplified 83.2%

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

   7. Taylor expanded in k around 0 72.8%

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

      1. unpow2 73.0%

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

   9. Simplified 72.8%

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

       1. associate-*l/ 84.4%

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

       2. associate-/l* 88.7%

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

   11. Applied egg-rr 73.0%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.22 \cdot 10^{-99} \lor \neg \left(t \leq 5.6 \cdot 10^{-95}\right):\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{2}{{t}^{-3}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \end{array} \]

Alternative 11: 69.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{2}{{t}^{-3}}\right)}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\ell}}}{\frac{2 \cdot \frac{k}{\ell}}{{t}^{-3}}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t -3.5e-99)
   (/ 2.0 (* (/ k l) (* (/ k l) (/ 2.0 (pow t -3.0)))))
   (if (<= t 9.5e-97)
     (/ 2.0 (* (* k k) (/ (* t (/ k (/ l k))) l)))
     (/ (/ 2.0 (/ k l)) (/ (* 2.0 (/ k l)) (pow t -3.0))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.5e-99) {
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / pow(t, -3.0))));
	} else if (t <= 9.5e-97) {
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	} else {
		tmp = (2.0 / (k / l)) / ((2.0 * (k / l)) / pow(t, -3.0));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-3.5d-99)) then
        tmp = 2.0d0 / ((k / l) * ((k / l) * (2.0d0 / (t ** (-3.0d0)))))
    else if (t <= 9.5d-97) then
        tmp = 2.0d0 / ((k * k) * ((t * (k / (l / k))) / l))
    else
        tmp = (2.0d0 / (k / l)) / ((2.0d0 * (k / l)) / (t ** (-3.0d0)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.5e-99) {
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / Math.pow(t, -3.0))));
	} else if (t <= 9.5e-97) {
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	} else {
		tmp = (2.0 / (k / l)) / ((2.0 * (k / l)) / Math.pow(t, -3.0));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if t <= -3.5e-99:
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / math.pow(t, -3.0))))
	elif t <= 9.5e-97:
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l))
	else:
		tmp = (2.0 / (k / l)) / ((2.0 * (k / l)) / math.pow(t, -3.0))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (t <= -3.5e-99)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(k / l) * Float64(2.0 / (t ^ -3.0)))));
	elseif (t <= 9.5e-97)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t * Float64(k / Float64(l / k))) / l)));
	else
		tmp = Float64(Float64(2.0 / Float64(k / l)) / Float64(Float64(2.0 * Float64(k / l)) / (t ^ -3.0)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -3.5e-99)
		tmp = 2.0 / ((k / l) * ((k / l) * (2.0 / (t ^ -3.0))));
	elseif (t <= 9.5e-97)
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	else
		tmp = (2.0 / (k / l)) / ((2.0 * (k / l)) / (t ^ -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, -3.5e-99], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(2.0 / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.5e-97], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * N[(k / l), $MachinePrecision]), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.4999999999999999e-99

   1. Initial program 62.2%

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

      1. associate-*l* 62.2%

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

      2. +-commutative 62.2%

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

   3. Simplified 62.2%

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

   4. Taylor expanded in k around 0 59.1%

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

   5. Taylor expanded in k around 0 61.2%

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

      1. associate-/l* 59.1%

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

      2. unpow2 59.1%

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

      3. associate-*r/ 61.9%

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

   7. Simplified 61.9%

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

      1. expm1-log1p-u 17.9%

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

      2. expm1-udef 1.3%

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

      3. associate-*l/ 1.2%

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

      4. times-frac 1.7%

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

      5. *-commutative 1.7%

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

      6. div-inv 1.7%

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

      7. pow-flip 1.7%

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

      8. metadata-eval 1.7%

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

   9. Applied egg-rr 1.7%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{k}{\ell} \cdot \frac{k \cdot 2}{\ell \cdot {t}^{-3}}\right)} - 1}} \]
   10. Step-by-step derivation

       1. expm1-def 19.8%

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

       2. expm1-log1p 66.7%

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

       3. times-frac 67.0%

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

   11. Simplified 67.0%

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

   if -3.4999999999999999e-99 < t < 9.5000000000000001e-97

   1. Initial program 32.5%

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

      1. *-commutative 32.5%

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

      2. associate-*l* 32.5%

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

      3. associate-*r* 32.5%

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

      4. +-commutative 32.5%

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

      5. associate-+r+ 32.5%

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

      6. metadata-eval 32.5%

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

   3. Simplified 32.5%

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

   4. Taylor expanded in k around inf 65.2%

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

      1. *-commutative 65.2%

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

      2. unpow2 65.2%

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

      3. times-frac 83.2%

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

      4. unpow2 83.2%

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

   6. Simplified 83.2%

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

   7. Taylor expanded in k around 0 72.8%

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

      1. unpow2 73.0%

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

   9. Simplified 72.8%

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

       1. associate-*l/ 84.4%

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

       2. associate-/l* 88.7%

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

   11. Applied egg-rr 73.0%

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

   if 9.5000000000000001e-97 < t

   1. Initial program 52.5%

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

      1. associate-*l* 52.5%

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

      2. +-commutative 52.5%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in k around 0 47.4%

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

   5. Taylor expanded in k around 0 51.3%

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

      1. associate-/l* 50.1%

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

      2. unpow2 50.1%

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

      3. associate-*r/ 54.7%

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

   7. Simplified 54.7%

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

      1. expm1-log1p-u 54.7%

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

      2. expm1-udef 50.8%

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

      3. associate-*l/ 47.5%

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

      4. times-frac 54.0%

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

      5. *-commutative 54.0%

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

      6. div-inv 54.0%

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

      7. pow-flip 54.0%

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

      8. metadata-eval 54.0%

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

   9. Applied egg-rr 54.0%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{k}{\ell} \cdot \frac{k \cdot 2}{\ell \cdot {t}^{-3}}\right)} - 1}} \]
   10. Step-by-step derivation

       1. expm1-def 57.9%

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

       2. expm1-log1p 58.2%

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

       3. times-frac 59.6%

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

   11. Simplified 59.6%

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

       1. expm1-log1p-u 59.4%

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

       2. expm1-udef 54.2%

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

   13. Applied egg-rr 54.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{2}{{t}^{-3}}\right)}\right)} - 1} \]
   14. Step-by-step derivation

       1. expm1-def 59.4%

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

       2. expm1-log1p 59.6%

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

       3. associate-/r* 59.6%

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

       4. associate-*r/ 59.6%

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

       5. *-commutative 59.6%

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

   15. Simplified 59.6%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{2}{{t}^{-3}}\right)}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\ell}}}{\frac{2 \cdot \frac{k}{\ell}}{{t}^{-3}}}\\ \end{array} \]

Alternative 12: 68.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-96} \lor \neg \left(t \leq 1.3 \cdot 10^{-95}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot {t}^{-3}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.05e-96) (not (<= t 1.3e-95)))
   (* (/ l k) (/ (* l (pow t -3.0)) k))
   (/ 2.0 (* (* k k) (/ (* t (/ k (/ l k))) l)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.05e-96) || !(t <= 1.3e-95)) {
		tmp = (l / k) * ((l * pow(t, -3.0)) / k);
	} else {
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-1.05d-96)) .or. (.not. (t <= 1.3d-95))) then
        tmp = (l / k) * ((l * (t ** (-3.0d0))) / k)
    else
        tmp = 2.0d0 / ((k * k) * ((t * (k / (l / k))) / l))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.05e-96) || !(t <= 1.3e-95)) {
		tmp = (l / k) * ((l * Math.pow(t, -3.0)) / k);
	} else {
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if (t <= -1.05e-96) or not (t <= 1.3e-95):
		tmp = (l / k) * ((l * math.pow(t, -3.0)) / k)
	else:
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.05e-96) || !(t <= 1.3e-95))
		tmp = Float64(Float64(l / k) * Float64(Float64(l * (t ^ -3.0)) / k));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t * Float64(k / Float64(l / k))) / l)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.05e-96) || ~((t <= 1.3e-95)))
		tmp = (l / k) * ((l * (t ^ -3.0)) / k);
	else
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -1.05e-96], N[Not[LessEqual[t, 1.3e-95]], $MachinePrecision]], N[(N[(l / k), $MachinePrecision] * N[(N[(l * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.05000000000000001e-96 or 1.3e-95 < t

   1. Initial program 58.3%

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

      1. associate-*l* 58.3%

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

      2. associate-/l/ 58.3%

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

      3. *-commutative 58.3%

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

      4. associate-*r/ 60.0%

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

      5. associate-/l* 58.3%

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

      6. associate-/r/ 52.9%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in k around 0 52.4%

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

      1. unpow2 52.4%

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

      2. unpow2 52.4%

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

   6. Simplified 52.4%

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

      1. expm1-log1p-u 42.9%

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

      2. expm1-udef 41.8%

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

      3. times-frac 43.7%

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

      4. div-inv 43.7%

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

      5. pow-flip 43.7%

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

      6. metadata-eval 43.7%

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

   8. Applied egg-rr 43.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\ell}{k \cdot k} \cdot \left(\ell \cdot {t}^{-3}\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 45.3%

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

      2. expm1-log1p 55.4%

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

      3. unpow2 55.4%

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

      4. associate-*l/ 54.3%

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

      5. unpow2 54.3%

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

      6. times-frac 63.3%

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

   10. Simplified 63.3%

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

   if -1.05000000000000001e-96 < t < 1.3e-95

   1. Initial program 32.2%

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

      1. *-commutative 32.2%

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

      2. associate-*l* 32.2%

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

      3. associate-*r* 32.2%

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

      4. +-commutative 32.2%

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

      5. associate-+r+ 32.2%

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

      6. metadata-eval 32.2%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in k around inf 64.5%

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

      1. *-commutative 64.5%

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

      2. unpow2 64.5%

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

      3. times-frac 82.4%

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

      4. unpow2 82.4%

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

   6. Simplified 82.4%

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

   7. Taylor expanded in k around 0 72.1%

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

      1. unpow2 72.2%

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

   9. Simplified 72.1%

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

       1. associate-*l/ 83.6%

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

       2. associate-/l* 88.8%

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

   11. Applied egg-rr 72.2%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-96} \lor \neg \left(t \leq 1.3 \cdot 10^{-95}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell \cdot {t}^{-3}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \end{array} \]

Alternative 13: 58.2% accurate, 28.1× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (* (/ k l) (/ t (/ l k))) (* k k))))

C

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

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((k / l) * (t / (l / k))) * (k * k))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.7%

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

   1. *-commutative 48.7%

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

   2. associate-*l* 46.1%

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

   3. associate-*r* 46.1%

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

   4. +-commutative 46.1%

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

   5. associate-+r+ 46.1%

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

   6. metadata-eval 46.1%

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

3. Simplified 46.1%

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

4. Taylor expanded in k around inf 55.6%

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

   1. *-commutative 55.6%

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

   2. unpow2 55.6%

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

   3. times-frac 66.1%

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

   4. unpow2 66.1%

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

6. Simplified 66.1%

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

   1. associate-*l/ 65.6%

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

   2. associate-/l* 72.1%

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

8. Applied egg-rr 72.1%

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

   1. associate-/l* 70.5%

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

   2. associate-/r/ 70.5%

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

10. Simplified 70.5%

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

    1. *-un-lft-identity 70.5%

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

    2. add-sqr-sqrt 70.4%

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

    3. times-frac 70.4%

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

    4. *-un-lft-identity 70.4%

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

    5. frac-times 70.4%

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

    6. clear-num 70.5%

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

    7. sqrt-unprod 43.3%

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

    8. add-sqr-sqrt 49.3%

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

    9. clear-num 49.3%

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

    10. remove-double-div 49.3%

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

    11. *-un-lft-identity 49.3%

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

    12. frac-times 50.0%

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

    13. clear-num 50.0%

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

    14. sqrt-unprod 47.1%

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

    15. add-sqr-sqrt 77.2%

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

12. Applied egg-rr 77.2%

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

13. Taylor expanded in k around 0 57.0%

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

    1. unpow2 57.0%

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

15. Simplified 57.0%

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

16. Final simplification 57.0%

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

Alternative 14: 58.5% accurate, 28.1× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (* k k) (* (/ t l) (/ (* k k) l)))))

C

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

Fortran

NOTE: k should be positive before calling this function
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 * k) * ((t / l) * ((k * k) / l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.7%

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

   1. *-commutative 48.7%

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

   2. associate-*l* 46.1%

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

   3. associate-*r* 46.1%

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

   4. +-commutative 46.1%

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

   5. associate-+r+ 46.1%

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

   6. metadata-eval 46.1%

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

3. Simplified 46.1%

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

4. Taylor expanded in k around inf 55.6%

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

   1. *-commutative 55.6%

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

   2. unpow2 55.6%

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

   3. times-frac 66.1%

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

   4. unpow2 66.1%

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

6. Simplified 66.1%

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

7. Taylor expanded in k around 0 57.6%

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

   1. unpow2 57.0%

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

9. Simplified 57.6%

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

10. Final simplification 57.6%

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

Alternative 15: 58.5% accurate, 28.1× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (* k k) (/ (* (/ t l) (* k k)) l))))

C

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

Fortran

NOTE: k should be positive before calling this function
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 * k) * (((t / l) * (k * k)) / l))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(N[(t / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.7%

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

   1. *-commutative 48.7%

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

   2. associate-*l* 46.1%

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

   3. associate-*r* 46.1%

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

   4. +-commutative 46.1%

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

   5. associate-+r+ 46.1%

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

   6. metadata-eval 46.1%

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

3. Simplified 46.1%

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

4. Taylor expanded in k around inf 55.6%

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

   1. *-commutative 55.6%

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

   2. unpow2 55.6%

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

   3. times-frac 66.1%

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

   4. unpow2 66.1%

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

6. Simplified 66.1%

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

7. Taylor expanded in k around 0 57.6%

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

   1. unpow2 57.0%

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

9. Simplified 57.6%

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

    1. associate-*r/ 57.6%

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

11. Applied egg-rr 57.6%

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

12. Final simplification 57.6%

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

Reproduce

herbie shell --seed 2023227 
(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))))