Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.7% → 85.6%

Time: 36.9s

Alternatives: 16

Speedup: 28.1×

• 27.7% 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: 55.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ t_2 := \frac{1}{\frac{t}{{\left(\sqrt[3]{\frac{\ell}{k}}\right)}^{2}}}\\ \mathbf{if}\;k \leq -6.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{\ell \cdot \frac{\ell}{k}} \cdot t_1}\\ \mathbf{elif}\;k \leq 5.7 \cdot 10^{-5}:\\ \;\;\;\;t_2 \cdot {t_2}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \frac{k \cdot \frac{k \cdot t}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))) (t_2 (/ 1.0 (/ t (pow (cbrt (/ l k)) 2.0)))))
   (if (<= k -6.5e+47)
     (/ 2.0 (* (/ (* k t) (* l (/ l k))) t_1))
     (if (<= k 5.7e-5)
       (* t_2 (pow t_2 2.0))
       (/ 2.0 (* t_1 (/ (* k (/ (* k t) l)) l)))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double t_2 = 1.0 / (t / pow(cbrt((l / k)), 2.0));
	double tmp;
	if (k <= -6.5e+47) {
		tmp = 2.0 / (((k * t) / (l * (l / k))) * t_1);
	} else if (k <= 5.7e-5) {
		tmp = t_2 * pow(t_2, 2.0);
	} else {
		tmp = 2.0 / (t_1 * ((k * ((k * t) / l)) / l));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.tan(k);
	double t_2 = 1.0 / (t / Math.pow(Math.cbrt((l / k)), 2.0));
	double tmp;
	if (k <= -6.5e+47) {
		tmp = 2.0 / (((k * t) / (l * (l / k))) * t_1);
	} else if (k <= 5.7e-5) {
		tmp = t_2 * Math.pow(t_2, 2.0);
	} else {
		tmp = 2.0 / (t_1 * ((k * ((k * t) / l)) / l));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	t_2 = Float64(1.0 / Float64(t / (cbrt(Float64(l / k)) ^ 2.0)))
	tmp = 0.0
	if (k <= -6.5e+47)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * t) / Float64(l * Float64(l / k))) * t_1));
	elseif (k <= 5.7e-5)
		tmp = Float64(t_2 * (t_2 ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(k * Float64(Float64(k * t) / l)) / l)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(t / N[Power[N[Power[N[(l / k), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -6.5e+47], N[(2.0 / N[(N[(N[(k * t), $MachinePrecision] / N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.7e-5], N[(t$95$2 * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[(k * N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < -6.49999999999999988e47

   1. Initial program 37.9%

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

      1. *-commutative 37.9%

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

      2. associate-*l* 37.9%

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

      3. associate-*r* 37.9%

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

      4. +-commutative 37.9%

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

      5. associate-+r+ 37.9%

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

      6. metadata-eval 37.9%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in k around inf 74.5%

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

      1. *-commutative 74.5%

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

      2. unpow2 74.5%

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

      3. times-frac 71.8%

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

      4. unpow2 71.8%

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

   6. Simplified 71.8%

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

      1. associate-*l/ 71.9%

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

      2. associate-/l* 77.0%

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

   8. Applied egg-rr 77.0%

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

      1. associate-*r/ 83.5%

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

   10. Applied egg-rr 83.5%

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

       1. *-un-lft-identity 83.5%

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

       2. associate-/l/ 84.8%

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

   12. Applied egg-rr 84.8%

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

   if -6.49999999999999988e47 < k < 5.7000000000000003e-5

   1. Initial program 69.8%

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

      1. associate-*l* 69.8%

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

      2. +-commutative 69.8%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in k around 0 62.3%

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

      1. *-commutative 62.3%

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

      2. unpow2 62.3%

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

      3. times-frac 66.5%

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

      4. unpow2 66.5%

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

   6. Simplified 66.5%

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

      1. associate-*l/ 66.6%

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

      2. associate-/l* 75.0%

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

   8. Applied egg-rr 75.0%

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

      1. add-cube-cbrt 74.9%

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

   10. Applied egg-rr 90.6%

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

   if 5.7000000000000003e-5 < k

   1. Initial program 38.5%

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

      1. *-commutative 38.5%

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

      2. associate-*l* 38.5%

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

      3. associate-*r* 38.5%

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

      4. +-commutative 38.5%

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

      5. associate-+r+ 38.5%

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

      6. metadata-eval 38.5%

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

   3. Simplified 38.5%

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

   4. Taylor expanded in k around inf 64.7%

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

      1. *-commutative 64.7%

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

      2. unpow2 64.7%

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

      3. times-frac 65.1%

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

      4. unpow2 65.1%

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

   6. Simplified 65.1%

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

      1. associate-*l/ 72.2%

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

      2. associate-/l* 84.8%

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

   8. Applied egg-rr 84.8%

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

      1. associate-*r/ 90.3%

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

   10. Applied egg-rr 90.3%

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

       1. associate-/r/ 90.3%

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

   12. Applied egg-rr 90.3%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -6.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{\ell \cdot \frac{\ell}{k}} \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{elif}\;k \leq 5.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{t}{{\left(\sqrt[3]{\frac{\ell}{k}}\right)}^{2}}} \cdot {\left(\frac{1}{\frac{t}{{\left(\sqrt[3]{\frac{\ell}{k}}\right)}^{2}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \frac{k \cdot \frac{k \cdot t}{\ell}}{\ell}}\\ \end{array} \]

Alternative 2: 85.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq -6.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{\ell \cdot \frac{\ell}{k}} \cdot t_1}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;{\left(\frac{1}{\frac{t}{{\left(\sqrt[3]{\frac{\ell}{k}}\right)}^{2}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \frac{k \cdot \frac{k \cdot t}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= k -6.5e+47)
     (/ 2.0 (* (/ (* k t) (* l (/ l k))) t_1))
     (if (<= k 3.8e-5)
       (pow (/ 1.0 (/ t (pow (cbrt (/ l k)) 2.0))) 3.0)
       (/ 2.0 (* t_1 (/ (* k (/ (* k t) l)) l)))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (k <= -6.5e+47) {
		tmp = 2.0 / (((k * t) / (l * (l / k))) * t_1);
	} else if (k <= 3.8e-5) {
		tmp = pow((1.0 / (t / pow(cbrt((l / k)), 2.0))), 3.0);
	} else {
		tmp = 2.0 / (t_1 * ((k * ((k * t) / l)) / l));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (k <= -6.5e+47) {
		tmp = 2.0 / (((k * t) / (l * (l / k))) * t_1);
	} else if (k <= 3.8e-5) {
		tmp = Math.pow((1.0 / (t / Math.pow(Math.cbrt((l / k)), 2.0))), 3.0);
	} else {
		tmp = 2.0 / (t_1 * ((k * ((k * t) / l)) / l));
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (k <= -6.5e+47)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * t) / Float64(l * Float64(l / k))) * t_1));
	elseif (k <= 3.8e-5)
		tmp = Float64(1.0 / Float64(t / (cbrt(Float64(l / k)) ^ 2.0))) ^ 3.0;
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(k * Float64(Float64(k * t) / l)) / l)));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -6.5e+47], N[(2.0 / N[(N[(N[(k * t), $MachinePrecision] / N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.8e-5], N[Power[N[(1.0 / N[(t / N[Power[N[Power[N[(l / k), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[(k * N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < -6.49999999999999988e47

   1. Initial program 37.9%

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

      1. *-commutative 37.9%

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

      2. associate-*l* 37.9%

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

      3. associate-*r* 37.9%

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

      4. +-commutative 37.9%

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

      5. associate-+r+ 37.9%

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

      6. metadata-eval 37.9%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in k around inf 74.5%

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

      1. *-commutative 74.5%

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

      2. unpow2 74.5%

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

      3. times-frac 71.8%

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

      4. unpow2 71.8%

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

   6. Simplified 71.8%

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

      1. associate-*l/ 71.9%

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

      2. associate-/l* 77.0%

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

   8. Applied egg-rr 77.0%

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

      1. associate-*r/ 83.5%

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

   10. Applied egg-rr 83.5%

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

       1. *-un-lft-identity 83.5%

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

       2. associate-/l/ 84.8%

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

   12. Applied egg-rr 84.8%

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

   if -6.49999999999999988e47 < k < 3.8000000000000002e-5

   1. Initial program 69.8%

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

      1. associate-*l* 69.8%

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

      2. +-commutative 69.8%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in k around 0 62.3%

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

      1. *-commutative 62.3%

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

      2. unpow2 62.3%

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

      3. times-frac 66.5%

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

      4. unpow2 66.5%

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

   6. Simplified 66.5%

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

      1. associate-*l/ 66.6%

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

      2. associate-/l* 75.0%

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

   8. Applied egg-rr 75.0%

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

      1. add-cube-cbrt 74.9%

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

      2. pow3 74.9%

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

   10. Applied egg-rr 90.6%

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

   if 3.8000000000000002e-5 < k

   1. Initial program 38.5%

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

      1. *-commutative 38.5%

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

      2. associate-*l* 38.5%

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

      3. associate-*r* 38.5%

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

      4. +-commutative 38.5%

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

      5. associate-+r+ 38.5%

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

      6. metadata-eval 38.5%

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

   3. Simplified 38.5%

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

   4. Taylor expanded in k around inf 64.7%

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

      1. *-commutative 64.7%

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

      2. unpow2 64.7%

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

      3. times-frac 65.1%

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

      4. unpow2 65.1%

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

   6. Simplified 65.1%

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

      1. associate-*l/ 72.2%

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

      2. associate-/l* 84.8%

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

   8. Applied egg-rr 84.8%

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

      1. associate-*r/ 90.3%

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

   10. Applied egg-rr 90.3%

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

       1. associate-/r/ 90.3%

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

   12. Applied egg-rr 90.3%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -6.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{\ell \cdot \frac{\ell}{k}} \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;{\left(\frac{1}{\frac{t}{{\left(\sqrt[3]{\frac{\ell}{k}}\right)}^{2}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \frac{k \cdot \frac{k \cdot t}{\ell}}{\ell}}\\ \end{array} \]

Alternative 3: 72.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.034:\\ \;\;\;\;\frac{2}{2 \cdot \frac{\frac{k \cdot {t}^{3}}{\frac{\ell}{k}}}{\ell}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-78}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \frac{1}{{t}^{1.5}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -0.034)
   (/ 2.0 (* 2.0 (/ (/ (* k (pow t 3.0)) (/ l k)) l)))
   (if (<= t 3.8e-78)
     (* (* l l) (/ 2.0 (* (tan k) (* (* k k) (* t (sin k))))))
     (pow (* (/ l k) (/ 1.0 (pow t 1.5))) 2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -0.034) {
		tmp = 2.0 / (2.0 * (((k * pow(t, 3.0)) / (l / k)) / l));
	} else if (t <= 3.8e-78) {
		tmp = (l * l) * (2.0 / (tan(k) * ((k * k) * (t * sin(k)))));
	} else {
		tmp = pow(((l / k) * (1.0 / pow(t, 1.5))), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-0.034d0)) then
        tmp = 2.0d0 / (2.0d0 * (((k * (t ** 3.0d0)) / (l / k)) / l))
    else if (t <= 3.8d-78) then
        tmp = (l * l) * (2.0d0 / (tan(k) * ((k * k) * (t * sin(k)))))
    else
        tmp = ((l / k) * (1.0d0 / (t ** 1.5d0))) ** 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -0.034) {
		tmp = 2.0 / (2.0 * (((k * Math.pow(t, 3.0)) / (l / k)) / l));
	} else if (t <= 3.8e-78) {
		tmp = (l * l) * (2.0 / (Math.tan(k) * ((k * k) * (t * Math.sin(k)))));
	} else {
		tmp = Math.pow(((l / k) * (1.0 / Math.pow(t, 1.5))), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -0.034:
		tmp = 2.0 / (2.0 * (((k * math.pow(t, 3.0)) / (l / k)) / l))
	elif t <= 3.8e-78:
		tmp = (l * l) * (2.0 / (math.tan(k) * ((k * k) * (t * math.sin(k)))))
	else:
		tmp = math.pow(((l / k) * (1.0 / math.pow(t, 1.5))), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -0.034)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(Float64(k * (t ^ 3.0)) / Float64(l / k)) / l)));
	elseif (t <= 3.8e-78)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * Float64(Float64(k * k) * Float64(t * sin(k))))));
	else
		tmp = Float64(Float64(l / k) * Float64(1.0 / (t ^ 1.5))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -0.034)
		tmp = 2.0 / (2.0 * (((k * (t ^ 3.0)) / (l / k)) / l));
	elseif (t <= 3.8e-78)
		tmp = (l * l) * (2.0 / (tan(k) * ((k * k) * (t * sin(k)))));
	else
		tmp = ((l / k) * (1.0 / (t ^ 1.5))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -0.034], N[(2.0 / N[(2.0 * N[(N[(N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.8e-78], 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], N[Power[N[(N[(l / k), $MachinePrecision] * N[(1.0 / N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.034000000000000002

   1. Initial program 66.6%

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

      1. associate-*l* 66.6%

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

      2. +-commutative 66.6%

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

   3. Simplified 66.6%

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

   4. Taylor expanded in k around 0 54.8%

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

      1. *-commutative 54.8%

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

      2. unpow2 54.8%

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

      3. times-frac 57.1%

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

      4. unpow2 57.1%

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

   6. Simplified 57.1%

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

      1. associate-*l/ 58.6%

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

      2. associate-/l* 66.8%

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

   8. Applied egg-rr 66.8%

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

      1. associate-*r/ 74.8%

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

   10. Applied egg-rr 74.8%

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

   if -0.034000000000000002 < t < 3.7999999999999999e-78

   1. Initial program 34.5%

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

      1. associate-/l/ 34.5%

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

      2. associate-*l/ 33.6%

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

      3. associate-*l/ 33.6%

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

      4. associate-/r/ 34.2%

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

      5. *-commutative 34.2%

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

      6. associate-/l/ 34.2%

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

      7. associate-*r* 34.2%

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

      8. *-commutative 34.2%

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

      9. associate-*r* 34.2%

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

      10. *-commutative 34.2%

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

   3. Simplified 34.2%

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

   4. Taylor expanded in k around inf 73.4%

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

      1. unpow2 73.4%

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

   6. Simplified 73.4%

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

   if 3.7999999999999999e-78 < t

   1. Initial program 71.2%

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

      1. associate-*l* 71.2%

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

      2. +-commutative 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in k around 0 62.5%

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

      1. *-commutative 62.5%

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

      2. unpow2 62.5%

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

      3. times-frac 66.6%

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

      4. unpow2 66.6%

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

   6. Simplified 66.6%

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

      1. associate-*l/ 66.6%

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

      2. associate-/l* 74.0%

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

   8. Applied egg-rr 74.0%

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

      1. expm1-log1p-u 73.9%

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

      2. expm1-udef 69.5%

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

   10. Applied egg-rr 74.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1}{\frac{{t}^{1.5}}{\frac{\ell}{k}}}\right)}^{2}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 80.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1}{\frac{{t}^{1.5}}{\frac{\ell}{k}}}\right)}^{2}\right)\right)} \]

       2. expm1-log1p 81.2%

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

       3. associate-/r/ 81.2%

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

   12. Simplified 81.2%

       \[\leadsto \color{blue}{{\left(\frac{1}{{t}^{1.5}} \cdot \frac{\ell}{k}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.034:\\ \;\;\;\;\frac{2}{2 \cdot \frac{\frac{k \cdot {t}^{3}}{\frac{\ell}{k}}}{\ell}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-78}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \frac{1}{{t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 4: 79.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{+36}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \frac{t}{\frac{\ell}{k} \cdot \frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \frac{1}{{t}^{1.5}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -8.6e+36)
   (/ l (/ (* k (* k (pow t 3.0))) l))
   (if (<= t 6.2e-77)
     (/ 2.0 (* (sin k) (* (tan k) (/ t (* (/ l k) (/ l k))))))
     (pow (* (/ l k) (/ 1.0 (pow t 1.5))) 2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -8.6e+36) {
		tmp = l / ((k * (k * pow(t, 3.0))) / l);
	} else if (t <= 6.2e-77) {
		tmp = 2.0 / (sin(k) * (tan(k) * (t / ((l / k) * (l / k)))));
	} else {
		tmp = pow(((l / k) * (1.0 / pow(t, 1.5))), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-8.6d+36)) then
        tmp = l / ((k * (k * (t ** 3.0d0))) / l)
    else if (t <= 6.2d-77) then
        tmp = 2.0d0 / (sin(k) * (tan(k) * (t / ((l / k) * (l / k)))))
    else
        tmp = ((l / k) * (1.0d0 / (t ** 1.5d0))) ** 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -8.6e+36) {
		tmp = l / ((k * (k * Math.pow(t, 3.0))) / l);
	} else if (t <= 6.2e-77) {
		tmp = 2.0 / (Math.sin(k) * (Math.tan(k) * (t / ((l / k) * (l / k)))));
	} else {
		tmp = Math.pow(((l / k) * (1.0 / Math.pow(t, 1.5))), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -8.6e+36:
		tmp = l / ((k * (k * math.pow(t, 3.0))) / l)
	elif t <= 6.2e-77:
		tmp = 2.0 / (math.sin(k) * (math.tan(k) * (t / ((l / k) * (l / k)))))
	else:
		tmp = math.pow(((l / k) * (1.0 / math.pow(t, 1.5))), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -8.6e+36)
		tmp = Float64(l / Float64(Float64(k * Float64(k * (t ^ 3.0))) / l));
	elseif (t <= 6.2e-77)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(t / Float64(Float64(l / k) * Float64(l / k))))));
	else
		tmp = Float64(Float64(l / k) * Float64(1.0 / (t ^ 1.5))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -8.6e+36)
		tmp = l / ((k * (k * (t ^ 3.0))) / l);
	elseif (t <= 6.2e-77)
		tmp = 2.0 / (sin(k) * (tan(k) * (t / ((l / k) * (l / k)))));
	else
		tmp = ((l / k) * (1.0 / (t ^ 1.5))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -8.6e+36], N[(l / N[(N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.2e-77], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t / N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(l / k), $MachinePrecision] * N[(1.0 / N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.6000000000000001e36

   1. Initial program 65.3%

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

      1. associate-/l/ 65.3%

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

      2. associate-*l/ 69.0%

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

      3. associate-*l/ 70.4%

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

      4. associate-/r/ 70.4%

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

      5. *-commutative 70.4%

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

      6. associate-/l/ 70.4%

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

      7. associate-*r* 70.4%

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

      8. *-commutative 70.4%

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

      9. associate-*r* 70.4%

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

      10. *-commutative 70.4%

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

   3. Simplified 70.4%

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

   4. Taylor expanded in k around 0 56.7%

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

      1. unpow2 56.7%

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

      2. unpow2 56.7%

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

   6. Simplified 56.7%

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

      1. *-un-lft-identity 56.7%

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

      2. associate-/l* 63.0%

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

      3. associate-*l* 78.4%

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

   8. Applied egg-rr 78.4%

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

   if -8.6000000000000001e36 < t < 6.20000000000000016e-77

   1. Initial program 37.9%

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

      1. *-commutative 37.9%

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

      2. associate-*l* 37.1%

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

      3. associate-*r* 37.1%

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

      4. +-commutative 37.1%

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

      5. associate-+r+ 37.1%

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

      6. metadata-eval 37.1%

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

   3. Simplified 37.1%

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

   4. Taylor expanded in k around inf 71.8%

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

      1. *-commutative 71.8%

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

      2. unpow2 71.8%

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

      3. times-frac 75.0%

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

      4. unpow2 75.0%

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

   6. Simplified 75.0%

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

      1. associate-*l/ 78.1%

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

      2. associate-/l* 86.8%

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

   8. Applied egg-rr 86.8%

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

      1. associate-/l* 78.1%

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

      2. associate-*l/ 75.0%

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

      3. associate-*l/ 80.9%

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

      4. associate-*r* 80.6%

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

      5. expm1-log1p-u 63.9%

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

      6. expm1-udef 27.6%

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

   10. Applied egg-rr 27.7%

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

       1. expm1-def 71.8%

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

       2. expm1-log1p 86.9%

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

       3. associate-*l* 87.4%

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

   12. Simplified 87.4%

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

       1. unpow2 87.4%

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

   14. Applied egg-rr 87.4%

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

   if 6.20000000000000016e-77 < t

   1. Initial program 71.2%

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

      1. associate-*l* 71.2%

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

      2. +-commutative 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in k around 0 62.5%

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

      1. *-commutative 62.5%

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

      2. unpow2 62.5%

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

      3. times-frac 66.6%

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

      4. unpow2 66.6%

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

   6. Simplified 66.6%

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

      1. associate-*l/ 66.6%

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

      2. associate-/l* 74.0%

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

   8. Applied egg-rr 74.0%

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

      1. expm1-log1p-u 73.9%

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

      2. expm1-udef 69.5%

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

   10. Applied egg-rr 74.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1}{\frac{{t}^{1.5}}{\frac{\ell}{k}}}\right)}^{2}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 80.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1}{\frac{{t}^{1.5}}{\frac{\ell}{k}}}\right)}^{2}\right)\right)} \]

       2. expm1-log1p 81.2%

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

       3. associate-/r/ 81.2%

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

   12. Simplified 81.2%

       \[\leadsto \color{blue}{{\left(\frac{1}{{t}^{1.5}} \cdot \frac{\ell}{k}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.6 \cdot 10^{+36}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \frac{t}{\frac{\ell}{k} \cdot \frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \frac{1}{{t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 5: 82.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+37}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \frac{k \cdot \frac{k \cdot t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \frac{1}{{t}^{1.5}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -5.8e+37)
   (/ l (/ (* k (* k (pow t 3.0))) l))
   (if (<= t 6.2e-77)
     (/ 2.0 (* (* (sin k) (tan k)) (/ (* k (/ (* k t) l)) l)))
     (pow (* (/ l k) (/ 1.0 (pow t 1.5))) 2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -5.8e+37) {
		tmp = l / ((k * (k * pow(t, 3.0))) / l);
	} else if (t <= 6.2e-77) {
		tmp = 2.0 / ((sin(k) * tan(k)) * ((k * ((k * t) / l)) / l));
	} else {
		tmp = pow(((l / k) * (1.0 / pow(t, 1.5))), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-5.8d+37)) then
        tmp = l / ((k * (k * (t ** 3.0d0))) / l)
    else if (t <= 6.2d-77) then
        tmp = 2.0d0 / ((sin(k) * tan(k)) * ((k * ((k * t) / l)) / l))
    else
        tmp = ((l / k) * (1.0d0 / (t ** 1.5d0))) ** 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -5.8e+37) {
		tmp = l / ((k * (k * Math.pow(t, 3.0))) / l);
	} else if (t <= 6.2e-77) {
		tmp = 2.0 / ((Math.sin(k) * Math.tan(k)) * ((k * ((k * t) / l)) / l));
	} else {
		tmp = Math.pow(((l / k) * (1.0 / Math.pow(t, 1.5))), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -5.8e+37:
		tmp = l / ((k * (k * math.pow(t, 3.0))) / l)
	elif t <= 6.2e-77:
		tmp = 2.0 / ((math.sin(k) * math.tan(k)) * ((k * ((k * t) / l)) / l))
	else:
		tmp = math.pow(((l / k) * (1.0 / math.pow(t, 1.5))), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -5.8e+37)
		tmp = Float64(l / Float64(Float64(k * Float64(k * (t ^ 3.0))) / l));
	elseif (t <= 6.2e-77)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * tan(k)) * Float64(Float64(k * Float64(Float64(k * t) / l)) / l)));
	else
		tmp = Float64(Float64(l / k) * Float64(1.0 / (t ^ 1.5))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -5.8e+37)
		tmp = l / ((k * (k * (t ^ 3.0))) / l);
	elseif (t <= 6.2e-77)
		tmp = 2.0 / ((sin(k) * tan(k)) * ((k * ((k * t) / l)) / l));
	else
		tmp = ((l / k) * (1.0 / (t ^ 1.5))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -5.8e+37], N[(l / N[(N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.2e-77], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(l / k), $MachinePrecision] * N[(1.0 / N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.79999999999999957e37

   1. Initial program 65.3%

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

      1. associate-/l/ 65.3%

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

      2. associate-*l/ 69.0%

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

      3. associate-*l/ 70.4%

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

      4. associate-/r/ 70.4%

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

      5. *-commutative 70.4%

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

      6. associate-/l/ 70.4%

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

      7. associate-*r* 70.4%

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

      8. *-commutative 70.4%

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

      9. associate-*r* 70.4%

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

      10. *-commutative 70.4%

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

   3. Simplified 70.4%

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

   4. Taylor expanded in k around 0 56.7%

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

      1. unpow2 56.7%

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

      2. unpow2 56.7%

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

   6. Simplified 56.7%

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

      1. *-un-lft-identity 56.7%

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

      2. associate-/l* 63.0%

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

      3. associate-*l* 78.4%

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

   8. Applied egg-rr 78.4%

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

   if -5.79999999999999957e37 < t < 6.20000000000000016e-77

   1. Initial program 37.9%

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

      1. *-commutative 37.9%

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

      2. associate-*l* 37.1%

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

      3. associate-*r* 37.1%

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

      4. +-commutative 37.1%

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

      5. associate-+r+ 37.1%

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

      6. metadata-eval 37.1%

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

   3. Simplified 37.1%

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

   4. Taylor expanded in k around inf 71.8%

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

      1. *-commutative 71.8%

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

      2. unpow2 71.8%

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

      3. times-frac 75.0%

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

      4. unpow2 75.0%

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

   6. Simplified 75.0%

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

      1. associate-*l/ 78.1%

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

      2. associate-/l* 86.8%

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

   8. Applied egg-rr 86.8%

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

      1. associate-*r/ 92.1%

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

   10. Applied egg-rr 92.1%

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

       1. associate-/r/ 92.1%

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

   12. Applied egg-rr 92.1%

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

   if 6.20000000000000016e-77 < t

   1. Initial program 71.2%

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

      1. associate-*l* 71.2%

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

      2. +-commutative 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in k around 0 62.5%

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

      1. *-commutative 62.5%

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

      2. unpow2 62.5%

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

      3. times-frac 66.6%

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

      4. unpow2 66.6%

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

   6. Simplified 66.6%

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

      1. associate-*l/ 66.6%

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

      2. associate-/l* 74.0%

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

   8. Applied egg-rr 74.0%

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

      1. expm1-log1p-u 73.9%

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

      2. expm1-udef 69.5%

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

   10. Applied egg-rr 74.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1}{\frac{{t}^{1.5}}{\frac{\ell}{k}}}\right)}^{2}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 80.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1}{\frac{{t}^{1.5}}{\frac{\ell}{k}}}\right)}^{2}\right)\right)} \]

       2. expm1-log1p 81.2%

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

       3. associate-/r/ 81.2%

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

   12. Simplified 81.2%

       \[\leadsto \color{blue}{{\left(\frac{1}{{t}^{1.5}} \cdot \frac{\ell}{k}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+37}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \frac{k \cdot \frac{k \cdot t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \frac{1}{{t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 6: 72.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.0055:\\ \;\;\;\;\frac{2}{2 \cdot \frac{\frac{k \cdot {t}^{3}}{\frac{\ell}{k}}}{\ell}}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{k \cdot k}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \frac{1}{{t}^{1.5}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -0.0055)
   (/ 2.0 (* 2.0 (/ (/ (* k (pow t 3.0)) (/ l k)) l)))
   (if (<= t 6.2e-77)
     (/ 2.0 (* (* (/ t l) (/ (* k k) l)) (/ (* k k) (cos k))))
     (pow (* (/ l k) (/ 1.0 (pow t 1.5))) 2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -0.0055) {
		tmp = 2.0 / (2.0 * (((k * pow(t, 3.0)) / (l / k)) / l));
	} else if (t <= 6.2e-77) {
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * ((k * k) / cos(k)));
	} else {
		tmp = pow(((l / k) * (1.0 / pow(t, 1.5))), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-0.0055d0)) then
        tmp = 2.0d0 / (2.0d0 * (((k * (t ** 3.0d0)) / (l / k)) / l))
    else if (t <= 6.2d-77) then
        tmp = 2.0d0 / (((t / l) * ((k * k) / l)) * ((k * k) / cos(k)))
    else
        tmp = ((l / k) * (1.0d0 / (t ** 1.5d0))) ** 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -0.0055) {
		tmp = 2.0 / (2.0 * (((k * Math.pow(t, 3.0)) / (l / k)) / l));
	} else if (t <= 6.2e-77) {
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * ((k * k) / Math.cos(k)));
	} else {
		tmp = Math.pow(((l / k) * (1.0 / Math.pow(t, 1.5))), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -0.0055:
		tmp = 2.0 / (2.0 * (((k * math.pow(t, 3.0)) / (l / k)) / l))
	elif t <= 6.2e-77:
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * ((k * k) / math.cos(k)))
	else:
		tmp = math.pow(((l / k) * (1.0 / math.pow(t, 1.5))), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -0.0055)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(Float64(k * (t ^ 3.0)) / Float64(l / k)) / l)));
	elseif (t <= 6.2e-77)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * Float64(Float64(k * k) / l)) * Float64(Float64(k * k) / cos(k))));
	else
		tmp = Float64(Float64(l / k) * Float64(1.0 / (t ^ 1.5))) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -0.0055)
		tmp = 2.0 / (2.0 * (((k * (t ^ 3.0)) / (l / k)) / l));
	elseif (t <= 6.2e-77)
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * ((k * k) / cos(k)));
	else
		tmp = ((l / k) * (1.0 / (t ^ 1.5))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -0.0055], N[(2.0 / N[(2.0 * N[(N[(N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.2e-77], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(l / k), $MachinePrecision] * N[(1.0 / N[Power[t, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.0054999999999999997

   1. Initial program 66.6%

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

      1. associate-*l* 66.6%

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

      2. +-commutative 66.6%

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

   3. Simplified 66.6%

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

   4. Taylor expanded in k around 0 54.8%

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

      1. *-commutative 54.8%

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

      2. unpow2 54.8%

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

      3. times-frac 57.1%

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

      4. unpow2 57.1%

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

   6. Simplified 57.1%

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

      1. associate-*l/ 58.6%

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

      2. associate-/l* 66.8%

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

   8. Applied egg-rr 66.8%

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

      1. associate-*r/ 74.8%

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

   10. Applied egg-rr 74.8%

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

   if -0.0054999999999999997 < t < 6.20000000000000016e-77

   1. Initial program 34.5%

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

      1. *-commutative 34.5%

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

      2. associate-*l* 34.5%

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

      3. associate-*r* 34.5%

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

      4. +-commutative 34.5%

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

      5. associate-+r+ 34.5%

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

      6. metadata-eval 34.5%

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

   3. Simplified 34.5%

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

   4. Taylor expanded in k around inf 73.4%

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

      1. *-commutative 73.4%

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

      2. unpow2 73.4%

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

      3. times-frac 76.0%

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

      4. unpow2 76.0%

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

   6. Simplified 76.0%

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

   7. Taylor expanded in k around inf 76.0%

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

   8. Taylor expanded in k around 0 67.4%

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

      1. unpow2 67.4%

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

   10. Simplified 67.4%

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

   if 6.20000000000000016e-77 < t

   1. Initial program 71.2%

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

      1. associate-*l* 71.2%

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

      2. +-commutative 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in k around 0 62.5%

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

      1. *-commutative 62.5%

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

      2. unpow2 62.5%

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

      3. times-frac 66.6%

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

      4. unpow2 66.6%

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

   6. Simplified 66.6%

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

      1. associate-*l/ 66.6%

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

      2. associate-/l* 74.0%

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

   8. Applied egg-rr 74.0%

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

      1. expm1-log1p-u 73.9%

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

      2. expm1-udef 69.5%

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

   10. Applied egg-rr 74.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{1}{\frac{{t}^{1.5}}{\frac{\ell}{k}}}\right)}^{2}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 80.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1}{\frac{{t}^{1.5}}{\frac{\ell}{k}}}\right)}^{2}\right)\right)} \]

       2. expm1-log1p 81.2%

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

       3. associate-/r/ 81.2%

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

   12. Simplified 81.2%

       \[\leadsto \color{blue}{{\left(\frac{1}{{t}^{1.5}} \cdot \frac{\ell}{k}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.0055:\\ \;\;\;\;\frac{2}{2 \cdot \frac{\frac{k \cdot {t}^{3}}{\frac{\ell}{k}}}{\ell}}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{k \cdot k}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot \frac{1}{{t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 7: 69.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.0055:\\ \;\;\;\;\frac{2}{2 \cdot \frac{\frac{k \cdot {t}^{3}}{\frac{\ell}{k}}}{\ell}}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{k \cdot k}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left({t}^{3} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -0.0055)
   (/ 2.0 (* 2.0 (/ (/ (* k (pow t 3.0)) (/ l k)) l)))
   (if (<= t 1.15e-96)
     (/ 2.0 (* (* (/ t l) (/ (* k k) l)) (/ (* k k) (cos k))))
     (/ 2.0 (* 2.0 (* (pow t 3.0) (* (/ k l) (/ k l))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -0.0055) {
		tmp = 2.0 / (2.0 * (((k * pow(t, 3.0)) / (l / k)) / l));
	} else if (t <= 1.15e-96) {
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * ((k * k) / cos(k)));
	} else {
		tmp = 2.0 / (2.0 * (pow(t, 3.0) * ((k / l) * (k / l))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-0.0055d0)) then
        tmp = 2.0d0 / (2.0d0 * (((k * (t ** 3.0d0)) / (l / k)) / l))
    else if (t <= 1.15d-96) then
        tmp = 2.0d0 / (((t / l) * ((k * k) / l)) * ((k * k) / cos(k)))
    else
        tmp = 2.0d0 / (2.0d0 * ((t ** 3.0d0) * ((k / l) * (k / l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -0.0055) {
		tmp = 2.0 / (2.0 * (((k * Math.pow(t, 3.0)) / (l / k)) / l));
	} else if (t <= 1.15e-96) {
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * ((k * k) / Math.cos(k)));
	} else {
		tmp = 2.0 / (2.0 * (Math.pow(t, 3.0) * ((k / l) * (k / l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -0.0055:
		tmp = 2.0 / (2.0 * (((k * math.pow(t, 3.0)) / (l / k)) / l))
	elif t <= 1.15e-96:
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * ((k * k) / math.cos(k)))
	else:
		tmp = 2.0 / (2.0 * (math.pow(t, 3.0) * ((k / l) * (k / l))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -0.0055)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(Float64(k * (t ^ 3.0)) / Float64(l / k)) / l)));
	elseif (t <= 1.15e-96)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * Float64(Float64(k * k) / l)) * Float64(Float64(k * k) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64((t ^ 3.0) * Float64(Float64(k / l) * Float64(k / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -0.0055)
		tmp = 2.0 / (2.0 * (((k * (t ^ 3.0)) / (l / k)) / l));
	elseif (t <= 1.15e-96)
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * ((k * k) / cos(k)));
	else
		tmp = 2.0 / (2.0 * ((t ^ 3.0) * ((k / l) * (k / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -0.0055], N[(2.0 / N[(2.0 * N[(N[(N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.15e-96], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(N[Power[t, 3.0], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.0054999999999999997

   1. Initial program 66.6%

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

      1. associate-*l* 66.6%

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

      2. +-commutative 66.6%

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

   3. Simplified 66.6%

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

   4. Taylor expanded in k around 0 54.8%

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

      1. *-commutative 54.8%

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

      2. unpow2 54.8%

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

      3. times-frac 57.1%

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

      4. unpow2 57.1%

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

   6. Simplified 57.1%

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

      1. associate-*l/ 58.6%

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

      2. associate-/l* 66.8%

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

   8. Applied egg-rr 66.8%

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

      1. associate-*r/ 74.8%

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

   10. Applied egg-rr 74.8%

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

   if -0.0054999999999999997 < t < 1.15e-96

   1. Initial program 32.9%

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

      1. *-commutative 32.9%

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

      2. associate-*l* 32.9%

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

      3. associate-*r* 32.9%

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

      4. +-commutative 32.9%

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

      5. associate-+r+ 32.9%

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

      6. metadata-eval 32.9%

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

   3. Simplified 32.9%

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

   4. Taylor expanded in k around inf 74.0%

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

      1. *-commutative 74.0%

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

      2. unpow2 74.0%

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

      3. times-frac 76.8%

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

      4. unpow2 76.8%

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

   6. Simplified 76.8%

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

   7. Taylor expanded in k around inf 76.8%

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

   8. Taylor expanded in k around 0 68.2%

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

      1. unpow2 68.2%

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

   10. Simplified 68.2%

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

   if 1.15e-96 < t

   1. Initial program 69.0%

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

      1. associate-*l* 69.0%

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

      2. +-commutative 69.0%

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

   3. Simplified 69.0%

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

   4. Taylor expanded in k around 0 61.0%

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

      1. *-commutative 61.0%

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

      2. unpow2 61.0%

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

      3. times-frac 65.7%

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

      4. unpow2 65.7%

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

   6. Simplified 65.7%

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

   7. Taylor expanded in t around 0 61.0%

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

      1. unpow2 61.0%

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

      2. associate-/l* 61.1%

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

      3. associate-/r/ 63.0%

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

      4. unpow2 63.0%

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

      5. times-frac 76.3%

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

   9. Simplified 76.3%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.0055:\\ \;\;\;\;\frac{2}{2 \cdot \frac{\frac{k \cdot {t}^{3}}{\frac{\ell}{k}}}{\ell}}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{k \cdot k}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left({t}^{3} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternative 8: 67.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -24000:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left({t}^{3} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -24000.0)
   (/ l (/ (* k (* k (pow t 3.0))) l))
   (if (<= t 8e-100)
     (/ 2.0 (* (* k k) (/ (* t (/ k (/ l k))) l)))
     (/ 2.0 (* 2.0 (* (pow t 3.0) (* (/ k l) (/ k l))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -24000.0) {
		tmp = l / ((k * (k * pow(t, 3.0))) / l);
	} else if (t <= 8e-100) {
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	} else {
		tmp = 2.0 / (2.0 * (pow(t, 3.0) * ((k / l) * (k / l))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-24000.0d0)) then
        tmp = l / ((k * (k * (t ** 3.0d0))) / l)
    else if (t <= 8d-100) then
        tmp = 2.0d0 / ((k * k) * ((t * (k / (l / k))) / l))
    else
        tmp = 2.0d0 / (2.0d0 * ((t ** 3.0d0) * ((k / l) * (k / l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -24000.0) {
		tmp = l / ((k * (k * Math.pow(t, 3.0))) / l);
	} else if (t <= 8e-100) {
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	} else {
		tmp = 2.0 / (2.0 * (Math.pow(t, 3.0) * ((k / l) * (k / l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -24000.0:
		tmp = l / ((k * (k * math.pow(t, 3.0))) / l)
	elif t <= 8e-100:
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l))
	else:
		tmp = 2.0 / (2.0 * (math.pow(t, 3.0) * ((k / l) * (k / l))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -24000.0)
		tmp = Float64(l / Float64(Float64(k * Float64(k * (t ^ 3.0))) / l));
	elseif (t <= 8e-100)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t * Float64(k / Float64(l / k))) / l)));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64((t ^ 3.0) * Float64(Float64(k / l) * Float64(k / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -24000.0)
		tmp = l / ((k * (k * (t ^ 3.0))) / l);
	elseif (t <= 8e-100)
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	else
		tmp = 2.0 / (2.0 * ((t ^ 3.0) * ((k / l) * (k / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -24000.0], N[(l / N[(N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e-100], 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[(2.0 * N[(N[Power[t, 3.0], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -24000

   1. Initial program 67.8%

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

      1. associate-/l/ 67.8%

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

      2. associate-*l/ 71.1%

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

      3. associate-*l/ 71.6%

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

      4. associate-/r/ 70.6%

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

      5. *-commutative 70.6%

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

      6. associate-/l/ 70.5%

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

      7. associate-*r* 70.5%

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

      8. *-commutative 70.5%

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

      9. associate-*r* 70.6%

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

      10. *-commutative 70.6%

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

   3. Simplified 70.6%

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

   4. Taylor expanded in k around 0 55.2%

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

      1. unpow2 55.2%

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

      2. unpow2 55.2%

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

   6. Simplified 55.2%

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

      1. *-un-lft-identity 55.2%

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

      2. associate-/l* 60.7%

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

      3. associate-*l* 75.3%

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

   8. Applied egg-rr 75.3%

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

   if -24000 < t < 8.0000000000000002e-100

   1. Initial program 33.5%

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

      1. *-commutative 33.5%

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

      2. associate-*l* 33.5%

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

      3. associate-*r* 33.5%

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

      4. +-commutative 33.5%

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

      5. associate-+r+ 33.5%

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

      6. metadata-eval 33.5%

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

   3. Simplified 33.5%

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

   4. Taylor expanded in k around inf 73.1%

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

      1. *-commutative 73.1%

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

      2. unpow2 73.1%

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

      3. times-frac 75.8%

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

      4. unpow2 75.8%

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

   6. Simplified 75.8%

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

      1. associate-*l/ 79.4%

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

      2. associate-/l* 87.7%

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

   8. Applied egg-rr 87.7%

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

   9. Taylor expanded in k around 0 62.7%

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

       1. unpow2 62.6%

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

   11. Simplified 62.7%

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

   if 8.0000000000000002e-100 < t

   1. Initial program 69.0%

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

      1. associate-*l* 69.0%

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

      2. +-commutative 69.0%

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

   3. Simplified 69.0%

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

   4. Taylor expanded in k around 0 61.0%

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

      1. *-commutative 61.0%

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

      2. unpow2 61.0%

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

      3. times-frac 65.7%

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

      4. unpow2 65.7%

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

   6. Simplified 65.7%

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

   7. Taylor expanded in t around 0 61.0%

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

      1. unpow2 61.0%

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

      2. associate-/l* 61.1%

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

      3. associate-/r/ 63.0%

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

      4. unpow2 63.0%

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

      5. times-frac 76.3%

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

   9. Simplified 76.3%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -24000:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left({t}^{3} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternative 9: 68.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.0055:\\ \;\;\;\;\frac{2}{2 \cdot \frac{\frac{k \cdot {t}^{3}}{\frac{\ell}{k}}}{\ell}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left({t}^{3} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -0.0055)
   (/ 2.0 (* 2.0 (/ (/ (* k (pow t 3.0)) (/ l k)) l)))
   (if (<= t 4.2e-100)
     (/ 2.0 (* (* k k) (/ (* t (/ k (/ l k))) l)))
     (/ 2.0 (* 2.0 (* (pow t 3.0) (* (/ k l) (/ k l))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -0.0055) {
		tmp = 2.0 / (2.0 * (((k * pow(t, 3.0)) / (l / k)) / l));
	} else if (t <= 4.2e-100) {
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	} else {
		tmp = 2.0 / (2.0 * (pow(t, 3.0) * ((k / l) * (k / l))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-0.0055d0)) then
        tmp = 2.0d0 / (2.0d0 * (((k * (t ** 3.0d0)) / (l / k)) / l))
    else if (t <= 4.2d-100) then
        tmp = 2.0d0 / ((k * k) * ((t * (k / (l / k))) / l))
    else
        tmp = 2.0d0 / (2.0d0 * ((t ** 3.0d0) * ((k / l) * (k / l))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -0.0055) {
		tmp = 2.0 / (2.0 * (((k * Math.pow(t, 3.0)) / (l / k)) / l));
	} else if (t <= 4.2e-100) {
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	} else {
		tmp = 2.0 / (2.0 * (Math.pow(t, 3.0) * ((k / l) * (k / l))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -0.0055:
		tmp = 2.0 / (2.0 * (((k * math.pow(t, 3.0)) / (l / k)) / l))
	elif t <= 4.2e-100:
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l))
	else:
		tmp = 2.0 / (2.0 * (math.pow(t, 3.0) * ((k / l) * (k / l))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -0.0055)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(Float64(k * (t ^ 3.0)) / Float64(l / k)) / l)));
	elseif (t <= 4.2e-100)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t * Float64(k / Float64(l / k))) / l)));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64((t ^ 3.0) * Float64(Float64(k / l) * Float64(k / l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -0.0055)
		tmp = 2.0 / (2.0 * (((k * (t ^ 3.0)) / (l / k)) / l));
	elseif (t <= 4.2e-100)
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	else
		tmp = 2.0 / (2.0 * ((t ^ 3.0) * ((k / l) * (k / l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -0.0055], N[(2.0 / N[(2.0 * N[(N[(N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e-100], 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[(2.0 * N[(N[Power[t, 3.0], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -0.0054999999999999997

   1. Initial program 66.6%

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

      1. associate-*l* 66.6%

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

      2. +-commutative 66.6%

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

   3. Simplified 66.6%

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

   4. Taylor expanded in k around 0 54.8%

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

      1. *-commutative 54.8%

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

      2. unpow2 54.8%

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

      3. times-frac 57.1%

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

      4. unpow2 57.1%

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

   6. Simplified 57.1%

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

      1. associate-*l/ 58.6%

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

      2. associate-/l* 66.8%

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

   8. Applied egg-rr 66.8%

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

      1. associate-*r/ 74.8%

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

   10. Applied egg-rr 74.8%

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

   if -0.0054999999999999997 < t < 4.20000000000000019e-100

   1. Initial program 32.9%

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

      1. *-commutative 32.9%

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

      2. associate-*l* 32.9%

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

      3. associate-*r* 32.9%

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

      4. +-commutative 32.9%

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

      5. associate-+r+ 32.9%

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

      6. metadata-eval 32.9%

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

   3. Simplified 32.9%

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

   4. Taylor expanded in k around inf 74.0%

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

      1. *-commutative 74.0%

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

      2. unpow2 74.0%

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

      3. times-frac 76.8%

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

      4. unpow2 76.8%

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

   6. Simplified 76.8%

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

      1. associate-*l/ 80.5%

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

      2. associate-/l* 87.2%

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

   8. Applied egg-rr 87.2%

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

   9. Taylor expanded in k around 0 63.2%

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

       1. unpow2 63.0%

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

   11. Simplified 63.2%

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

   if 4.20000000000000019e-100 < t

   1. Initial program 69.0%

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

      1. associate-*l* 69.0%

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

      2. +-commutative 69.0%

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

   3. Simplified 69.0%

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

   4. Taylor expanded in k around 0 61.0%

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

      1. *-commutative 61.0%

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

      2. unpow2 61.0%

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

      3. times-frac 65.7%

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

      4. unpow2 65.7%

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

   6. Simplified 65.7%

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

   7. Taylor expanded in t around 0 61.0%

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

      1. unpow2 61.0%

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

      2. associate-/l* 61.1%

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

      3. associate-/r/ 63.0%

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

      4. unpow2 63.0%

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

      5. times-frac 76.3%

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

   9. Simplified 76.3%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.0055:\\ \;\;\;\;\frac{2}{2 \cdot \frac{\frac{k \cdot {t}^{3}}{\frac{\ell}{k}}}{\ell}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left({t}^{3} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternative 10: 62.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -56 \lor \neg \left(t \leq 1.9 \cdot 10^{-107}\right):\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -56.0) (not (<= t 1.9e-107)))
   (* l (/ (/ l (* k k)) (pow t 3.0)))
   (/ 2.0 (* (* k k) (/ (* t (/ k (/ l k))) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -56.0) || !(t <= 1.9e-107)) {
		tmp = l * ((l / (k * k)) / pow(t, 3.0));
	} else {
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-56.0d0)) .or. (.not. (t <= 1.9d-107))) then
        tmp = l * ((l / (k * k)) / (t ** 3.0d0))
    else
        tmp = 2.0d0 / ((k * k) * ((t * (k / (l / k))) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -56.0) || !(t <= 1.9e-107)) {
		tmp = l * ((l / (k * k)) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -56.0) or not (t <= 1.9e-107):
		tmp = l * ((l / (k * k)) / math.pow(t, 3.0))
	else:
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -56.0) || !(t <= 1.9e-107))
		tmp = Float64(l * Float64(Float64(l / Float64(k * k)) / (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

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -56.0) || ~((t <= 1.9e-107)))
		tmp = l * ((l / (k * k)) / (t ^ 3.0));
	else
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -56.0], N[Not[LessEqual[t, 1.9e-107]], $MachinePrecision]], N[(l * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $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 < -56 or 1.9000000000000001e-107 < t

   1. Initial program 68.0%

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

      1. associate-/l/ 68.0%

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

      2. associate-*l/ 68.7%

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

      3. associate-*l/ 68.9%

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

      4. associate-/r/ 68.4%

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

      5. *-commutative 68.4%

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

      6. associate-/l/ 68.4%

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

      7. associate-*r* 68.4%

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

      8. *-commutative 68.4%

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

      9. associate-*r* 68.4%

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

      10. *-commutative 68.4%

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

   3. Simplified 68.4%

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

   4. Taylor expanded in k around 0 58.5%

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

      1. unpow2 58.5%

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

      2. unpow2 58.5%

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

   6. Simplified 58.5%

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

   7. Taylor expanded in l around 0 58.5%

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

      1. unpow2 58.5%

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

      2. unpow2 58.5%

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

      3. associate-*r* 65.2%

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

      4. associate-*l/ 70.9%

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

      5. *-commutative 70.9%

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

      6. associate-*r* 63.0%

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

      7. associate-/r* 63.6%

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

   9. Simplified 63.6%

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

   if -56 < t < 1.9000000000000001e-107

   1. Initial program 32.9%

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

      1. *-commutative 32.9%

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

      2. associate-*l* 32.9%

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

      3. associate-*r* 32.9%

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

      4. +-commutative 32.9%

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

      5. associate-+r+ 32.9%

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

      6. metadata-eval 32.9%

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

   3. Simplified 32.9%

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

   4. Taylor expanded in k around inf 74.0%

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

      1. *-commutative 74.0%

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

      2. unpow2 74.0%

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

      3. times-frac 76.7%

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

      4. unpow2 76.7%

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

   6. Simplified 76.7%

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

      1. associate-*l/ 80.4%

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

      2. associate-/l* 88.1%

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

   8. Applied egg-rr 88.1%

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

   9. Taylor expanded in k around 0 63.1%

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

       1. unpow2 63.0%

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

   11. Simplified 63.1%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -56 \lor \neg \left(t \leq 1.9 \cdot 10^{-107}\right):\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{k \cdot k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \end{array} \]

Alternative 11: 65.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -28000 \lor \neg \left(t \leq 3.2 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -28000.0) (not (<= t 3.2e-49)))
   (/ (* l l) (* k (* k (pow t 3.0))))
   (/ 2.0 (* (* k k) (/ (* t (/ k (/ l k))) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -28000.0) || !(t <= 3.2e-49)) {
		tmp = (l * l) / (k * (k * pow(t, 3.0)));
	} else {
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-28000.0d0)) .or. (.not. (t <= 3.2d-49))) then
        tmp = (l * l) / (k * (k * (t ** 3.0d0)))
    else
        tmp = 2.0d0 / ((k * k) * ((t * (k / (l / k))) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -28000.0) || !(t <= 3.2e-49)) {
		tmp = (l * l) / (k * (k * Math.pow(t, 3.0)));
	} else {
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -28000.0) or not (t <= 3.2e-49):
		tmp = (l * l) / (k * (k * math.pow(t, 3.0)))
	else:
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -28000.0) || !(t <= 3.2e-49))
		tmp = Float64(Float64(l * l) / Float64(k * Float64(k * (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

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -28000.0) || ~((t <= 3.2e-49)))
		tmp = (l * l) / (k * (k * (t ^ 3.0)));
	else
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -28000.0], N[Not[LessEqual[t, 3.2e-49]], $MachinePrecision]], N[(N[(l * l), $MachinePrecision] / N[(k * N[(k * N[Power[t, 3.0], $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 < -28000 or 3.20000000000000002e-49 < t

   1. Initial program 70.5%

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

      1. associate-/l/ 70.5%

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

      2. associate-*l/ 71.9%

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

      3. associate-*l/ 72.2%

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

      4. associate-/r/ 71.1%

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

      5. *-commutative 71.1%

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

      6. associate-/l/ 71.0%

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

      7. associate-*r* 71.0%

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

      8. *-commutative 71.0%

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

      9. associate-*r* 71.1%

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

      10. *-commutative 71.1%

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

   3. Simplified 71.1%

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

   4. Taylor expanded in k around 0 60.6%

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

      1. unpow2 60.6%

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

      2. unpow2 60.6%

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

   6. Simplified 60.6%

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

   7. Taylor expanded in k around 0 60.6%

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

      1. unpow2 60.6%

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

      2. associate-*r* 70.0%

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

   9. Simplified 70.0%

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

   if -28000 < t < 3.20000000000000002e-49

   1. Initial program 35.0%

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

      1. *-commutative 35.0%

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

      2. associate-*l* 35.0%

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

      3. associate-*r* 35.0%

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

      4. +-commutative 35.0%

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

      5. associate-+r+ 35.0%

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

      6. metadata-eval 35.0%

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

   3. Simplified 35.0%

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

   4. Taylor expanded in k around inf 70.8%

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

      1. *-commutative 70.8%

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

      2. unpow2 70.8%

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

      3. times-frac 73.4%

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

      4. unpow2 73.4%

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

   6. Simplified 73.4%

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

      1. associate-*l/ 76.6%

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

      2. associate-/l* 86.5%

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

   8. Applied egg-rr 86.5%

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

   9. Taylor expanded in k around 0 61.0%

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

       1. unpow2 60.9%

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

   11. Simplified 61.0%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -28000 \lor \neg \left(t \leq 3.2 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \end{array} \]

Alternative 12: 67.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -24000 \lor \neg \left(t \leq 3.1 \cdot 10^{-77}\right):\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -24000.0) (not (<= t 3.1e-77)))
   (/ l (/ (* k (* k (pow t 3.0))) l))
   (/ 2.0 (* (* k k) (/ (* t (/ k (/ l k))) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -24000.0) || !(t <= 3.1e-77)) {
		tmp = l / ((k * (k * pow(t, 3.0))) / l);
	} else {
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-24000.0d0)) .or. (.not. (t <= 3.1d-77))) then
        tmp = l / ((k * (k * (t ** 3.0d0))) / l)
    else
        tmp = 2.0d0 / ((k * k) * ((t * (k / (l / k))) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -24000.0) || !(t <= 3.1e-77)) {
		tmp = l / ((k * (k * Math.pow(t, 3.0))) / l);
	} else {
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -24000.0) or not (t <= 3.1e-77):
		tmp = l / ((k * (k * math.pow(t, 3.0))) / l)
	else:
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -24000.0) || !(t <= 3.1e-77))
		tmp = Float64(l / Float64(Float64(k * Float64(k * (t ^ 3.0))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t * Float64(k / Float64(l / k))) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -24000.0) || ~((t <= 3.1e-77)))
		tmp = l / ((k * (k * (t ^ 3.0))) / l);
	else
		tmp = 2.0 / ((k * k) * ((t * (k / (l / k))) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -24000.0], N[Not[LessEqual[t, 3.1e-77]], $MachinePrecision]], N[(l / N[(N[(k * N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $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 < -24000 or 3.10000000000000008e-77 < t

   1. Initial program 69.7%

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

      1. associate-/l/ 69.7%

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

      2. associate-*l/ 70.5%

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

      3. associate-*l/ 70.7%

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

      4. associate-/r/ 69.6%

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

      5. *-commutative 69.6%

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

      6. associate-/l/ 69.6%

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

      7. associate-*r* 69.6%

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

      8. *-commutative 69.6%

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

      9. associate-*r* 69.6%

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

      10. *-commutative 69.6%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in k around 0 59.4%

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

      1. unpow2 59.4%

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

      2. unpow2 59.4%

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

   6. Simplified 59.4%

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

      1. *-un-lft-identity 59.4%

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

      2. associate-/l* 64.4%

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

      3. associate-*l* 74.8%

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

   8. Applied egg-rr 74.8%

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

   if -24000 < t < 3.10000000000000008e-77

   1. Initial program 35.0%

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

      1. *-commutative 35.0%

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

      2. associate-*l* 35.0%

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

      3. associate-*r* 35.0%

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

      4. +-commutative 35.0%

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

      5. associate-+r+ 35.0%

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

      6. metadata-eval 35.0%

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

   3. Simplified 35.0%

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

   4. Taylor expanded in k around inf 72.6%

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

      1. *-commutative 72.6%

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

      2. unpow2 72.6%

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

      3. times-frac 75.2%

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

      4. unpow2 75.2%

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

   6. Simplified 75.2%

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

      1. associate-*l/ 78.4%

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

      2. associate-/l* 87.7%

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

   8. Applied egg-rr 87.7%

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

   9. Taylor expanded in k around 0 62.4%

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

       1. unpow2 62.3%

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

   11. Simplified 62.4%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -24000 \lor \neg \left(t \leq 3.1 \cdot 10^{-77}\right):\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \end{array} \]

Alternative 13: 57.9% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(t, l, k):
	return 2.0 / ((k * k) * (t * ((k / l) * (k / l))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

   1. *-commutative 54.2%

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

   2. associate-*l* 49.6%

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

   3. associate-*r* 49.6%

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

   4. +-commutative 49.6%

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

   5. associate-+r+ 49.6%

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

   6. metadata-eval 49.6%

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

3. Simplified 49.6%

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

4. Taylor expanded in k around inf 60.8%

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

   1. *-commutative 60.8%

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

   2. unpow2 60.8%

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

   3. times-frac 62.9%

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

   4. unpow2 62.9%

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

6. Simplified 62.9%

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

7. Taylor expanded in k around 0 57.2%

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

   1. unpow2 57.2%

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

9. Simplified 57.2%

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

10. Taylor expanded in t around 0 53.6%

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

    1. unpow2 53.6%

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

    2. *-commutative 53.6%

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

    3. associate-*r/ 53.3%

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

    4. unpow2 53.3%

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

    5. times-frac 56.5%

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

12. Simplified 56.5%

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

13. Final simplification 56.5%

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

Alternative 14: 57.9% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(t, l, k):
	return 2.0 / ((k * k) * (t * ((k * (k / l)) / l)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

   1. *-commutative 54.2%

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

   2. associate-*l* 49.6%

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

   3. associate-*r* 49.6%

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

   4. +-commutative 49.6%

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

   5. associate-+r+ 49.6%

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

   6. metadata-eval 49.6%

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

3. Simplified 49.6%

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

4. Taylor expanded in k around inf 60.8%

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

   1. *-commutative 60.8%

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

   2. unpow2 60.8%

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

   3. times-frac 62.9%

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

   4. unpow2 62.9%

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

6. Simplified 62.9%

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

7. Taylor expanded in k around 0 57.2%

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

   1. unpow2 57.2%

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

9. Simplified 57.2%

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

10. Taylor expanded in t around 0 53.6%

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

    1. unpow2 53.6%

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

    2. *-commutative 53.6%

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

    3. associate-*r/ 53.3%

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

    4. unpow2 53.3%

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

    5. times-frac 56.5%

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

12. Simplified 56.5%

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

    1. associate-*l/ 56.5%

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

14. Applied egg-rr 56.5%

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

15. Final simplification 56.5%

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

Alternative 15: 59.3% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(t, l, k):
	return 2.0 / ((k * k) * ((t / l) * ((k * k) / l)))

Julia

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

MATLAB

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

Wolfram

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 54.2%

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

   1. *-commutative 54.2%

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

   2. associate-*l* 49.6%

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

   3. associate-*r* 49.6%

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

   4. +-commutative 49.6%

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

   5. associate-+r+ 49.6%

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

   6. metadata-eval 49.6%

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

3. Simplified 49.6%

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

4. Taylor expanded in k around inf 60.8%

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

   1. *-commutative 60.8%

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

   2. unpow2 60.8%

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

   3. times-frac 62.9%

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

   4. unpow2 62.9%

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

6. Simplified 62.9%

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

7. Taylor expanded in k around 0 57.2%

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

   1. unpow2 57.2%

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

9. Simplified 57.2%

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

10. Final simplification 57.2%

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

Alternative 16: 58.7% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(t, l, k):
	return 2.0 / ((k * k) * ((t * (k / (l / k))) / l))

Julia

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

MATLAB

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

Wolfram

code[t_, l_, k_] := 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. Initial program 54.2%

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

   1. *-commutative 54.2%

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

   2. associate-*l* 49.6%

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

   3. associate-*r* 49.6%

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

   4. +-commutative 49.6%

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

   5. associate-+r+ 49.6%

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

   6. metadata-eval 49.6%

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

3. Simplified 49.6%

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

4. Taylor expanded in k around inf 60.8%

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

   1. *-commutative 60.8%

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

   2. unpow2 60.8%

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

   3. times-frac 62.9%

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

   4. unpow2 62.9%

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

6. Simplified 62.9%

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

   1. associate-*l/ 64.9%

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

   2. associate-/l* 69.4%

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

8. Applied egg-rr 69.4%

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

9. Taylor expanded in k around 0 57.3%

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

    1. unpow2 57.2%

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

11. Simplified 57.3%

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

12. Final simplification 57.3%

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

Reproduce

herbie shell --seed 2023187 
(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))))