Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.2% → 83.5%

Time: 17.7s

Alternatives: 15

Speedup: 28.1×

• 28.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.2% 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: 83.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{+32}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(2 \cdot \left({t}^{3} \cdot k\right)\right)}\right)\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \tan k}}{\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot {t}^{-1.5}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -8.8e+32)
   (* l (* l (/ 2.0 (* (tan k) (* 2.0 (* (pow t 3.0) k))))))
   (if (<= t 6.8e+21)
     (/ (/ 2.0 (* (sin k) (tan k))) (* (/ k l) (* t (/ k l))))
     (pow (* (/ l k) (pow t -1.5)) 2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -8.8e+32) {
		tmp = l * (l * (2.0 / (tan(k) * (2.0 * (pow(t, 3.0) * k)))));
	} else if (t <= 6.8e+21) {
		tmp = (2.0 / (sin(k) * tan(k))) / ((k / l) * (t * (k / l)));
	} else {
		tmp = pow(((l / k) * 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 <= (-8.8d+32)) then
        tmp = l * (l * (2.0d0 / (tan(k) * (2.0d0 * ((t ** 3.0d0) * k)))))
    else if (t <= 6.8d+21) then
        tmp = (2.0d0 / (sin(k) * tan(k))) / ((k / l) * (t * (k / l)))
    else
        tmp = ((l / k) * (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 <= -8.8e+32) {
		tmp = l * (l * (2.0 / (Math.tan(k) * (2.0 * (Math.pow(t, 3.0) * k)))));
	} else if (t <= 6.8e+21) {
		tmp = (2.0 / (Math.sin(k) * Math.tan(k))) / ((k / l) * (t * (k / l)));
	} else {
		tmp = Math.pow(((l / k) * Math.pow(t, -1.5)), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -8.8e+32:
		tmp = l * (l * (2.0 / (math.tan(k) * (2.0 * (math.pow(t, 3.0) * k)))))
	elif t <= 6.8e+21:
		tmp = (2.0 / (math.sin(k) * math.tan(k))) / ((k / l) * (t * (k / l)))
	else:
		tmp = math.pow(((l / k) * math.pow(t, -1.5)), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -8.8e+32)
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(2.0 * Float64((t ^ 3.0) * k))))));
	elseif (t <= 6.8e+21)
		tmp = Float64(Float64(2.0 / Float64(sin(k) * tan(k))) / Float64(Float64(k / l) * Float64(t * Float64(k / l))));
	else
		tmp = Float64(Float64(l / k) * (t ^ -1.5)) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -8.8e+32)
		tmp = l * (l * (2.0 / (tan(k) * (2.0 * ((t ^ 3.0) * k)))));
	elseif (t <= 6.8e+21)
		tmp = (2.0 / (sin(k) * tan(k))) / ((k / l) * (t * (k / l)));
	else
		tmp = ((l / k) * (t ^ -1.5)) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -8.8e+32], N[(l * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(2.0 * N[(N[Power[t, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.8e+21], N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / l), $MachinePrecision] * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(l / k), $MachinePrecision] * N[Power[t, -1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.80000000000000004e32

   1. Initial program 71.9%

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

      1. associate-/l/ 71.9%

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

      2. associate-*l/ 75.1%

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

      3. associate-*l/ 75.1%

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

      4. associate-/r/ 75.1%

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

      5. *-commutative 75.1%

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

      6. associate-/l/ 75.1%

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

      7. associate-*r* 75.1%

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

      8. *-commutative 75.1%

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

      9. associate-*r* 75.1%

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

      10. *-commutative 75.1%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in k around 0 75.1%

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

      1. pow1 75.1%

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

      2. associate-*l* 82.0%

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

   6. Applied egg-rr 82.0%

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

   if -8.80000000000000004e32 < t < 6.8e21

   1. Initial program 52.8%

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

      1. *-commutative 52.8%

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

      2. associate-*l* 52.8%

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

      3. associate-*r* 52.8%

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

      4. +-commutative 52.8%

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

      5. associate-+r+ 52.8%

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

      6. metadata-eval 52.8%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in k around inf 76.8%

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

      1. *-commutative 76.8%

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

      2. unpow2 76.8%

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

      3. times-frac 83.9%

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

      4. unpow2 83.9%

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

   6. Simplified 83.9%

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

      1. *-un-lft-identity 83.9%

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

      2. associate-/l* 86.3%

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

   8. Applied egg-rr 86.3%

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

      1. expm1-log1p-u 58.5%

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

      2. expm1-udef 49.8%

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

      3. *-commutative 49.8%

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

      4. *-un-lft-identity 49.8%

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

      5. associate-/r/ 49.8%

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

   10. Applied egg-rr 49.8%

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

       1. expm1-def 58.5%

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

       2. expm1-log1p 86.3%

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

       3. associate-/r* 86.4%

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

       4. associate-/r/ 87.1%

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

       5. associate-/l/ 89.9%

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

       6. associate-/l* 85.5%

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

       7. associate-*r/ 89.8%

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

       8. unpow2 89.8%

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

   12. Simplified 89.8%

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

       1. div-inv 89.8%

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

       2. metadata-eval 89.8%

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

       3. unpow2 89.8%

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

       4. frac-times 89.8%

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

       5. clear-num 89.9%

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

       6. clear-num 89.8%

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

       7. *-commutative 89.8%

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

       8. associate-*l* 93.4%

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

   14. Applied egg-rr 93.4%

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

   if 6.8e21 < t

   1. Initial program 47.7%

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

      1. associate-*l* 47.7%

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

      2. associate-/l/ 47.7%

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

      3. *-commutative 47.7%

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

      4. associate-*r/ 50.8%

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

      5. associate-/l* 48.3%

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

      6. associate-/r/ 44.0%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in k around 0 42.6%

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

      1. associate-/r* 42.2%

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

      2. unpow2 42.2%

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

      3. unpow2 42.2%

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

      4. times-frac 50.4%

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

   6. Simplified 50.4%

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

      1. expm1-log1p-u 50.1%

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

      2. expm1-udef 48.6%

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

      3. div-inv 48.6%

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

      4. pow2 48.6%

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

      5. pow-flip 48.7%

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

      6. metadata-eval 48.7%

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

   8. Applied egg-rr 48.7%

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

      1. expm1-def 50.3%

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

      2. expm1-log1p 50.5%

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

   10. Simplified 50.5%

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

       1. add-sqr-sqrt 50.5%

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

       2. pow2 50.5%

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

       3. sqrt-prod 50.5%

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

       4. unpow2 50.5%

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

       5. sqrt-prod 34.3%

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

       6. add-sqr-sqrt 53.0%

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

       7. sqrt-pow1 62.2%

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

       8. metadata-eval 62.2%

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

   12. Applied egg-rr 62.2%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{+32}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(2 \cdot \left({t}^{3} \cdot k\right)\right)}\right)\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \tan k}}{\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot {t}^{-1.5}\right)}^{2}\\ \end{array} \]

Alternative 2: 83.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ 1.0 (+ 1.0 t_1)))
        INFINITY)
     (* l (* l (/ 2.0 (* (tan k) (* (* (pow t 3.0) (sin k)) (+ 2.0 t_1))))))
     (/ (/ 2.0 (* (sin k) (tan k))) (* (/ k l) (* t (/ k l)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if (((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1))) <= ((double) INFINITY)) {
		tmp = l * (l * (2.0 / (tan(k) * ((pow(t, 3.0) * sin(k)) * (2.0 + t_1)))));
	} else {
		tmp = (2.0 / (sin(k) * tan(k))) / ((k / l) * (t * (k / l)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if (((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + t_1))) <= Double.POSITIVE_INFINITY) {
		tmp = l * (l * (2.0 / (Math.tan(k) * ((Math.pow(t, 3.0) * Math.sin(k)) * (2.0 + t_1)))));
	} else {
		tmp = (2.0 / (Math.sin(k) * Math.tan(k))) / ((k / l) * (t * (k / l)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	tmp = 0
	if ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * (1.0 + (1.0 + t_1))) <= math.inf:
		tmp = l * (l * (2.0 / (math.tan(k) * ((math.pow(t, 3.0) * math.sin(k)) * (2.0 + t_1)))))
	else:
		tmp = (2.0 / (math.sin(k) * math.tan(k))) / ((k / l) * (t * (k / l)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1))) <= Inf)
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(Float64((t ^ 3.0) * sin(k)) * Float64(2.0 + t_1))))));
	else
		tmp = Float64(Float64(2.0 / Float64(sin(k) * tan(k))) / Float64(Float64(k / l) * Float64(t * Float64(k / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	tmp = 0.0;
	if ((((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1))) <= Inf)
		tmp = l * (l * (2.0 / (tan(k) * (((t ^ 3.0) * sin(k)) * (2.0 + t_1)))));
	else
		tmp = (2.0 / (sin(k) * tan(k))) / ((k / l) * (t * (k / l)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < +inf.0

   1. Initial program 81.4%

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

      1. associate-/l/ 81.4%

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

      2. associate-*l/ 83.6%

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

      3. associate-*l/ 83.5%

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

      4. associate-/r/ 83.0%

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

      5. *-commutative 83.0%

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

      6. associate-/l/ 82.9%

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

      7. associate-*r* 82.9%

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

      8. *-commutative 82.9%

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

      9. associate-*r* 82.9%

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

      10. *-commutative 82.9%

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

   3. Simplified 82.9%

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

      1. expm1-log1p-u 67.3%

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

      2. expm1-udef 63.1%

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

      3. associate-*l* 63.2%

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

      4. *-commutative 63.2%

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

   5. Applied egg-rr 63.2%

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

      1. expm1-def 68.5%

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

      2. expm1-log1p 85.8%

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

      3. *-commutative 85.8%

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

      4. *-commutative 85.8%

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

   7. Simplified 85.8%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*l* 0.0%

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

      3. associate-*r* 0.0%

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

      4. +-commutative 0.0%

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

      5. associate-+r+ 0.0%

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

      6. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in k around inf 47.5%

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

      1. *-commutative 47.5%

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

      2. unpow2 47.5%

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

      3. times-frac 58.5%

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

      4. unpow2 58.5%

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

   6. Simplified 58.5%

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

      1. *-un-lft-identity 58.5%

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

      2. associate-/l* 66.1%

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

   8. Applied egg-rr 66.1%

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

      1. expm1-log1p-u 47.9%

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

      2. expm1-udef 38.0%

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

      3. *-commutative 38.0%

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

      4. *-un-lft-identity 38.0%

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

      5. associate-/r/ 38.0%

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

   10. Applied egg-rr 38.0%

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

       1. expm1-def 47.9%

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

       2. expm1-log1p 66.1%

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

       3. associate-/r* 66.1%

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

       4. associate-/r/ 67.4%

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

       5. associate-/l/ 69.8%

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

       6. associate-/l* 62.5%

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

       7. associate-*r/ 69.7%

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

       8. unpow2 69.7%

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

   12. Simplified 69.7%

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

       1. div-inv 69.7%

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

       2. metadata-eval 69.7%

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

       3. unpow2 69.7%

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

       4. frac-times 69.7%

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

       5. clear-num 69.8%

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

       6. clear-num 69.7%

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

       7. *-commutative 69.7%

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

       8. associate-*l* 75.7%

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

   14. Applied egg-rr 75.7%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \tan k}}{\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 3: 71.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}\right)}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-112}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot {t}^{-1.5}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -9.8e-17)
   (/ 2.0 (* 2.0 (* (/ k l) (/ (/ k l) (pow t -3.0)))))
   (if (<= t 3e-112)
     (* (* l l) (/ 2.0 (* (tan k) (* (* k k) (* t (sin k))))))
     (pow (* (/ l k) (pow t -1.5)) 2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -9.8e-17) {
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / pow(t, -3.0))));
	} else if (t <= 3e-112) {
		tmp = (l * l) * (2.0 / (tan(k) * ((k * k) * (t * sin(k)))));
	} else {
		tmp = pow(((l / k) * 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 <= (-9.8d-17)) then
        tmp = 2.0d0 / (2.0d0 * ((k / l) * ((k / l) / (t ** (-3.0d0)))))
    else if (t <= 3d-112) then
        tmp = (l * l) * (2.0d0 / (tan(k) * ((k * k) * (t * sin(k)))))
    else
        tmp = ((l / k) * (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 <= -9.8e-17) {
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / Math.pow(t, -3.0))));
	} else if (t <= 3e-112) {
		tmp = (l * l) * (2.0 / (Math.tan(k) * ((k * k) * (t * Math.sin(k)))));
	} else {
		tmp = Math.pow(((l / k) * Math.pow(t, -1.5)), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -9.8e-17:
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / math.pow(t, -3.0))))
	elif t <= 3e-112:
		tmp = (l * l) * (2.0 / (math.tan(k) * ((k * k) * (t * math.sin(k)))))
	else:
		tmp = math.pow(((l / k) * math.pow(t, -1.5)), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -9.8e-17)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k / l) * Float64(Float64(k / l) / (t ^ -3.0)))));
	elseif (t <= 3e-112)
		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) * (t ^ -1.5)) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -9.8e-17)
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / (t ^ -3.0))));
	elseif (t <= 3e-112)
		tmp = (l * l) * (2.0 / (tan(k) * ((k * k) * (t * sin(k)))));
	else
		tmp = ((l / k) * (t ^ -1.5)) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -9.8e-17], N[(2.0 / N[(2.0 * N[(N[(k / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3e-112], 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[Power[t, -1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.80000000000000024e-17

   1. Initial program 71.0%

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

      1. *-commutative 71.0%

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

      2. associate-*l* 69.4%

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

      3. associate-*r* 69.4%

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

      4. +-commutative 69.4%

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

      5. associate-+r+ 69.4%

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

      6. metadata-eval 69.4%

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

   3. Simplified 69.4%

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

   4. Taylor expanded in k around 0 65.7%

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

      1. associate-/l* 64.4%

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

      2. unpow2 64.4%

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

      3. unpow2 64.4%

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

      4. associate-*r/ 68.7%

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

   6. Simplified 68.7%

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

      1. times-frac 76.7%

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

      2. div-inv 76.7%

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

      3. pow-flip 76.7%

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

      4. metadata-eval 76.7%

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

   8. Applied egg-rr 76.7%

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

      1. associate-/r* 78.5%

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

   10. Simplified 78.5%

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

   if -9.80000000000000024e-17 < t < 3.0000000000000001e-112

   1. Initial program 47.1%

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

      1. associate-/l/ 47.1%

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

      2. associate-*l/ 47.1%

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

      3. associate-*l/ 47.1%

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

      4. associate-/r/ 48.1%

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

      5. *-commutative 48.1%

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

      6. associate-/l/ 48.1%

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

      7. associate-*r* 48.1%

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

      8. *-commutative 48.1%

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

      9. associate-*r* 48.1%

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

      10. *-commutative 48.1%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in k around inf 80.9%

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

      1. unpow2 80.9%

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

      2. *-commutative 80.9%

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

   6. Simplified 80.9%

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

   if 3.0000000000000001e-112 < t

   1. Initial program 53.7%

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

      1. associate-*l* 53.7%

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

      2. associate-/l/ 53.7%

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

      3. *-commutative 53.7%

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

      4. associate-*r/ 55.9%

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

      5. associate-/l* 54.1%

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

      6. associate-/r/ 51.0%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in k around 0 46.8%

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

      1. associate-/r* 46.6%

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

      2. unpow2 46.6%

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

      3. unpow2 46.6%

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

      4. times-frac 55.9%

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

   6. Simplified 55.9%

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

      1. expm1-log1p-u 55.6%

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

      2. expm1-udef 53.4%

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

      3. div-inv 53.4%

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

      4. pow2 53.4%

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

      5. pow-flip 53.5%

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

      6. metadata-eval 53.5%

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

   8. Applied egg-rr 53.5%

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

      1. expm1-def 55.7%

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

      2. expm1-log1p 56.0%

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

   10. Simplified 56.0%

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

       1. add-sqr-sqrt 56.0%

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

       2. pow2 56.0%

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

       3. sqrt-prod 56.0%

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

       4. unpow2 56.0%

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

       5. sqrt-prod 34.1%

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

       6. add-sqr-sqrt 56.7%

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

       7. sqrt-pow1 64.4%

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

       8. metadata-eval 64.4%

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

   12. Applied egg-rr 64.4%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}\right)}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-112}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot {t}^{-1.5}\right)}^{2}\\ \end{array} \]

Alternative 4: 79.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{+31}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}\right)}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot {t}^{-1.5}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -1.02e+31)
   (/ 2.0 (* 2.0 (* (/ k l) (/ (/ k l) (pow t -3.0)))))
   (if (<= t 4.6e+18)
     (/ 2.0 (* (* (sin k) (tan k)) (* t (* (/ k l) (/ k l)))))
     (pow (* (/ l k) (pow t -1.5)) 2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.02e+31) {
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / pow(t, -3.0))));
	} else if (t <= 4.6e+18) {
		tmp = 2.0 / ((sin(k) * tan(k)) * (t * ((k / l) * (k / l))));
	} else {
		tmp = pow(((l / k) * 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 <= (-1.02d+31)) then
        tmp = 2.0d0 / (2.0d0 * ((k / l) * ((k / l) / (t ** (-3.0d0)))))
    else if (t <= 4.6d+18) then
        tmp = 2.0d0 / ((sin(k) * tan(k)) * (t * ((k / l) * (k / l))))
    else
        tmp = ((l / k) * (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 <= -1.02e+31) {
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / Math.pow(t, -3.0))));
	} else if (t <= 4.6e+18) {
		tmp = 2.0 / ((Math.sin(k) * Math.tan(k)) * (t * ((k / l) * (k / l))));
	} else {
		tmp = Math.pow(((l / k) * Math.pow(t, -1.5)), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -1.02e+31:
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / math.pow(t, -3.0))))
	elif t <= 4.6e+18:
		tmp = 2.0 / ((math.sin(k) * math.tan(k)) * (t * ((k / l) * (k / l))))
	else:
		tmp = math.pow(((l / k) * math.pow(t, -1.5)), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -1.02e+31)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k / l) * Float64(Float64(k / l) / (t ^ -3.0)))));
	elseif (t <= 4.6e+18)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * tan(k)) * Float64(t * Float64(Float64(k / l) * Float64(k / l)))));
	else
		tmp = Float64(Float64(l / k) * (t ^ -1.5)) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -1.02e+31)
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / (t ^ -3.0))));
	elseif (t <= 4.6e+18)
		tmp = 2.0 / ((sin(k) * tan(k)) * (t * ((k / l) * (k / l))));
	else
		tmp = ((l / k) * (t ^ -1.5)) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -1.02e+31], N[(2.0 / N[(2.0 * N[(N[(k / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.6e+18], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(l / k), $MachinePrecision] * N[Power[t, -1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.02000000000000007e31

   1. Initial program 71.9%

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

      1. *-commutative 71.9%

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

      2. associate-*l* 69.9%

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

      3. associate-*r* 69.9%

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

      4. +-commutative 69.9%

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

      5. associate-+r+ 69.9%

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

      6. metadata-eval 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in k around 0 71.6%

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

      1. associate-/l* 69.9%

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

      2. unpow2 69.9%

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

      3. unpow2 69.9%

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

      4. associate-*r/ 75.1%

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

   6. Simplified 75.1%

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

      1. times-frac 79.6%

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

      2. div-inv 79.6%

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

      3. pow-flip 79.6%

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

      4. metadata-eval 79.6%

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

   8. Applied egg-rr 79.6%

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

      1. associate-/r* 81.8%

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

   10. Simplified 81.8%

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

   if -1.02000000000000007e31 < t < 4.6e18

   1. Initial program 52.8%

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

      1. *-commutative 52.8%

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

      2. associate-*l* 52.8%

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

      3. associate-*r* 52.8%

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

      4. +-commutative 52.8%

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

      5. associate-+r+ 52.8%

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

      6. metadata-eval 52.8%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in k around inf 76.8%

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

      1. *-commutative 76.8%

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

      2. unpow2 76.8%

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

      3. times-frac 83.9%

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

      4. unpow2 83.9%

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

   6. Simplified 83.9%

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

   7. Taylor expanded in t around 0 76.8%

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

      1. associate-/l* 63.1%

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

      2. associate-/r/ 63.1%

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

      3. unpow2 63.1%

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

      4. unpow2 63.1%

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

      5. times-frac 67.0%

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

   9. Simplified 89.9%

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

   if 4.6e18 < t

   1. Initial program 47.7%

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

      1. associate-*l* 47.7%

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

      2. associate-/l/ 47.7%

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

      3. *-commutative 47.7%

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

      4. associate-*r/ 50.8%

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

      5. associate-/l* 48.3%

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

      6. associate-/r/ 44.0%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in k around 0 42.6%

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

      1. associate-/r* 42.2%

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

      2. unpow2 42.2%

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

      3. unpow2 42.2%

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

      4. times-frac 50.4%

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

   6. Simplified 50.4%

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

      1. expm1-log1p-u 50.1%

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

      2. expm1-udef 48.6%

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

      3. div-inv 48.6%

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

      4. pow2 48.6%

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

      5. pow-flip 48.7%

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

      6. metadata-eval 48.7%

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

   8. Applied egg-rr 48.7%

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

      1. expm1-def 50.3%

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

      2. expm1-log1p 50.5%

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

   10. Simplified 50.5%

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

       1. add-sqr-sqrt 50.5%

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

       2. pow2 50.5%

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

       3. sqrt-prod 50.5%

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

       4. unpow2 50.5%

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

       5. sqrt-prod 34.3%

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

       6. add-sqr-sqrt 53.0%

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

       7. sqrt-pow1 62.2%

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

       8. metadata-eval 62.2%

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

   12. Applied egg-rr 62.2%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{+31}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}\right)}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot {t}^{-1.5}\right)}^{2}\\ \end{array} \]

Alternative 5: 79.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}\right)}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{+22}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot {t}^{-1.5}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -2.1e+31)
   (/ 2.0 (* 2.0 (* (/ k l) (/ (/ k l) (pow t -3.0)))))
   (if (<= t 1.65e+22)
     (/ 2.0 (* (* (sin k) (tan k)) (/ (* t (/ k (/ l k))) l)))
     (pow (* (/ l k) (pow t -1.5)) 2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -2.1e+31) {
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / pow(t, -3.0))));
	} else if (t <= 1.65e+22) {
		tmp = 2.0 / ((sin(k) * tan(k)) * ((t * (k / (l / k))) / l));
	} else {
		tmp = pow(((l / k) * 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 <= (-2.1d+31)) then
        tmp = 2.0d0 / (2.0d0 * ((k / l) * ((k / l) / (t ** (-3.0d0)))))
    else if (t <= 1.65d+22) then
        tmp = 2.0d0 / ((sin(k) * tan(k)) * ((t * (k / (l / k))) / l))
    else
        tmp = ((l / k) * (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 <= -2.1e+31) {
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / Math.pow(t, -3.0))));
	} else if (t <= 1.65e+22) {
		tmp = 2.0 / ((Math.sin(k) * Math.tan(k)) * ((t * (k / (l / k))) / l));
	} else {
		tmp = Math.pow(((l / k) * Math.pow(t, -1.5)), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -2.1e+31:
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / math.pow(t, -3.0))))
	elif t <= 1.65e+22:
		tmp = 2.0 / ((math.sin(k) * math.tan(k)) * ((t * (k / (l / k))) / l))
	else:
		tmp = math.pow(((l / k) * math.pow(t, -1.5)), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -2.1e+31)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k / l) * Float64(Float64(k / l) / (t ^ -3.0)))));
	elseif (t <= 1.65e+22)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * tan(k)) * Float64(Float64(t * Float64(k / Float64(l / k))) / l)));
	else
		tmp = Float64(Float64(l / k) * (t ^ -1.5)) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -2.1e+31)
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / (t ^ -3.0))));
	elseif (t <= 1.65e+22)
		tmp = 2.0 / ((sin(k) * tan(k)) * ((t * (k / (l / k))) / l));
	else
		tmp = ((l / k) * (t ^ -1.5)) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -2.1e+31], N[(2.0 / N[(2.0 * N[(N[(k / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.65e+22], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(l / k), $MachinePrecision] * N[Power[t, -1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.09999999999999979e31

   1. Initial program 71.9%

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

      1. *-commutative 71.9%

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

      2. associate-*l* 69.9%

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

      3. associate-*r* 69.9%

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

      4. +-commutative 69.9%

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

      5. associate-+r+ 69.9%

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

      6. metadata-eval 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in k around 0 71.6%

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

      1. associate-/l* 69.9%

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

      2. unpow2 69.9%

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

      3. unpow2 69.9%

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

      4. associate-*r/ 75.1%

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

   6. Simplified 75.1%

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

      1. times-frac 79.6%

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

      2. div-inv 79.6%

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

      3. pow-flip 79.6%

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

      4. metadata-eval 79.6%

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

   8. Applied egg-rr 79.6%

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

      1. associate-/r* 81.8%

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

   10. Simplified 81.8%

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

   if -2.09999999999999979e31 < t < 1.6499999999999999e22

   1. Initial program 52.8%

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

      1. *-commutative 52.8%

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

      2. associate-*l* 52.8%

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

      3. associate-*r* 52.8%

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

      4. +-commutative 52.8%

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

      5. associate-+r+ 52.8%

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

      6. metadata-eval 52.8%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in k around inf 76.8%

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

      1. *-commutative 76.8%

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

      2. unpow2 76.8%

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

      3. times-frac 83.9%

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

      4. unpow2 83.9%

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

   6. Simplified 83.9%

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

      1. associate-*l/ 68.4%

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

      2. associate-/l* 68.5%

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

   8. Applied egg-rr 90.6%

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

   if 1.6499999999999999e22 < t

   1. Initial program 47.7%

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

      1. associate-*l* 47.7%

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

      2. associate-/l/ 47.7%

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

      3. *-commutative 47.7%

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

      4. associate-*r/ 50.8%

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

      5. associate-/l* 48.3%

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

      6. associate-/r/ 44.0%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in k around 0 42.6%

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

      1. associate-/r* 42.2%

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

      2. unpow2 42.2%

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

      3. unpow2 42.2%

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

      4. times-frac 50.4%

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

   6. Simplified 50.4%

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

      1. expm1-log1p-u 50.1%

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

      2. expm1-udef 48.6%

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

      3. div-inv 48.6%

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

      4. pow2 48.6%

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

      5. pow-flip 48.7%

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

      6. metadata-eval 48.7%

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

   8. Applied egg-rr 48.7%

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

      1. expm1-def 50.3%

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

      2. expm1-log1p 50.5%

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

   10. Simplified 50.5%

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

       1. add-sqr-sqrt 50.5%

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

       2. pow2 50.5%

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

       3. sqrt-prod 50.5%

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

       4. unpow2 50.5%

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

       5. sqrt-prod 34.3%

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

       6. add-sqr-sqrt 53.0%

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

       7. sqrt-pow1 62.2%

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

       8. metadata-eval 62.2%

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

   12. Applied egg-rr 62.2%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}\right)}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{+22}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot {t}^{-1.5}\right)}^{2}\\ \end{array} \]

Alternative 6: 83.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+33}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}\right)}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \tan k}}{\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot {t}^{-1.5}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -8e+33)
   (/ 2.0 (* 2.0 (* (/ k l) (/ (/ k l) (pow t -3.0)))))
   (if (<= t 5.2e+19)
     (/ (/ 2.0 (* (sin k) (tan k))) (* (/ k l) (* t (/ k l))))
     (pow (* (/ l k) (pow t -1.5)) 2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -8e+33) {
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / pow(t, -3.0))));
	} else if (t <= 5.2e+19) {
		tmp = (2.0 / (sin(k) * tan(k))) / ((k / l) * (t * (k / l)));
	} else {
		tmp = pow(((l / k) * 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 <= (-8d+33)) then
        tmp = 2.0d0 / (2.0d0 * ((k / l) * ((k / l) / (t ** (-3.0d0)))))
    else if (t <= 5.2d+19) then
        tmp = (2.0d0 / (sin(k) * tan(k))) / ((k / l) * (t * (k / l)))
    else
        tmp = ((l / k) * (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 <= -8e+33) {
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / Math.pow(t, -3.0))));
	} else if (t <= 5.2e+19) {
		tmp = (2.0 / (Math.sin(k) * Math.tan(k))) / ((k / l) * (t * (k / l)));
	} else {
		tmp = Math.pow(((l / k) * Math.pow(t, -1.5)), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -8e+33:
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / math.pow(t, -3.0))))
	elif t <= 5.2e+19:
		tmp = (2.0 / (math.sin(k) * math.tan(k))) / ((k / l) * (t * (k / l)))
	else:
		tmp = math.pow(((l / k) * math.pow(t, -1.5)), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -8e+33)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k / l) * Float64(Float64(k / l) / (t ^ -3.0)))));
	elseif (t <= 5.2e+19)
		tmp = Float64(Float64(2.0 / Float64(sin(k) * tan(k))) / Float64(Float64(k / l) * Float64(t * Float64(k / l))));
	else
		tmp = Float64(Float64(l / k) * (t ^ -1.5)) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -8e+33)
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / (t ^ -3.0))));
	elseif (t <= 5.2e+19)
		tmp = (2.0 / (sin(k) * tan(k))) / ((k / l) * (t * (k / l)));
	else
		tmp = ((l / k) * (t ^ -1.5)) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -8e+33], N[(2.0 / N[(2.0 * N[(N[(k / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.2e+19], N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / l), $MachinePrecision] * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(l / k), $MachinePrecision] * N[Power[t, -1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.9999999999999996e33

   1. Initial program 71.9%

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

      1. *-commutative 71.9%

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

      2. associate-*l* 69.9%

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

      3. associate-*r* 69.9%

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

      4. +-commutative 69.9%

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

      5. associate-+r+ 69.9%

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

      6. metadata-eval 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in k around 0 71.6%

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

      1. associate-/l* 69.9%

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

      2. unpow2 69.9%

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

      3. unpow2 69.9%

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

      4. associate-*r/ 75.1%

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

   6. Simplified 75.1%

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

      1. times-frac 79.6%

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

      2. div-inv 79.6%

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

      3. pow-flip 79.6%

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

      4. metadata-eval 79.6%

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

   8. Applied egg-rr 79.6%

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

      1. associate-/r* 81.8%

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

   10. Simplified 81.8%

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

   if -7.9999999999999996e33 < t < 5.2e19

   1. Initial program 52.8%

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

      1. *-commutative 52.8%

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

      2. associate-*l* 52.8%

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

      3. associate-*r* 52.8%

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

      4. +-commutative 52.8%

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

      5. associate-+r+ 52.8%

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

      6. metadata-eval 52.8%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in k around inf 76.8%

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

      1. *-commutative 76.8%

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

      2. unpow2 76.8%

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

      3. times-frac 83.9%

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

      4. unpow2 83.9%

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

   6. Simplified 83.9%

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

      1. *-un-lft-identity 83.9%

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

      2. associate-/l* 86.3%

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

   8. Applied egg-rr 86.3%

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

      1. expm1-log1p-u 58.5%

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

      2. expm1-udef 49.8%

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

      3. *-commutative 49.8%

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

      4. *-un-lft-identity 49.8%

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

      5. associate-/r/ 49.8%

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

   10. Applied egg-rr 49.8%

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

       1. expm1-def 58.5%

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

       2. expm1-log1p 86.3%

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

       3. associate-/r* 86.4%

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

       4. associate-/r/ 87.1%

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

       5. associate-/l/ 89.9%

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

       6. associate-/l* 85.5%

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

       7. associate-*r/ 89.8%

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

       8. unpow2 89.8%

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

   12. Simplified 89.8%

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

       1. div-inv 89.8%

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

       2. metadata-eval 89.8%

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

       3. unpow2 89.8%

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

       4. frac-times 89.8%

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

       5. clear-num 89.9%

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

       6. clear-num 89.8%

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

       7. *-commutative 89.8%

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

       8. associate-*l* 93.4%

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

   14. Applied egg-rr 93.4%

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

   if 5.2e19 < t

   1. Initial program 47.7%

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

      1. associate-*l* 47.7%

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

      2. associate-/l/ 47.7%

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

      3. *-commutative 47.7%

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

      4. associate-*r/ 50.8%

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

      5. associate-/l* 48.3%

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

      6. associate-/r/ 44.0%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in k around 0 42.6%

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

      1. associate-/r* 42.2%

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

      2. unpow2 42.2%

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

      3. unpow2 42.2%

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

      4. times-frac 50.4%

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

   6. Simplified 50.4%

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

      1. expm1-log1p-u 50.1%

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

      2. expm1-udef 48.6%

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

      3. div-inv 48.6%

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

      4. pow2 48.6%

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

      5. pow-flip 48.7%

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

      6. metadata-eval 48.7%

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

   8. Applied egg-rr 48.7%

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

      1. expm1-def 50.3%

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

      2. expm1-log1p 50.5%

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

   10. Simplified 50.5%

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

       1. add-sqr-sqrt 50.5%

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

       2. pow2 50.5%

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

       3. sqrt-prod 50.5%

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

       4. unpow2 50.5%

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

       5. sqrt-prod 34.3%

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

       6. add-sqr-sqrt 53.0%

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

       7. sqrt-pow1 62.2%

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

       8. metadata-eval 62.2%

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

   12. Applied egg-rr 62.2%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+33}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}\right)}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \tan k}}{\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot {t}^{-1.5}\right)}^{2}\\ \end{array} \]

Alternative 7: 70.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}\right)}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-112}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot {t}^{-1.5}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -5.6e-30)
   (/ 2.0 (* 2.0 (* (/ k l) (/ (/ k l) (pow t -3.0)))))
   (if (<= t 2.6e-112)
     (/ 2.0 (* (/ (* t (/ k (/ l k))) l) (* k k)))
     (pow (* (/ l k) (pow t -1.5)) 2.0))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -5.6e-30) {
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / pow(t, -3.0))));
	} else if (t <= 2.6e-112) {
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	} else {
		tmp = pow(((l / k) * 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 <= (-5.6d-30)) then
        tmp = 2.0d0 / (2.0d0 * ((k / l) * ((k / l) / (t ** (-3.0d0)))))
    else if (t <= 2.6d-112) then
        tmp = 2.0d0 / (((t * (k / (l / k))) / l) * (k * k))
    else
        tmp = ((l / k) * (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 <= -5.6e-30) {
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / Math.pow(t, -3.0))));
	} else if (t <= 2.6e-112) {
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	} else {
		tmp = Math.pow(((l / k) * Math.pow(t, -1.5)), 2.0);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -5.6e-30:
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / math.pow(t, -3.0))))
	elif t <= 2.6e-112:
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k))
	else:
		tmp = math.pow(((l / k) * math.pow(t, -1.5)), 2.0)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -5.6e-30)
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k / l) * Float64(Float64(k / l) / (t ^ -3.0)))));
	elseif (t <= 2.6e-112)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(k / Float64(l / k))) / l) * Float64(k * k)));
	else
		tmp = Float64(Float64(l / k) * (t ^ -1.5)) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -5.6e-30)
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / (t ^ -3.0))));
	elseif (t <= 2.6e-112)
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	else
		tmp = ((l / k) * (t ^ -1.5)) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -5.6e-30], N[(2.0 / N[(2.0 * N[(N[(k / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.6e-112], N[(2.0 / N[(N[(N[(t * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(l / k), $MachinePrecision] * N[Power[t, -1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.59999999999999977e-30

   1. Initial program 72.1%

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

      1. *-commutative 72.1%

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

      2. associate-*l* 70.5%

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

      3. associate-*r* 70.5%

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

      4. +-commutative 70.5%

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

      5. associate-+r+ 70.5%

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

      6. metadata-eval 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in k around 0 64.7%

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

      1. associate-/l* 63.4%

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

      2. unpow2 63.4%

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

      3. unpow2 63.4%

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

      4. associate-*r/ 67.3%

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

   6. Simplified 67.3%

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

      1. times-frac 74.8%

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

      2. div-inv 74.8%

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

      3. pow-flip 74.7%

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

      4. metadata-eval 74.7%

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

   8. Applied egg-rr 74.7%

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

      1. associate-/r* 76.5%

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

   10. Simplified 76.5%

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

   if -5.59999999999999977e-30 < t < 2.59999999999999992e-112

   1. Initial program 44.7%

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

      1. *-commutative 44.7%

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

      2. associate-*l* 44.7%

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

      3. associate-*r* 44.7%

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

      4. +-commutative 44.7%

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

      5. associate-+r+ 44.7%

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

      6. metadata-eval 44.7%

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

   3. Simplified 44.7%

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

   4. Taylor expanded in k around inf 81.6%

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

      1. *-commutative 81.6%

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

      2. unpow2 81.6%

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

      3. times-frac 90.9%

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

      4. unpow2 90.9%

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

   6. Simplified 90.9%

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

   7. Taylor expanded in k around 0 75.2%

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

      1. unpow2 75.2%

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

   9. Simplified 75.2%

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

       1. associate-*l/ 75.3%

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

       2. associate-/l* 75.4%

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

   11. Applied egg-rr 75.4%

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

   if 2.59999999999999992e-112 < t

   1. Initial program 53.7%

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

      1. associate-*l* 53.7%

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

      2. associate-/l/ 53.7%

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

      3. *-commutative 53.7%

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

      4. associate-*r/ 55.9%

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

      5. associate-/l* 54.1%

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

      6. associate-/r/ 51.0%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in k around 0 46.8%

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

      1. associate-/r* 46.6%

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

      2. unpow2 46.6%

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

      3. unpow2 46.6%

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

      4. times-frac 55.9%

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

   6. Simplified 55.9%

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

      1. expm1-log1p-u 55.6%

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

      2. expm1-udef 53.4%

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

      3. div-inv 53.4%

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

      4. pow2 53.4%

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

      5. pow-flip 53.5%

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

      6. metadata-eval 53.5%

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

   8. Applied egg-rr 53.5%

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

      1. expm1-def 55.7%

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

      2. expm1-log1p 56.0%

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

   10. Simplified 56.0%

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

       1. add-sqr-sqrt 56.0%

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

       2. pow2 56.0%

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

       3. sqrt-prod 56.0%

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

       4. unpow2 56.0%

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

       5. sqrt-prod 34.1%

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

       6. add-sqr-sqrt 56.7%

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

       7. sqrt-pow1 64.4%

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

       8. metadata-eval 64.4%

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

   12. Applied egg-rr 64.4%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}\right)}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-112}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k} \cdot {t}^{-1.5}\right)}^{2}\\ \end{array} \]

Alternative 8: 69.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-30} \lor \neg \left(t \leq 2.4 \cdot 10^{-104}\right):\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell} \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -7e-30) (not (<= t 2.4e-104)))
   (/ 2.0 (* 2.0 (* (/ k l) (/ (/ k l) (pow t -3.0)))))
   (/ 2.0 (* (/ (* t (/ k (/ l k))) l) (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -7e-30) || !(t <= 2.4e-104)) {
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / pow(t, -3.0))));
	} else {
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-7d-30)) .or. (.not. (t <= 2.4d-104))) then
        tmp = 2.0d0 / (2.0d0 * ((k / l) * ((k / l) / (t ** (-3.0d0)))))
    else
        tmp = 2.0d0 / (((t * (k / (l / k))) / l) * (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -7e-30) || !(t <= 2.4e-104)) {
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / Math.pow(t, -3.0))));
	} else {
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -7e-30) or not (t <= 2.4e-104):
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / math.pow(t, -3.0))))
	else:
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -7e-30) || !(t <= 2.4e-104))
		tmp = Float64(2.0 / Float64(2.0 * Float64(Float64(k / l) * Float64(Float64(k / l) / (t ^ -3.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(k / Float64(l / k))) / l) * Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -7e-30) || ~((t <= 2.4e-104)))
		tmp = 2.0 / (2.0 * ((k / l) * ((k / l) / (t ^ -3.0))));
	else
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -7e-30], N[Not[LessEqual[t, 2.4e-104]], $MachinePrecision]], N[(2.0 / N[(2.0 * N[(N[(k / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] / N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.0000000000000006e-30 or 2.4000000000000001e-104 < t

   1. Initial program 62.5%

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

      1. *-commutative 62.5%

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

      2. associate-*l* 60.3%

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

      3. associate-*r* 60.3%

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

      4. +-commutative 60.3%

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

      5. associate-+r+ 60.3%

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

      6. metadata-eval 60.3%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in k around 0 55.3%

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

      1. associate-/l* 54.6%

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

      2. unpow2 54.6%

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

      3. unpow2 54.6%

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

      4. associate-*r/ 57.5%

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

   6. Simplified 57.5%

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

      1. times-frac 64.5%

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

      2. div-inv 64.5%

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

      3. pow-flip 64.6%

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

      4. metadata-eval 64.6%

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

   8. Applied egg-rr 64.6%

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

      1. associate-/r* 66.2%

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

   10. Simplified 66.2%

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

   if -7.0000000000000006e-30 < t < 2.4000000000000001e-104

   1. Initial program 44.8%

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

      1. *-commutative 44.8%

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

      2. associate-*l* 44.8%

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

      3. associate-*r* 44.8%

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

      4. +-commutative 44.8%

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

      5. associate-+r+ 44.8%

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

      6. metadata-eval 44.8%

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

   3. Simplified 44.8%

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

   4. Taylor expanded in k around inf 82.0%

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

      1. *-commutative 82.0%

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

      2. unpow2 82.0%

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

      3. times-frac 91.0%

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

      4. unpow2 91.0%

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

   6. Simplified 91.0%

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

   7. Taylor expanded in k around 0 75.7%

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

      1. unpow2 75.7%

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

   9. Simplified 75.7%

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

       1. associate-*l/ 75.8%

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

       2. associate-/l* 75.9%

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

   11. Applied egg-rr 75.9%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-30} \lor \neg \left(t \leq 2.4 \cdot 10^{-104}\right):\\ \;\;\;\;\frac{2}{2 \cdot \left(\frac{k}{\ell} \cdot \frac{\frac{k}{\ell}}{{t}^{-3}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell} \cdot \left(k \cdot k\right)}\\ \end{array} \]

Alternative 9: 66.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{-120}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-104}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;{t}^{-3} \cdot \frac{-\ell}{\frac{k}{\ell} \cdot \left(-k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -5.4e-120)
   (/ (* (/ l k) (/ l k)) (pow t 3.0))
   (if (<= t 2.4e-104)
     (/ 2.0 (* (/ (* t (/ k (/ l k))) l) (* k k)))
     (* (pow t -3.0) (/ (- l) (* (/ k l) (- k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -5.4e-120) {
		tmp = ((l / k) * (l / k)) / pow(t, 3.0);
	} else if (t <= 2.4e-104) {
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	} else {
		tmp = pow(t, -3.0) * (-l / ((k / l) * -k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-5.4d-120)) then
        tmp = ((l / k) * (l / k)) / (t ** 3.0d0)
    else if (t <= 2.4d-104) then
        tmp = 2.0d0 / (((t * (k / (l / k))) / l) * (k * k))
    else
        tmp = (t ** (-3.0d0)) * (-l / ((k / l) * -k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -5.4e-120) {
		tmp = ((l / k) * (l / k)) / Math.pow(t, 3.0);
	} else if (t <= 2.4e-104) {
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	} else {
		tmp = Math.pow(t, -3.0) * (-l / ((k / l) * -k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -5.4e-120:
		tmp = ((l / k) * (l / k)) / math.pow(t, 3.0)
	elif t <= 2.4e-104:
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k))
	else:
		tmp = math.pow(t, -3.0) * (-l / ((k / l) * -k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -5.4e-120)
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) / (t ^ 3.0));
	elseif (t <= 2.4e-104)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(k / Float64(l / k))) / l) * Float64(k * k)));
	else
		tmp = Float64((t ^ -3.0) * Float64(Float64(-l) / Float64(Float64(k / l) * Float64(-k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -5.4e-120)
		tmp = ((l / k) * (l / k)) / (t ^ 3.0);
	elseif (t <= 2.4e-104)
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	else
		tmp = (t ^ -3.0) * (-l / ((k / l) * -k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -5.4e-120], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.4e-104], N[(2.0 / N[(N[(N[(t * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[t, -3.0], $MachinePrecision] * N[((-l) / N[(N[(k / l), $MachinePrecision] * (-k)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.3999999999999997e-120

   1. Initial program 72.8%

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

      1. associate-*l* 72.8%

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

      2. associate-/l/ 72.8%

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

      3. *-commutative 72.8%

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

      4. associate-*r/ 74.6%

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

      5. associate-/l* 72.7%

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

      6. associate-/r/ 71.5%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in k around 0 64.8%

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

      1. associate-/r* 65.7%

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

      2. unpow2 65.7%

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

      3. unpow2 65.7%

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

      4. times-frac 73.2%

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

   6. Simplified 73.2%

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

   if -5.3999999999999997e-120 < t < 2.4000000000000001e-104

   1. Initial program 30.2%

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

      1. *-commutative 30.2%

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

      2. associate-*l* 30.2%

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

      3. associate-*r* 30.2%

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

      4. +-commutative 30.2%

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

      5. associate-+r+ 30.2%

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

      6. metadata-eval 30.2%

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

   3. Simplified 30.2%

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

   4. Taylor expanded in k around inf 81.1%

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

      1. *-commutative 81.1%

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

      2. unpow2 81.1%

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

      3. times-frac 93.0%

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

      4. unpow2 93.0%

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

   6. Simplified 93.0%

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

   7. Taylor expanded in k around 0 77.7%

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

      1. unpow2 77.7%

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

   9. Simplified 77.7%

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

       1. associate-*l/ 77.8%

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

       2. associate-/l* 77.9%

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

   11. Applied egg-rr 77.9%

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

   if 2.4000000000000001e-104 < t

   1. Initial program 53.8%

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

      1. associate-*l* 53.8%

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

      2. associate-/l/ 53.8%

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

      3. *-commutative 53.8%

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

      4. associate-*r/ 56.0%

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

      5. associate-/l* 54.2%

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

      6. associate-/r/ 51.0%

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

   3. Simplified 53.9%

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

   4. Taylor expanded in k around 0 46.7%

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

      1. associate-/r* 46.5%

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

      2. unpow2 46.5%

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

      3. unpow2 46.5%

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

      4. times-frac 56.0%

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

   6. Simplified 56.0%

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

      1. expm1-log1p-u 55.7%

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

      2. expm1-udef 53.5%

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

      3. div-inv 53.5%

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

      4. pow2 53.5%

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

      5. pow-flip 53.6%

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

      6. metadata-eval 53.6%

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

   8. Applied egg-rr 53.6%

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

      1. expm1-def 55.8%

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

      2. expm1-log1p 56.2%

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

   10. Simplified 56.2%

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

       1. unpow2 56.2%

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

       2. clear-num 56.2%

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

       3. frac-2neg 56.2%

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

       4. frac-times 56.2%

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

       5. *-un-lft-identity 56.2%

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

   12. Applied egg-rr 56.2%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{-120}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-104}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;{t}^{-3} \cdot \frac{-\ell}{\frac{k}{\ell} \cdot \left(-k\right)}\\ \end{array} \]

Alternative 10: 67.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{-120} \lor \neg \left(t \leq 2.4 \cdot 10^{-104}\right):\\ \;\;\;\;{t}^{-3} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell} \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -5.4e-120) (not (<= t 2.4e-104)))
   (* (pow t -3.0) (* (/ l k) (/ l k)))
   (/ 2.0 (* (/ (* t (/ k (/ l k))) l) (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -5.4e-120) || !(t <= 2.4e-104)) {
		tmp = pow(t, -3.0) * ((l / k) * (l / k));
	} else {
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-5.4d-120)) .or. (.not. (t <= 2.4d-104))) then
        tmp = (t ** (-3.0d0)) * ((l / k) * (l / k))
    else
        tmp = 2.0d0 / (((t * (k / (l / k))) / l) * (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -5.4e-120) || !(t <= 2.4e-104)) {
		tmp = Math.pow(t, -3.0) * ((l / k) * (l / k));
	} else {
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -5.4e-120) or not (t <= 2.4e-104):
		tmp = math.pow(t, -3.0) * ((l / k) * (l / k))
	else:
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -5.4e-120) || !(t <= 2.4e-104))
		tmp = Float64((t ^ -3.0) * Float64(Float64(l / k) * Float64(l / k)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(k / Float64(l / k))) / l) * Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -5.4e-120) || ~((t <= 2.4e-104)))
		tmp = (t ^ -3.0) * ((l / k) * (l / k));
	else
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -5.4e-120], N[Not[LessEqual[t, 2.4e-104]], $MachinePrecision]], N[(N[Power[t, -3.0], $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.3999999999999997e-120 or 2.4000000000000001e-104 < t

   1. Initial program 64.4%

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

      1. associate-*l* 64.4%

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

      2. associate-/l/ 64.4%

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

      3. *-commutative 64.4%

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

      4. associate-*r/ 66.4%

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

      5. associate-/l* 64.6%

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

      6. associate-/r/ 62.5%

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

   3. Simplified 65.8%

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

   4. Taylor expanded in k around 0 56.9%

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

      1. associate-/r* 57.2%

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

      2. unpow2 57.2%

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

      3. unpow2 57.2%

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

      4. times-frac 65.7%

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

   6. Simplified 65.7%

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

      1. expm1-log1p-u 54.4%

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

      2. expm1-udef 51.5%

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

      3. div-inv 51.5%

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

      4. pow2 51.5%

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

      5. pow-flip 51.5%

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

      6. metadata-eval 51.5%

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

   8. Applied egg-rr 51.5%

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

      1. expm1-def 54.5%

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

      2. expm1-log1p 65.7%

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

   10. Simplified 65.7%

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

       1. unpow2 65.7%

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

   12. Applied egg-rr 65.7%

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

   if -5.3999999999999997e-120 < t < 2.4000000000000001e-104

   1. Initial program 30.2%

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

      1. *-commutative 30.2%

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

      2. associate-*l* 30.2%

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

      3. associate-*r* 30.2%

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

      4. +-commutative 30.2%

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

      5. associate-+r+ 30.2%

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

      6. metadata-eval 30.2%

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

   3. Simplified 30.2%

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

   4. Taylor expanded in k around inf 81.1%

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

      1. *-commutative 81.1%

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

      2. unpow2 81.1%

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

      3. times-frac 93.0%

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

      4. unpow2 93.0%

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

   6. Simplified 93.0%

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

   7. Taylor expanded in k around 0 77.7%

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

      1. unpow2 77.7%

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

   9. Simplified 77.7%

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

       1. associate-*l/ 77.8%

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

       2. associate-/l* 77.9%

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

   11. Applied egg-rr 77.9%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{-120} \lor \neg \left(t \leq 2.4 \cdot 10^{-104}\right):\\ \;\;\;\;{t}^{-3} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell} \cdot \left(k \cdot k\right)}\\ \end{array} \]

Alternative 11: 67.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ \mathbf{if}\;t \leq -5.4 \cdot 10^{-120}:\\ \;\;\;\;\frac{t_1}{{t}^{3}}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-104}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;{t}^{-3} \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ l k) (/ l k))))
   (if (<= t -5.4e-120)
     (/ t_1 (pow t 3.0))
     (if (<= t 2.4e-104)
       (/ 2.0 (* (/ (* t (/ k (/ l k))) l) (* k k)))
       (* (pow t -3.0) t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = (l / k) * (l / k);
	double tmp;
	if (t <= -5.4e-120) {
		tmp = t_1 / pow(t, 3.0);
	} else if (t <= 2.4e-104) {
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	} else {
		tmp = pow(t, -3.0) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (l / k) * (l / k)
    if (t <= (-5.4d-120)) then
        tmp = t_1 / (t ** 3.0d0)
    else if (t <= 2.4d-104) then
        tmp = 2.0d0 / (((t * (k / (l / k))) / l) * (k * k))
    else
        tmp = (t ** (-3.0d0)) * t_1
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (l / k) * (l / k);
	double tmp;
	if (t <= -5.4e-120) {
		tmp = t_1 / Math.pow(t, 3.0);
	} else if (t <= 2.4e-104) {
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	} else {
		tmp = Math.pow(t, -3.0) * t_1;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (l / k) * (l / k)
	tmp = 0
	if t <= -5.4e-120:
		tmp = t_1 / math.pow(t, 3.0)
	elif t <= 2.4e-104:
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k))
	else:
		tmp = math.pow(t, -3.0) * t_1
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(l / k) * Float64(l / k))
	tmp = 0.0
	if (t <= -5.4e-120)
		tmp = Float64(t_1 / (t ^ 3.0));
	elseif (t <= 2.4e-104)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(k / Float64(l / k))) / l) * Float64(k * k)));
	else
		tmp = Float64((t ^ -3.0) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (l / k) * (l / k);
	tmp = 0.0;
	if (t <= -5.4e-120)
		tmp = t_1 / (t ^ 3.0);
	elseif (t <= 2.4e-104)
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	else
		tmp = (t ^ -3.0) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.4e-120], N[(t$95$1 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.4e-104], N[(2.0 / N[(N[(N[(t * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[t, -3.0], $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.3999999999999997e-120

   1. Initial program 72.8%

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

      1. associate-*l* 72.8%

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

      2. associate-/l/ 72.8%

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

      3. *-commutative 72.8%

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

      4. associate-*r/ 74.6%

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

      5. associate-/l* 72.7%

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

      6. associate-/r/ 71.5%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in k around 0 64.8%

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

      1. associate-/r* 65.7%

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

      2. unpow2 65.7%

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

      3. unpow2 65.7%

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

      4. times-frac 73.2%

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

   6. Simplified 73.2%

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

   if -5.3999999999999997e-120 < t < 2.4000000000000001e-104

   1. Initial program 30.2%

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

      1. *-commutative 30.2%

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

      2. associate-*l* 30.2%

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

      3. associate-*r* 30.2%

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

      4. +-commutative 30.2%

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

      5. associate-+r+ 30.2%

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

      6. metadata-eval 30.2%

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

   3. Simplified 30.2%

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

   4. Taylor expanded in k around inf 81.1%

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

      1. *-commutative 81.1%

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

      2. unpow2 81.1%

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

      3. times-frac 93.0%

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

      4. unpow2 93.0%

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

   6. Simplified 93.0%

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

   7. Taylor expanded in k around 0 77.7%

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

      1. unpow2 77.7%

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

   9. Simplified 77.7%

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

       1. associate-*l/ 77.8%

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

       2. associate-/l* 77.9%

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

   11. Applied egg-rr 77.9%

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

   if 2.4000000000000001e-104 < t

   1. Initial program 53.8%

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

      1. associate-*l* 53.8%

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

      2. associate-/l/ 53.8%

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

      3. *-commutative 53.8%

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

      4. associate-*r/ 56.0%

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

      5. associate-/l* 54.2%

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

      6. associate-/r/ 51.0%

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

   3. Simplified 53.9%

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

   4. Taylor expanded in k around 0 46.7%

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

      1. associate-/r* 46.5%

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

      2. unpow2 46.5%

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

      3. unpow2 46.5%

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

      4. times-frac 56.0%

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

   6. Simplified 56.0%

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

      1. expm1-log1p-u 55.7%

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

      2. expm1-udef 53.5%

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

      3. div-inv 53.5%

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

      4. pow2 53.5%

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

      5. pow-flip 53.6%

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

      6. metadata-eval 53.6%

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

   8. Applied egg-rr 53.6%

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

      1. expm1-def 55.8%

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

      2. expm1-log1p 56.2%

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

   10. Simplified 56.2%

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

       1. unpow2 56.2%

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

   12. Applied egg-rr 56.2%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{-120}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-104}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;{t}^{-3} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \end{array} \]

Alternative 12: 59.2% accurate, 24.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(t, l, k):
	return 2.0 / ((k * k) * ((t / l) * ((k * k) * (1.0 / l))))

Julia

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

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / ((k * k) * ((t / l) * ((k * k) * (1.0 / 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] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.0%

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

   1. *-commutative 56.0%

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

   2. associate-*l* 54.6%

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

   3. associate-*r* 54.6%

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

   4. +-commutative 54.6%

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

   5. associate-+r+ 54.6%

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

   6. metadata-eval 54.6%

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

3. Simplified 54.6%

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

4. Taylor expanded in k around inf 64.3%

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

   1. *-commutative 64.3%

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

   2. unpow2 64.3%

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

   3. times-frac 68.5%

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

   4. unpow2 68.5%

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

6. Simplified 68.5%

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

7. Taylor expanded in k around 0 60.7%

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

   1. unpow2 60.7%

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

9. Simplified 60.7%

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

    1. div-inv 60.7%

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

11. Applied egg-rr 60.7%

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

12. Final simplification 60.7%

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

Alternative 13: 57.8% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

   1. *-commutative 56.0%

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

   2. associate-*l* 54.6%

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

   3. associate-*r* 54.6%

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

   4. +-commutative 54.6%

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

   5. associate-+r+ 54.6%

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

   6. metadata-eval 54.6%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

3. Simplified 54.6%

   \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

4. Taylor expanded in k around inf 64.3%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
5. Step-by-step derivation

   1. *-commutative 64.3%

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   2. unpow2 64.3%

      \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

   3. times-frac 68.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   4. unpow2 68.5%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

6. Simplified 68.5%

   \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

7. Taylor expanded in k around 0 60.7%

   \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \color{blue}{{k}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 60.7%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

9. Simplified 60.7%

   \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

10. Taylor expanded in t around 0 57.4%

    \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(k \cdot k\right)} \]
11. Step-by-step derivation

    1. associate-/l* 58.6%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{t}}} \cdot \left(k \cdot k\right)} \]

    2. associate-/r/ 57.1%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot t\right)} \cdot \left(k \cdot k\right)} \]

    3. unpow2 57.1%

       \[\leadsto \frac{2}{\left(\frac{\color{blue}{k \cdot k}}{{\ell}^{2}} \cdot t\right) \cdot \left(k \cdot k\right)} \]

    4. unpow2 57.1%

       \[\leadsto \frac{2}{\left(\frac{k \cdot k}{\color{blue}{\ell \cdot \ell}} \cdot t\right) \cdot \left(k \cdot k\right)} \]

    5. times-frac 58.1%

       \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot t\right) \cdot \left(k \cdot k\right)} \]

12. Simplified 58.1%

    \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t\right)} \cdot \left(k \cdot k\right)} \]

13. Final simplification 58.1%

    \[\leadsto \frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \left(k \cdot k\right)} \]

Alternative 14: 59.2% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)} \end{array} \]

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(k * Float64(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[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 56.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

   2. associate-*l* 54.6%

      \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   3. associate-*r* 54.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. +-commutative 54.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. associate-+r+ 54.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. metadata-eval 54.6%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

3. Simplified 54.6%

   \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

4. Taylor expanded in k around inf 64.3%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
5. Step-by-step derivation

   1. *-commutative 64.3%

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   2. unpow2 64.3%

      \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

   3. times-frac 68.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   4. unpow2 68.5%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

6. Simplified 68.5%

   \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

7. Taylor expanded in k around 0 60.7%

   \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \color{blue}{{k}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 60.7%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

9. Simplified 60.7%

   \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

10. Taylor expanded in k around 0 60.7%

    \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{{k}^{2}}{\ell}}\right) \cdot \left(k \cdot k\right)} \]
11. Step-by-step derivation

    1. unpow2 60.7%

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(k \cdot k\right)} \]

    2. associate-*l/ 60.7%

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(\frac{k}{\ell} \cdot k\right)}\right) \cdot \left(k \cdot k\right)} \]

    3. *-commutative 60.7%

       \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(k \cdot \frac{k}{\ell}\right)}\right) \cdot \left(k \cdot k\right)} \]

12. Simplified 60.7%

    \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \color{blue}{\left(k \cdot \frac{k}{\ell}\right)}\right) \cdot \left(k \cdot k\right)} \]

13. Final simplification 60.7%

    \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)} \]

Alternative 15: 59.2% 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 56.0%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 56.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

   2. associate-*l* 54.6%

      \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   3. associate-*r* 54.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. +-commutative 54.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. associate-+r+ 54.6%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. metadata-eval 54.6%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

3. Simplified 54.6%

   \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

4. Taylor expanded in k around inf 64.3%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
5. Step-by-step derivation

   1. *-commutative 64.3%

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   2. unpow2 64.3%

      \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

   3. times-frac 68.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   4. unpow2 68.5%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

6. Simplified 68.5%

   \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

7. Taylor expanded in k around 0 60.7%

   \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \color{blue}{{k}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 60.7%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

9. Simplified 60.7%

   \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

10. Final simplification 60.7%

    \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \]

Reproduce

herbie shell --seed 2023189 
(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))))