Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.2% → 86.8%

Time: 21.7s

Alternatives: 16

Speedup: 28.1×

• 27.5% 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: 86.8% accurate, 0.6× speedup

Math

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

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)
     (/ (/ 2.0 (* (/ (sin k) (* (/ l t) (/ l t))) (* t (tan k)))) (+ 2.0 t_1))
     (* 2.0 (* (/ l k) (* (/ l k) (/ (/ (cos k) t) (pow (sin k) 2.0))))))))

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 = (2.0 / ((sin(k) / ((l / t) * (l / t))) * (t * tan(k)))) / (2.0 + t_1);
	} else {
		tmp = 2.0 * ((l / k) * ((l / k) * ((cos(k) / t) / pow(sin(k), 2.0))));
	}
	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 = (2.0 / ((Math.sin(k) / ((l / t) * (l / t))) * (t * Math.tan(k)))) / (2.0 + t_1);
	} else {
		tmp = 2.0 * ((l / k) * ((l / k) * ((Math.cos(k) / t) / Math.pow(Math.sin(k), 2.0))));
	}
	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 = (2.0 / ((math.sin(k) / ((l / t) * (l / t))) * (t * math.tan(k)))) / (2.0 + t_1)
	else:
		tmp = 2.0 * ((l / k) * ((l / k) * ((math.cos(k) / t) / math.pow(math.sin(k), 2.0))))
	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(Float64(2.0 / Float64(Float64(sin(k) / Float64(Float64(l / t) * Float64(l / t))) * Float64(t * tan(k)))) / Float64(2.0 + t_1));
	else
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(Float64(cos(k) / t) / (sin(k) ^ 2.0)))));
	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 = (2.0 / ((sin(k) / ((l / t) * (l / t))) * (t * tan(k)))) / (2.0 + t_1);
	else
		tmp = 2.0 * ((l / k) * ((l / k) * ((cos(k) / t) / (sin(k) ^ 2.0))));
	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[(N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] / N[(N[(l / t), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $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 81.8%

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

      1. associate-/r* 82.0%

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

      2. associate-*l* 72.9%

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

      3. sqr-neg 72.9%

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

      4. associate-*l* 82.0%

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

      5. *-commutative 82.0%

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

      6. sqr-neg 82.0%

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

      7. associate-/r* 82.0%

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

   3. Simplified 81.8%

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

      1. cube-mult 81.8%

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

      2. *-un-lft-identity 81.8%

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

      3. times-frac 83.5%

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

      4. associate-/l* 86.4%

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

   5. Applied egg-rr 86.4%

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

   6. Taylor expanded in t around 0 83.6%

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

      1. unpow2 83.6%

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

      2. times-frac 88.2%

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

      3. unpow2 88.2%

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

      4. associate-/l* 90.9%

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

   8. Simplified 90.9%

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

      1. div-inv 90.9%

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

      2. /-rgt-identity 90.9%

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

   10. Applied egg-rr 90.9%

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

       1. associate-*r/ 90.9%

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

       2. *-rgt-identity 90.9%

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

       3. associate-/l/ 90.9%

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

       4. *-commutative 90.9%

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

       5. associate-*l* 90.9%

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

       6. times-frac 91.5%

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

       7. *-commutative 91.5%

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

       8. associate-*l/ 89.1%

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

       9. unpow2 89.1%

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

       10. associate-/l* 87.5%

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

       11. unpow2 87.5%

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

       12. associate-*l/ 89.8%

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

       13. associate-*r/ 91.5%

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

   12. Simplified 91.5%

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

   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. associate-/r* 0.0%

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

      2. associate-*l* 0.0%

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

      3. sqr-neg 0.0%

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

      4. associate-*l* 0.0%

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

      5. *-commutative 0.0%

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

      6. sqr-neg 0.0%

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

      7. associate-*l/ 0.0%

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

      8. associate-*r/ 0.0%

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

      9. associate-/r/ 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in k around inf 51.8%

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

      1. times-frac 50.1%

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

      2. unpow2 50.1%

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

      3. unpow2 50.1%

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

   6. Simplified 50.1%

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

      1. *-un-lft-identity 50.1%

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

      2. associate-/r* 54.2%

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

   8. Applied egg-rr 54.2%

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

   9. Taylor expanded in l around 0 51.8%

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

       1. times-frac 50.1%

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

       2. unpow2 50.1%

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

       3. unpow2 50.1%

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

       4. times-frac 70.9%

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

       5. associate-*l* 82.2%

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

       6. associate-/r* 82.3%

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

   11. Simplified 82.3%

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

4. Final simplification 88.4%

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

Alternative 2: 88.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-45} \lor \neg \left(t \leq 4500000000000\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(t \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \frac{\sin k}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.3e-45) (not (<= t 4500000000000.0)))
   (/
    2.0
    (*
     (tan k)
     (* (+ 2.0 (pow (/ k t) 2.0)) (* t (* (* t (/ t l)) (/ (sin k) l))))))
   (* 2.0 (* (/ l k) (* (/ l k) (/ (/ (cos k) t) (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.3e-45) || !(t <= 4500000000000.0)) {
		tmp = 2.0 / (tan(k) * ((2.0 + pow((k / t), 2.0)) * (t * ((t * (t / l)) * (sin(k) / l)))));
	} else {
		tmp = 2.0 * ((l / k) * ((l / k) * ((cos(k) / t) / pow(sin(k), 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 <= (-1.3d-45)) .or. (.not. (t <= 4500000000000.0d0))) then
        tmp = 2.0d0 / (tan(k) * ((2.0d0 + ((k / t) ** 2.0d0)) * (t * ((t * (t / l)) * (sin(k) / l)))))
    else
        tmp = 2.0d0 * ((l / k) * ((l / k) * ((cos(k) / t) / (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.3e-45) || !(t <= 4500000000000.0)) {
		tmp = 2.0 / (Math.tan(k) * ((2.0 + Math.pow((k / t), 2.0)) * (t * ((t * (t / l)) * (Math.sin(k) / l)))));
	} else {
		tmp = 2.0 * ((l / k) * ((l / k) * ((Math.cos(k) / t) / Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -1.3e-45) or not (t <= 4500000000000.0):
		tmp = 2.0 / (math.tan(k) * ((2.0 + math.pow((k / t), 2.0)) * (t * ((t * (t / l)) * (math.sin(k) / l)))))
	else:
		tmp = 2.0 * ((l / k) * ((l / k) * ((math.cos(k) / t) / math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.3e-45) || !(t <= 4500000000000.0))
		tmp = Float64(2.0 / Float64(tan(k) * Float64(Float64(2.0 + (Float64(k / t) ^ 2.0)) * Float64(t * Float64(Float64(t * Float64(t / l)) * Float64(sin(k) / l))))));
	else
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(Float64(cos(k) / t) / (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.3e-45) || ~((t <= 4500000000000.0)))
		tmp = 2.0 / (tan(k) * ((2.0 + ((k / t) ^ 2.0)) * (t * ((t * (t / l)) * (sin(k) / l)))));
	else
		tmp = 2.0 * ((l / k) * ((l / k) * ((cos(k) / t) / (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -1.3e-45], N[Not[LessEqual[t, 4500000000000.0]], $MachinePrecision]], N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t * N[(N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.29999999999999993e-45 or 4.5e12 < t

   1. Initial program 66.4%

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

      1. associate-/r* 66.4%

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

      2. associate-*l* 53.7%

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

      3. sqr-neg 53.7%

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

      4. associate-*l* 66.4%

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

      5. *-commutative 66.4%

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

      6. sqr-neg 66.4%

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

      7. associate-/r* 66.4%

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

   3. Simplified 66.1%

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

      1. cube-mult 66.1%

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

      2. *-un-lft-identity 66.1%

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

      3. times-frac 69.4%

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

      4. associate-/l* 70.2%

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

   5. Applied egg-rr 70.2%

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

   6. Taylor expanded in t around 0 69.6%

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

      1. unpow2 69.6%

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

      2. times-frac 73.2%

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

      3. unpow2 73.2%

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

      4. associate-/l* 82.0%

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

   8. Simplified 82.0%

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

      1. *-un-lft-identity 82.0%

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

      2. associate-/l/ 82.0%

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

      3. /-rgt-identity 82.0%

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

   10. Applied egg-rr 82.0%

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

       1. *-lft-identity 82.0%

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

       2. associate-/l/ 82.1%

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

       3. associate-/r/ 82.0%

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

   12. Simplified 82.0%

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

   if -1.29999999999999993e-45 < t < 4.5e12

   1. Initial program 42.9%

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

      1. associate-/r* 43.1%

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

      2. associate-*l* 43.1%

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

      3. sqr-neg 43.1%

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

      4. associate-*l* 43.1%

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

      5. *-commutative 43.1%

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

      6. sqr-neg 43.1%

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

      7. associate-*l/ 43.1%

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

      8. associate-*r/ 43.1%

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

      9. associate-/r/ 43.1%

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

   3. Simplified 43.1%

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

   4. Taylor expanded in k around inf 73.2%

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

      1. times-frac 71.4%

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

      2. unpow2 71.4%

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

      3. unpow2 71.4%

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

   6. Simplified 71.4%

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

      1. *-un-lft-identity 71.4%

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

      2. associate-/r* 75.2%

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

   8. Applied egg-rr 75.2%

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

   9. Taylor expanded in l around 0 73.2%

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

       1. times-frac 71.4%

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

       2. unpow2 71.4%

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

       3. unpow2 71.4%

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

       4. times-frac 83.9%

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

       5. associate-*l* 92.7%

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

       6. associate-/r* 92.7%

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

   11. Simplified 92.7%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-45} \lor \neg \left(t \leq 4500000000000\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(t \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \frac{\sin k}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right)\\ \end{array} \]

Alternative 3: 74.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.35 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{t \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{k}{\ell}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 2.35e-20)
   (/
    (/ (/ 2.0 (tan k)) (* t (* (/ t (/ l t)) (/ k l))))
    (+ 2.0 (pow (/ k t) 2.0)))
   (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.35e-20) {
		tmp = ((2.0 / tan(k)) / (t * ((t / (l / t)) * (k / l)))) / (2.0 + pow((k / t), 2.0));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * pow(sin(k), 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 (k <= 2.35d-20) then
        tmp = ((2.0d0 / tan(k)) / (t * ((t / (l / t)) * (k / l)))) / (2.0d0 + ((k / t) ** 2.0d0))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.35e-20) {
		tmp = ((2.0 / Math.tan(k)) / (t * ((t / (l / t)) * (k / l)))) / (2.0 + Math.pow((k / t), 2.0));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 2.35e-20:
		tmp = ((2.0 / math.tan(k)) / (t * ((t / (l / t)) * (k / l)))) / (2.0 + math.pow((k / t), 2.0))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 2.35e-20)
		tmp = Float64(Float64(Float64(2.0 / tan(k)) / Float64(t * Float64(Float64(t / Float64(l / t)) * Float64(k / l)))) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.35e-20)
		tmp = ((2.0 / tan(k)) / (t * ((t / (l / t)) * (k / l)))) / (2.0 + ((k / t) ^ 2.0));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 2.35e-20], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(t * N[(N[(t / N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.35000000000000007e-20

   1. Initial program 59.4%

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

      1. associate-/r* 59.5%

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

      2. associate-*l* 51.1%

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

      3. sqr-neg 51.1%

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

      4. associate-*l* 59.5%

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

      5. *-commutative 59.5%

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

      6. sqr-neg 59.5%

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

      7. associate-/r* 59.5%

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

   3. Simplified 59.4%

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

      1. cube-mult 59.4%

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

      2. *-un-lft-identity 59.4%

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

      3. times-frac 62.3%

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

      4. associate-/l* 66.0%

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

   5. Applied egg-rr 66.0%

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

   6. Taylor expanded in k around 0 60.7%

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

      1. *-commutative 60.7%

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

      2. unpow2 60.7%

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

      3. times-frac 67.7%

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

      4. unpow2 67.7%

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

      5. associate-/l* 71.6%

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

   8. Simplified 71.6%

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

   if 2.35000000000000007e-20 < k

   1. Initial program 40.8%

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

      1. associate-/r* 40.8%

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

      2. associate-*l* 40.8%

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

      3. sqr-neg 40.8%

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

      4. associate-*l* 40.8%

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

      5. *-commutative 40.8%

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

      6. sqr-neg 40.8%

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

      7. associate-*l/ 40.8%

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

      8. associate-*r/ 40.8%

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

      9. associate-/r/ 40.8%

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

   3. Simplified 40.8%

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

   4. Taylor expanded in k around inf 68.2%

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

      1. times-frac 66.1%

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

      2. unpow2 66.1%

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

      3. unpow2 66.1%

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

   6. Simplified 66.1%

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

      1. times-frac 83.5%

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

   8. Applied egg-rr 83.5%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.35 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{t \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{k}{\ell}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 4: 76.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{t \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{k}{\ell}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 4.2e-21)
   (/
    (/ (/ 2.0 (tan k)) (* t (* (/ t (/ l t)) (/ k l))))
    (+ 2.0 (pow (/ k t) 2.0)))
   (* 2.0 (* (/ l k) (* (/ l k) (/ (/ (cos k) t) (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.2e-21) {
		tmp = ((2.0 / tan(k)) / (t * ((t / (l / t)) * (k / l)))) / (2.0 + pow((k / t), 2.0));
	} else {
		tmp = 2.0 * ((l / k) * ((l / k) * ((cos(k) / t) / pow(sin(k), 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 (k <= 4.2d-21) then
        tmp = ((2.0d0 / tan(k)) / (t * ((t / (l / t)) * (k / l)))) / (2.0d0 + ((k / t) ** 2.0d0))
    else
        tmp = 2.0d0 * ((l / k) * ((l / k) * ((cos(k) / t) / (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.2e-21) {
		tmp = ((2.0 / Math.tan(k)) / (t * ((t / (l / t)) * (k / l)))) / (2.0 + Math.pow((k / t), 2.0));
	} else {
		tmp = 2.0 * ((l / k) * ((l / k) * ((Math.cos(k) / t) / Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 4.2e-21:
		tmp = ((2.0 / math.tan(k)) / (t * ((t / (l / t)) * (k / l)))) / (2.0 + math.pow((k / t), 2.0))
	else:
		tmp = 2.0 * ((l / k) * ((l / k) * ((math.cos(k) / t) / math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 4.2e-21)
		tmp = Float64(Float64(Float64(2.0 / tan(k)) / Float64(t * Float64(Float64(t / Float64(l / t)) * Float64(k / l)))) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(Float64(cos(k) / t) / (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4.2e-21)
		tmp = ((2.0 / tan(k)) / (t * ((t / (l / t)) * (k / l)))) / (2.0 + ((k / t) ^ 2.0));
	else
		tmp = 2.0 * ((l / k) * ((l / k) * ((cos(k) / t) / (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 4.2e-21], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(t * N[(N[(t / N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.20000000000000025e-21

   1. Initial program 59.4%

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

      1. associate-/r* 59.5%

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

      2. associate-*l* 51.1%

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

      3. sqr-neg 51.1%

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

      4. associate-*l* 59.5%

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

      5. *-commutative 59.5%

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

      6. sqr-neg 59.5%

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

      7. associate-/r* 59.5%

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

   3. Simplified 59.4%

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

      1. cube-mult 59.4%

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

      2. *-un-lft-identity 59.4%

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

      3. times-frac 62.3%

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

      4. associate-/l* 66.0%

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

   5. Applied egg-rr 66.0%

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

   6. Taylor expanded in k around 0 60.7%

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

      1. *-commutative 60.7%

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

      2. unpow2 60.7%

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

      3. times-frac 67.7%

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

      4. unpow2 67.7%

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

      5. associate-/l* 71.6%

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

   8. Simplified 71.6%

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

   if 4.20000000000000025e-21 < k

   1. Initial program 40.8%

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

      1. associate-/r* 40.8%

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

      2. associate-*l* 40.8%

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

      3. sqr-neg 40.8%

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

      4. associate-*l* 40.8%

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

      5. *-commutative 40.8%

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

      6. sqr-neg 40.8%

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

      7. associate-*l/ 40.8%

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

      8. associate-*r/ 40.8%

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

      9. associate-/r/ 40.8%

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

   3. Simplified 40.8%

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

   4. Taylor expanded in k around inf 68.2%

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

      1. times-frac 66.1%

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

      2. unpow2 66.1%

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

      3. unpow2 66.1%

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

   6. Simplified 66.1%

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

      1. *-un-lft-identity 66.1%

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

      2. associate-/r* 71.9%

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

   8. Applied egg-rr 71.9%

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

   9. Taylor expanded in l around 0 68.2%

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

       1. times-frac 66.1%

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

       2. unpow2 66.1%

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

       3. unpow2 66.1%

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

       4. times-frac 83.5%

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

       5. associate-*l* 88.1%

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

       6. associate-/r* 88.2%

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

   11. Simplified 88.2%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{t \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{k}{\ell}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right)\\ \end{array} \]

Alternative 5: 72.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-153} \lor \neg \left(t \leq 1.05 \cdot 10^{-143}\right):\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{t \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \frac{k}{\ell}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -3.2e-153) (not (<= t 1.05e-143)))
   (/
    (/ (/ 2.0 (tan k)) (* t (* (* t (/ t l)) (/ k l))))
    (+ 2.0 (pow (/ k t) 2.0)))
   (* 2.0 (* (/ l t) (/ l (pow k 4.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.2e-153) || !(t <= 1.05e-143)) {
		tmp = ((2.0 / tan(k)) / (t * ((t * (t / l)) * (k / l)))) / (2.0 + pow((k / t), 2.0));
	} else {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-3.2d-153)) .or. (.not. (t <= 1.05d-143))) then
        tmp = ((2.0d0 / tan(k)) / (t * ((t * (t / l)) * (k / l)))) / (2.0d0 + ((k / t) ** 2.0d0))
    else
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3.2e-153) || !(t <= 1.05e-143)) {
		tmp = ((2.0 / Math.tan(k)) / (t * ((t * (t / l)) * (k / l)))) / (2.0 + Math.pow((k / t), 2.0));
	} else {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -3.2e-153) or not (t <= 1.05e-143):
		tmp = ((2.0 / math.tan(k)) / (t * ((t * (t / l)) * (k / l)))) / (2.0 + math.pow((k / t), 2.0))
	else:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -3.2e-153) || !(t <= 1.05e-143))
		tmp = Float64(Float64(Float64(2.0 / tan(k)) / Float64(t * Float64(Float64(t * Float64(t / l)) * Float64(k / l)))) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -3.2e-153) || ~((t <= 1.05e-143)))
		tmp = ((2.0 / tan(k)) / (t * ((t * (t / l)) * (k / l)))) / (2.0 + ((k / t) ^ 2.0));
	else
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -3.2e-153], N[Not[LessEqual[t, 1.05e-143]], $MachinePrecision]], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(t * N[(N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.1999999999999999e-153 or 1.0500000000000001e-143 < t

   1. Initial program 62.8%

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

      1. associate-/r* 63.0%

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

      2. associate-*l* 54.7%

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

      3. sqr-neg 54.7%

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

      4. associate-*l* 63.0%

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

      5. *-commutative 63.0%

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

      6. sqr-neg 63.0%

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

      7. associate-/r* 63.0%

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

   3. Simplified 62.8%

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

      1. cube-mult 62.8%

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

      2. *-un-lft-identity 62.8%

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

      3. times-frac 66.8%

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

      4. associate-/l* 70.0%

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

   5. Applied egg-rr 70.0%

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

   6. Taylor expanded in t around 0 66.9%

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

      1. unpow2 66.9%

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

      2. times-frac 75.0%

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

      3. unpow2 75.0%

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

      4. associate-/l* 80.7%

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

   8. Simplified 80.7%

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

   9. Taylor expanded in k around 0 63.2%

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

       1. *-commutative 63.2%

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

       2. unpow2 63.2%

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

       3. times-frac 70.3%

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

       4. unpow2 70.3%

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

       5. associate-*l/ 73.1%

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

   11. Simplified 73.1%

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

   if -3.1999999999999999e-153 < t < 1.0500000000000001e-143

   1. Initial program 31.0%

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

      1. associate-/r* 31.0%

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

      2. associate-*l* 31.0%

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

      3. sqr-neg 31.0%

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

      4. associate-*l* 31.0%

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

      5. *-commutative 31.0%

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

      6. sqr-neg 31.0%

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

      7. associate-*l/ 31.0%

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

      8. associate-*r/ 31.0%

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

      9. associate-/r/ 31.0%

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

   3. Simplified 31.0%

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

   4. Taylor expanded in k around inf 80.7%

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

      1. times-frac 78.6%

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

      2. unpow2 78.6%

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

      3. unpow2 78.6%

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

   6. Simplified 78.6%

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

   7. Taylor expanded in k around 0 59.5%

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

      1. unpow2 59.5%

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

      2. *-commutative 59.5%

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

      3. times-frac 66.3%

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

   9. Simplified 66.3%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-153} \lor \neg \left(t \leq 1.05 \cdot 10^{-143}\right):\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{t \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \frac{k}{\ell}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]

Alternative 6: 72.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{2}{\tan k}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{t_2}{t \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{k}{\ell}\right)}}{t_1}\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{-144}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_2}{t \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \frac{k}{\ell}\right)}}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 2.0 (pow (/ k t) 2.0))) (t_2 (/ 2.0 (tan k))))
   (if (<= t -5.2e-152)
     (/ (/ t_2 (* t (* (/ t (/ l t)) (/ k l)))) t_1)
     (if (<= t 4.9e-144)
       (* 2.0 (* (/ l t) (/ l (pow k 4.0))))
       (/ (/ t_2 (* t (* (* t (/ t l)) (/ k l)))) t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 + pow((k / t), 2.0);
	double t_2 = 2.0 / tan(k);
	double tmp;
	if (t <= -5.2e-152) {
		tmp = (t_2 / (t * ((t / (l / t)) * (k / l)))) / t_1;
	} else if (t <= 4.9e-144) {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	} else {
		tmp = (t_2 / (t * ((t * (t / l)) * (k / l)))) / 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) :: t_2
    real(8) :: tmp
    t_1 = 2.0d0 + ((k / t) ** 2.0d0)
    t_2 = 2.0d0 / tan(k)
    if (t <= (-5.2d-152)) then
        tmp = (t_2 / (t * ((t / (l / t)) * (k / l)))) / t_1
    else if (t <= 4.9d-144) then
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    else
        tmp = (t_2 / (t * ((t * (t / l)) * (k / l)))) / t_1
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 + Math.pow((k / t), 2.0);
	double t_2 = 2.0 / Math.tan(k);
	double tmp;
	if (t <= -5.2e-152) {
		tmp = (t_2 / (t * ((t / (l / t)) * (k / l)))) / t_1;
	} else if (t <= 4.9e-144) {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	} else {
		tmp = (t_2 / (t * ((t * (t / l)) * (k / l)))) / t_1;
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = 2.0 + math.pow((k / t), 2.0)
	t_2 = 2.0 / math.tan(k)
	tmp = 0
	if t <= -5.2e-152:
		tmp = (t_2 / (t * ((t / (l / t)) * (k / l)))) / t_1
	elif t <= 4.9e-144:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	else:
		tmp = (t_2 / (t * ((t * (t / l)) * (k / l)))) / t_1
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_2 = Float64(2.0 / tan(k))
	tmp = 0.0
	if (t <= -5.2e-152)
		tmp = Float64(Float64(t_2 / Float64(t * Float64(Float64(t / Float64(l / t)) * Float64(k / l)))) / t_1);
	elseif (t <= 4.9e-144)
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	else
		tmp = Float64(Float64(t_2 / Float64(t * Float64(Float64(t * Float64(t / l)) * Float64(k / l)))) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = 2.0 + ((k / t) ^ 2.0);
	t_2 = 2.0 / tan(k);
	tmp = 0.0;
	if (t <= -5.2e-152)
		tmp = (t_2 / (t * ((t / (l / t)) * (k / l)))) / t_1;
	elseif (t <= 4.9e-144)
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	else
		tmp = (t_2 / (t * ((t * (t / l)) * (k / l)))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.2e-152], N[(N[(t$95$2 / N[(t * N[(N[(t / N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t, 4.9e-144], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 / N[(t * N[(N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.20000000000000026e-152

   1. Initial program 60.9%

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

      1. associate-/r* 60.9%

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

      2. associate-*l* 52.8%

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

      3. sqr-neg 52.8%

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

      4. associate-*l* 60.9%

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

      5. *-commutative 60.9%

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

      6. sqr-neg 60.9%

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

      7. associate-/r* 60.9%

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

   3. Simplified 60.7%

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

      1. cube-mult 60.7%

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

      2. *-un-lft-identity 60.7%

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

      3. times-frac 66.5%

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

      4. associate-/l* 69.7%

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

   5. Applied egg-rr 69.7%

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

   6. Taylor expanded in k around 0 65.6%

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

      1. *-commutative 65.6%

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

      2. unpow2 65.6%

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

      3. times-frac 71.5%

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

      4. unpow2 71.5%

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

      5. associate-/l* 74.0%

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

   8. Simplified 74.0%

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

   if -5.20000000000000026e-152 < t < 4.9000000000000001e-144

   1. Initial program 31.0%

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

      1. associate-/r* 31.0%

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

      2. associate-*l* 31.0%

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

      3. sqr-neg 31.0%

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

      4. associate-*l* 31.0%

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

      5. *-commutative 31.0%

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

      6. sqr-neg 31.0%

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

      7. associate-*l/ 31.0%

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

      8. associate-*r/ 31.0%

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

      9. associate-/r/ 31.0%

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

   3. Simplified 31.0%

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

   4. Taylor expanded in k around inf 80.7%

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

      1. times-frac 78.6%

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

      2. unpow2 78.6%

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

      3. unpow2 78.6%

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

   6. Simplified 78.6%

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

   7. Taylor expanded in k around 0 59.5%

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

      1. unpow2 59.5%

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

      2. *-commutative 59.5%

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

      3. times-frac 66.3%

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

   9. Simplified 66.3%

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

   if 4.9000000000000001e-144 < t

   1. Initial program 64.6%

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

      1. associate-/r* 64.9%

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

      2. associate-*l* 56.4%

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

      3. sqr-neg 56.4%

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

      4. associate-*l* 64.9%

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

      5. *-commutative 64.9%

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

      6. sqr-neg 64.9%

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

      7. associate-/r* 64.9%

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

   3. Simplified 64.8%

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

      1. cube-mult 64.8%

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

      2. *-un-lft-identity 64.8%

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

      3. times-frac 67.1%

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

      4. associate-/l* 70.2%

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

   5. Applied egg-rr 70.2%

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

   6. Taylor expanded in t around 0 67.1%

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

      1. unpow2 67.1%

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

      2. times-frac 76.4%

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

      3. unpow2 76.4%

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

      4. associate-/l* 81.4%

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

   8. Simplified 81.4%

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

   9. Taylor expanded in k around 0 60.9%

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

       1. *-commutative 60.9%

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

       2. unpow2 60.9%

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

       3. times-frac 69.2%

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

       4. unpow2 69.2%

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

       5. associate-*l/ 72.3%

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

   11. Simplified 72.3%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{t \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot \frac{k}{\ell}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{-144}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{t \cdot \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \frac{k}{\ell}\right)}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 7: 67.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-103} \lor \neg \left(t \leq 4.6 \cdot 10^{-120}\right):\\ \;\;\;\;\frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -2.5e-103) (not (<= t 4.6e-120)))
   (/ (* (* (/ l k) (/ l k)) (/ 2.0 (pow t 3.0))) (+ 2.0 (pow (/ k t) 2.0)))
   (* 2.0 (/ (/ l t) (/ (pow k 4.0) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2.5e-103) || !(t <= 4.6e-120)) {
		tmp = (((l / k) * (l / k)) * (2.0 / pow(t, 3.0))) / (2.0 + pow((k / t), 2.0));
	} else {
		tmp = 2.0 * ((l / t) / (pow(k, 4.0) / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-2.5d-103)) .or. (.not. (t <= 4.6d-120))) then
        tmp = (((l / k) * (l / k)) * (2.0d0 / (t ** 3.0d0))) / (2.0d0 + ((k / t) ** 2.0d0))
    else
        tmp = 2.0d0 * ((l / t) / ((k ** 4.0d0) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -2.5e-103) || !(t <= 4.6e-120)) {
		tmp = (((l / k) * (l / k)) * (2.0 / Math.pow(t, 3.0))) / (2.0 + Math.pow((k / t), 2.0));
	} else {
		tmp = 2.0 * ((l / t) / (Math.pow(k, 4.0) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -2.5e-103) or not (t <= 4.6e-120):
		tmp = (((l / k) * (l / k)) * (2.0 / math.pow(t, 3.0))) / (2.0 + math.pow((k / t), 2.0))
	else:
		tmp = 2.0 * ((l / t) / (math.pow(k, 4.0) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -2.5e-103) || !(t <= 4.6e-120))
		tmp = Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(2.0 / (t ^ 3.0))) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(Float64(l / t) / Float64((k ^ 4.0) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -2.5e-103) || ~((t <= 4.6e-120)))
		tmp = (((l / k) * (l / k)) * (2.0 / (t ^ 3.0))) / (2.0 + ((k / t) ^ 2.0));
	else
		tmp = 2.0 * ((l / t) / ((k ^ 4.0) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -2.5e-103], N[Not[LessEqual[t, 4.6e-120]], $MachinePrecision]], N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / t), $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.49999999999999983e-103 or 4.59999999999999973e-120 < t

   1. Initial program 64.7%

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

      1. associate-/r* 64.9%

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

      2. associate-*l* 55.4%

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

      3. sqr-neg 55.4%

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

      4. associate-*l* 64.9%

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

      5. *-commutative 64.9%

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

      6. sqr-neg 64.9%

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

      7. associate-/r* 64.9%

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

   3. Simplified 64.7%

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

      1. cube-mult 64.7%

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

      2. *-un-lft-identity 64.7%

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

      3. times-frac 67.2%

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

      4. associate-/l* 70.9%

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

   5. Applied egg-rr 70.9%

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

   6. Taylor expanded in t around 0 67.4%

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

      1. unpow2 67.4%

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

      2. times-frac 73.7%

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

      3. unpow2 73.7%

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

      4. associate-/l* 80.3%

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

   8. Simplified 80.3%

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

   9. Taylor expanded in k around 0 50.4%

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

       1. associate-*r/ 50.4%

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

       2. *-commutative 50.4%

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

       3. times-frac 50.2%

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

       4. unpow2 50.2%

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

       5. unpow2 50.2%

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

       6. times-frac 64.9%

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

   11. Simplified 64.9%

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

   if -2.49999999999999983e-103 < t < 4.59999999999999973e-120

   1. Initial program 35.8%

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

      1. associate-/r* 35.8%

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

      2. associate-*l* 35.8%

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

      3. sqr-neg 35.8%

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

      4. associate-*l* 35.8%

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

      5. *-commutative 35.8%

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

      6. sqr-neg 35.8%

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

      7. associate-*l/ 35.8%

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

      8. associate-*r/ 35.8%

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

      9. associate-/r/ 35.8%

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

   3. Simplified 35.8%

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

   4. Taylor expanded in k around inf 77.4%

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

      1. times-frac 75.9%

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

      2. unpow2 75.9%

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

      3. unpow2 75.9%

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

   6. Simplified 75.9%

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

   7. Taylor expanded in k around 0 60.6%

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

      1. *-commutative 60.6%

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

      2. unpow2 60.6%

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

   9. Simplified 60.6%

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

   10. Taylor expanded in l around 0 60.5%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   11. Step-by-step derivation

       1. unpow2 60.5%

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

       2. times-frac 67.6%

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

       3. associate-*l/ 66.5%

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

       4. *-commutative 66.5%

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

       5. associate-/l* 67.6%

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

   12. Simplified 67.6%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-103} \lor \neg \left(t \leq 4.6 \cdot 10^{-120}\right):\\ \;\;\;\;\frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}\\ \end{array} \]

Alternative 8: 66.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-22} \lor \neg \left(t \leq 2050000\right):\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left({t}^{3} \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -3e-22) (not (<= t 2050000.0)))
   (/ l (/ (* k (* (pow t 3.0) k)) l))
   (* 2.0 (/ (/ l t) (/ (pow k 4.0) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3e-22) || !(t <= 2050000.0)) {
		tmp = l / ((k * (pow(t, 3.0) * k)) / l);
	} else {
		tmp = 2.0 * ((l / t) / (pow(k, 4.0) / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-3d-22)) .or. (.not. (t <= 2050000.0d0))) then
        tmp = l / ((k * ((t ** 3.0d0) * k)) / l)
    else
        tmp = 2.0d0 * ((l / t) / ((k ** 4.0d0) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -3e-22) || !(t <= 2050000.0)) {
		tmp = l / ((k * (Math.pow(t, 3.0) * k)) / l);
	} else {
		tmp = 2.0 * ((l / t) / (Math.pow(k, 4.0) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if (t <= -3e-22) or not (t <= 2050000.0):
		tmp = l / ((k * (math.pow(t, 3.0) * k)) / l)
	else:
		tmp = 2.0 * ((l / t) / (math.pow(k, 4.0) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if ((t <= -3e-22) || !(t <= 2050000.0))
		tmp = Float64(l / Float64(Float64(k * Float64((t ^ 3.0) * k)) / l));
	else
		tmp = Float64(2.0 * Float64(Float64(l / t) / Float64((k ^ 4.0) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -3e-22) || ~((t <= 2050000.0)))
		tmp = l / ((k * ((t ^ 3.0) * k)) / l);
	else
		tmp = 2.0 * ((l / t) / ((k ^ 4.0) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[Or[LessEqual[t, -3e-22], N[Not[LessEqual[t, 2050000.0]], $MachinePrecision]], N[(l / N[(N[(k * N[(N[Power[t, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / t), $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.9999999999999999e-22 or 2.05e6 < t

   1. Initial program 65.5%

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

      1. associate-/r* 65.5%

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

      2. associate-*l* 52.5%

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

      3. sqr-neg 52.5%

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

      4. associate-*l* 65.5%

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

      5. *-commutative 65.5%

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

      6. sqr-neg 65.5%

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

      7. associate-*l/ 65.5%

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

      8. associate-*r/ 64.1%

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

      9. associate-/r/ 63.8%

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

   3. Simplified 63.8%

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

   4. Taylor expanded in k around 0 50.1%

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

      1. unpow2 50.1%

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

      2. *-commutative 50.1%

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

      3. times-frac 52.2%

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

      4. unpow2 52.2%

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

   6. Simplified 52.2%

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

   7. Taylor expanded in l around 0 50.1%

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

      1. unpow2 50.1%

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

      2. *-commutative 50.1%

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

      3. unpow2 50.1%

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

      4. associate-/l* 52.6%

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

      5. unpow2 52.6%

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

      6. *-commutative 52.6%

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

      7. unpow2 52.6%

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

   9. Simplified 52.6%

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

   10. Taylor expanded in k around 0 52.6%

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

       1. unpow2 52.6%

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

       2. associate-*r* 64.2%

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

   12. Simplified 64.2%

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

   if -2.9999999999999999e-22 < t < 2.05e6

   1. Initial program 44.1%

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

      1. associate-/r* 44.4%

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

      2. associate-*l* 44.3%

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

      3. sqr-neg 44.3%

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

      4. associate-*l* 44.4%

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

      5. *-commutative 44.4%

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

      6. sqr-neg 44.4%

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

      7. associate-*l/ 44.3%

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

      8. associate-*r/ 44.3%

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

      9. associate-/r/ 44.3%

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

   3. Simplified 44.3%

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

   4. Taylor expanded in k around inf 73.7%

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

      1. times-frac 72.0%

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

      2. unpow2 72.0%

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

      3. unpow2 72.0%

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

   6. Simplified 72.0%

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

   7. Taylor expanded in k around 0 57.7%

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

      1. *-commutative 57.7%

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

      2. unpow2 57.7%

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

   9. Simplified 57.7%

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

   10. Taylor expanded in l around 0 57.6%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   11. Step-by-step derivation

       1. unpow2 57.6%

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

       2. times-frac 63.5%

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

       3. associate-*l/ 61.9%

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

       4. *-commutative 61.9%

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

       5. associate-/l* 63.5%

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

   12. Simplified 63.5%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-22} \lor \neg \left(t \leq 2050000\right):\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left({t}^{3} \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}\\ \end{array} \]

Alternative 9: 55.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.46 \cdot 10^{-265}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} - \frac{0.16666666666666666}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -1.46e-265)
   (*
    2.0
    (*
     (/ (* l l) (* k k))
     (- (/ 1.0 (* t (* k k))) (/ 0.16666666666666666 t))))
   (* 2.0 (* (/ l t) (/ l (pow k 4.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.46e-265) {
		tmp = 2.0 * (((l * l) / (k * k)) * ((1.0 / (t * (k * k))) - (0.16666666666666666 / t)));
	} else {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-1.46d-265)) then
        tmp = 2.0d0 * (((l * l) / (k * k)) * ((1.0d0 / (t * (k * k))) - (0.16666666666666666d0 / t)))
    else
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.46e-265) {
		tmp = 2.0 * (((l * l) / (k * k)) * ((1.0 / (t * (k * k))) - (0.16666666666666666 / t)));
	} else {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -1.46e-265:
		tmp = 2.0 * (((l * l) / (k * k)) * ((1.0 / (t * (k * k))) - (0.16666666666666666 / t)))
	else:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -1.46e-265)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * l) / Float64(k * k)) * Float64(Float64(1.0 / Float64(t * Float64(k * k))) - Float64(0.16666666666666666 / t))));
	else
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -1.46e-265)
		tmp = 2.0 * (((l * l) / (k * k)) * ((1.0 / (t * (k * k))) - (0.16666666666666666 / t)));
	else
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -1.46e-265], N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.46000000000000005e-265

   1. Initial program 54.4%

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

      1. associate-/r* 54.4%

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

      2. associate-*l* 47.8%

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

      3. sqr-neg 47.8%

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

      4. associate-*l* 54.4%

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

      5. *-commutative 54.4%

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

      6. sqr-neg 54.4%

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

      7. associate-*l/ 54.4%

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

      8. associate-*r/ 53.4%

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

      9. associate-/r/ 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in k around inf 56.5%

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

      1. times-frac 59.2%

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

      2. unpow2 59.2%

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

      3. unpow2 59.2%

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

   6. Simplified 59.2%

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

   7. Taylor expanded in k around 0 55.0%

      \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative 55.0%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \left(\frac{1}{\color{blue}{t \cdot {k}^{2}}} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      2. unpow2 55.0%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \left(\frac{1}{t \cdot \color{blue}{\left(k \cdot k\right)}} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      3. associate-*r/ 55.0%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

      4. metadata-eval 55.0%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

   9. Simplified 55.0%

      \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \color{blue}{\left(\frac{1}{t \cdot \left(k \cdot k\right)} - \frac{0.16666666666666666}{t}\right)}\right) \]

   if -1.46000000000000005e-265 < 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-/r* 53.9%

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

      2. associate-*l* 48.4%

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

      3. sqr-neg 48.4%

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

      4. associate-*l* 53.9%

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

      5. *-commutative 53.9%

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

      6. sqr-neg 53.9%

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

      7. associate-*l/ 53.9%

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

      8. associate-*r/ 53.4%

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

      9. associate-/r/ 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in k around inf 64.6%

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

      1. times-frac 63.0%

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

      2. unpow2 63.0%

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

      3. unpow2 63.0%

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

   6. Simplified 63.0%

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

   7. Taylor expanded in k around 0 54.6%

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

      1. unpow2 54.6%

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

      2. *-commutative 54.6%

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

      3. times-frac 59.1%

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

   9. Simplified 59.1%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.46 \cdot 10^{-265}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} - \frac{0.16666666666666666}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]

Alternative 10: 55.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-267}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} - \frac{0.16666666666666666}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -5.2e-267)
   (*
    2.0
    (*
     (/ (* l l) (* k k))
     (- (/ 1.0 (* t (* k k))) (/ 0.16666666666666666 t))))
   (* 2.0 (/ (/ l t) (/ (pow k 4.0) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -5.2e-267) {
		tmp = 2.0 * (((l * l) / (k * k)) * ((1.0 / (t * (k * k))) - (0.16666666666666666 / t)));
	} else {
		tmp = 2.0 * ((l / t) / (pow(k, 4.0) / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-5.2d-267)) then
        tmp = 2.0d0 * (((l * l) / (k * k)) * ((1.0d0 / (t * (k * k))) - (0.16666666666666666d0 / t)))
    else
        tmp = 2.0d0 * ((l / t) / ((k ** 4.0d0) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -5.2e-267) {
		tmp = 2.0 * (((l * l) / (k * k)) * ((1.0 / (t * (k * k))) - (0.16666666666666666 / t)));
	} else {
		tmp = 2.0 * ((l / t) / (Math.pow(k, 4.0) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -5.2e-267:
		tmp = 2.0 * (((l * l) / (k * k)) * ((1.0 / (t * (k * k))) - (0.16666666666666666 / t)))
	else:
		tmp = 2.0 * ((l / t) / (math.pow(k, 4.0) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -5.2e-267)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * l) / Float64(k * k)) * Float64(Float64(1.0 / Float64(t * Float64(k * k))) - Float64(0.16666666666666666 / t))));
	else
		tmp = Float64(2.0 * Float64(Float64(l / t) / Float64((k ^ 4.0) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -5.2e-267)
		tmp = 2.0 * (((l * l) / (k * k)) * ((1.0 / (t * (k * k))) - (0.16666666666666666 / t)));
	else
		tmp = 2.0 * ((l / t) / ((k ^ 4.0) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -5.2e-267], N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / t), $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.2000000000000003e-267

   1. Initial program 54.4%

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

      1. associate-/r* 54.4%

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

      2. associate-*l* 47.8%

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

      3. sqr-neg 47.8%

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

      4. associate-*l* 54.4%

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

      5. *-commutative 54.4%

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

      6. sqr-neg 54.4%

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

      7. associate-*l/ 54.4%

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

      8. associate-*r/ 53.4%

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

      9. associate-/r/ 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in k around inf 56.5%

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

      1. times-frac 59.2%

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

      2. unpow2 59.2%

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

      3. unpow2 59.2%

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

   6. Simplified 59.2%

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

   7. Taylor expanded in k around 0 55.0%

      \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative 55.0%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \left(\frac{1}{\color{blue}{t \cdot {k}^{2}}} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      2. unpow2 55.0%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \left(\frac{1}{t \cdot \color{blue}{\left(k \cdot k\right)}} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

      3. associate-*r/ 55.0%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

      4. metadata-eval 55.0%

         \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

   9. Simplified 55.0%

      \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \color{blue}{\left(\frac{1}{t \cdot \left(k \cdot k\right)} - \frac{0.16666666666666666}{t}\right)}\right) \]

   if -5.2000000000000003e-267 < 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-/r* 53.9%

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

      2. associate-*l* 48.4%

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

      3. sqr-neg 48.4%

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

      4. associate-*l* 53.9%

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

      5. *-commutative 53.9%

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

      6. sqr-neg 53.9%

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

      7. associate-*l/ 53.9%

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

      8. associate-*r/ 53.4%

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

      9. associate-/r/ 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in k around inf 64.6%

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

      1. times-frac 63.0%

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

      2. unpow2 63.0%

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

      3. unpow2 63.0%

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

   6. Simplified 63.0%

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

   7. Taylor expanded in k around 0 55.4%

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

      1. *-commutative 55.4%

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

      2. unpow2 55.4%

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

   9. Simplified 55.4%

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

   10. Taylor expanded in l around 0 54.6%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   11. Step-by-step derivation

       1. unpow2 54.6%

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

       2. times-frac 59.1%

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

       3. associate-*l/ 58.2%

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

       4. *-commutative 58.2%

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

       5. associate-/l* 59.1%

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

   12. Simplified 59.1%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-267}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} - \frac{0.16666666666666666}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}\\ \end{array} \]

Alternative 11: 59.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.46 \cdot 10^{-39}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -1.46e-39)
   (* (/ l (pow t 3.0)) (/ l (* k k)))
   (* 2.0 (/ (/ l t) (/ (pow k 4.0) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.46e-39) {
		tmp = (l / pow(t, 3.0)) * (l / (k * k));
	} else {
		tmp = 2.0 * ((l / t) / (pow(k, 4.0) / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-1.46d-39)) then
        tmp = (l / (t ** 3.0d0)) * (l / (k * k))
    else
        tmp = 2.0d0 * ((l / t) / ((k ** 4.0d0) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.46e-39) {
		tmp = (l / Math.pow(t, 3.0)) * (l / (k * k));
	} else {
		tmp = 2.0 * ((l / t) / (Math.pow(k, 4.0) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -1.46e-39:
		tmp = (l / math.pow(t, 3.0)) * (l / (k * k))
	else:
		tmp = 2.0 * ((l / t) / (math.pow(k, 4.0) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -1.46e-39)
		tmp = Float64(Float64(l / (t ^ 3.0)) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(2.0 * Float64(Float64(l / t) / Float64((k ^ 4.0) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -1.46e-39)
		tmp = (l / (t ^ 3.0)) * (l / (k * k));
	else
		tmp = 2.0 * ((l / t) / ((k ^ 4.0) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -1.46e-39], N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / t), $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.46000000000000001e-39

   1. Initial program 61.6%

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

      1. associate-/r* 61.6%

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

      2. associate-*l* 50.1%

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

      3. sqr-neg 50.1%

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

      4. associate-*l* 61.6%

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

      5. *-commutative 61.6%

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

      6. sqr-neg 61.6%

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

      7. associate-*l/ 61.6%

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

      8. associate-*r/ 59.9%

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

      9. associate-/r/ 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in k around 0 48.4%

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

      1. unpow2 48.4%

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

      2. *-commutative 48.4%

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

      3. times-frac 50.7%

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

      4. unpow2 50.7%

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

   6. Simplified 50.7%

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

   if -1.46000000000000001e-39 < t

   1. Initial program 51.5%

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

      1. associate-/r* 51.7%

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

      2. associate-*l* 47.4%

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

      3. sqr-neg 47.4%

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

      4. associate-*l* 51.7%

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

      5. *-commutative 51.7%

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

      6. sqr-neg 51.7%

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

      7. associate-*l/ 51.6%

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

      8. associate-*r/ 51.3%

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

      9. associate-/r/ 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in k around inf 67.7%

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

      1. times-frac 66.4%

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

      2. unpow2 66.4%

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

      3. unpow2 66.4%

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

   6. Simplified 66.4%

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

   7. Taylor expanded in k around 0 56.7%

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

      1. *-commutative 56.7%

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

      2. unpow2 56.7%

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

   9. Simplified 56.7%

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

   10. Taylor expanded in l around 0 56.1%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   11. Step-by-step derivation

       1. unpow2 56.1%

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

       2. times-frac 60.5%

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

       3. associate-*l/ 59.3%

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

       4. *-commutative 59.3%

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

       5. associate-/l* 60.5%

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

   12. Simplified 60.5%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.46 \cdot 10^{-39}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}\\ \end{array} \]

Alternative 12: 59.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.45 \cdot 10^{-41}:\\ \;\;\;\;\frac{\ell}{{t}^{3} \cdot \frac{k \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -2.45e-41)
   (/ l (* (pow t 3.0) (/ (* k k) l)))
   (* 2.0 (/ (/ l t) (/ (pow k 4.0) l)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -2.45e-41) {
		tmp = l / (pow(t, 3.0) * ((k * k) / l));
	} else {
		tmp = 2.0 * ((l / t) / (pow(k, 4.0) / l));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-2.45d-41)) then
        tmp = l / ((t ** 3.0d0) * ((k * k) / l))
    else
        tmp = 2.0d0 * ((l / t) / ((k ** 4.0d0) / l))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -2.45e-41) {
		tmp = l / (Math.pow(t, 3.0) * ((k * k) / l));
	} else {
		tmp = 2.0 * ((l / t) / (Math.pow(k, 4.0) / l));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -2.45e-41:
		tmp = l / (math.pow(t, 3.0) * ((k * k) / l))
	else:
		tmp = 2.0 * ((l / t) / (math.pow(k, 4.0) / l))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -2.45e-41)
		tmp = Float64(l / Float64((t ^ 3.0) * Float64(Float64(k * k) / l)));
	else
		tmp = Float64(2.0 * Float64(Float64(l / t) / Float64((k ^ 4.0) / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -2.45e-41)
		tmp = l / ((t ^ 3.0) * ((k * k) / l));
	else
		tmp = 2.0 * ((l / t) / ((k ^ 4.0) / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -2.45e-41], N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / t), $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.44999999999999977e-41

   1. Initial program 61.6%

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

      1. associate-/r* 61.6%

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

      2. associate-*l* 50.1%

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

      3. sqr-neg 50.1%

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

      4. associate-*l* 61.6%

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

      5. *-commutative 61.6%

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

      6. sqr-neg 61.6%

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

      7. associate-*l/ 61.6%

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

      8. associate-*r/ 59.9%

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

      9. associate-/r/ 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in k around 0 48.4%

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

      1. unpow2 48.4%

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

      2. *-commutative 48.4%

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

      3. times-frac 50.7%

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

      4. unpow2 50.7%

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

   6. Simplified 50.7%

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

   7. Taylor expanded in l around 0 48.4%

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

      1. unpow2 48.4%

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

      2. *-commutative 48.4%

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

      3. unpow2 48.4%

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

      4. associate-/l* 49.7%

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

      5. unpow2 49.7%

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

      6. *-commutative 49.7%

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

      7. unpow2 49.7%

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

   9. Simplified 49.7%

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

   10. Taylor expanded in k around 0 49.7%

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

       1. associate-*l/ 50.8%

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

       2. unpow2 50.8%

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

   12. Simplified 50.8%

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

   if -2.44999999999999977e-41 < t

   1. Initial program 51.5%

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

      1. associate-/r* 51.7%

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

      2. associate-*l* 47.4%

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

      3. sqr-neg 47.4%

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

      4. associate-*l* 51.7%

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

      5. *-commutative 51.7%

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

      6. sqr-neg 51.7%

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

      7. associate-*l/ 51.6%

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

      8. associate-*r/ 51.3%

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

      9. associate-/r/ 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in k around inf 67.7%

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

      1. times-frac 66.4%

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

      2. unpow2 66.4%

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

      3. unpow2 66.4%

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

   6. Simplified 66.4%

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

   7. Taylor expanded in k around 0 56.7%

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

      1. *-commutative 56.7%

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

      2. unpow2 56.7%

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

   9. Simplified 56.7%

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

   10. Taylor expanded in l around 0 56.1%

       \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   11. Step-by-step derivation

       1. unpow2 56.1%

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

       2. times-frac 60.5%

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

       3. associate-*l/ 59.3%

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

       4. *-commutative 59.3%

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

       5. associate-/l* 60.5%

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

   12. Simplified 60.5%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.45 \cdot 10^{-41}:\\ \;\;\;\;\frac{\ell}{{t}^{3} \cdot \frac{k \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}\\ \end{array} \]

Alternative 13: 53.3% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} - \frac{0.16666666666666666}{t}\right)\right) \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (*
  2.0
  (* (/ (* l l) (* k k)) (- (/ 1.0 (* t (* k k))) (/ 0.16666666666666666 t)))))

C

double code(double t, double l, double k) {
	return 2.0 * (((l * l) / (k * k)) * ((1.0 / (t * (k * k))) - (0.16666666666666666 / t)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (((l * l) / (k * k)) * ((1.0d0 / (t * (k * k))) - (0.16666666666666666d0 / t)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (((l * l) / (k * k)) * ((1.0 / (t * (k * k))) - (0.16666666666666666 / t)));
}

Python

def code(t, l, k):
	return 2.0 * (((l * l) / (k * k)) * ((1.0 / (t * (k * k))) - (0.16666666666666666 / t)))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(Float64(l * l) / Float64(k * k)) * Float64(Float64(1.0 / Float64(t * Float64(k * k))) - Float64(0.16666666666666666 / t))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (((l * l) / (k * k)) * ((1.0 / (t * (k * k))) - (0.16666666666666666 / t)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.0%

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

   1. associate-/r* 54.1%

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

   2. associate-*l* 48.1%

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

   3. sqr-neg 48.1%

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

   4. associate-*l* 54.1%

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

   5. *-commutative 54.1%

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

   6. sqr-neg 54.1%

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

   7. associate-*l/ 54.1%

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

   8. associate-*r/ 53.4%

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

   9. associate-/r/ 53.3%

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

3. Simplified 53.3%

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

4. Taylor expanded in k around inf 61.1%

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

   1. times-frac 61.3%

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

   2. unpow2 61.3%

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

   3. unpow2 61.3%

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

6. Simplified 61.3%

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

7. Taylor expanded in k around 0 55.5%

   \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]
8. Step-by-step derivation

   1. *-commutative 55.5%

      \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \left(\frac{1}{\color{blue}{t \cdot {k}^{2}}} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

   2. unpow2 55.5%

      \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \left(\frac{1}{t \cdot \color{blue}{\left(k \cdot k\right)}} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right) \]

   3. associate-*r/ 55.5%

      \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{t}}\right)\right) \]

   4. metadata-eval 55.5%

      \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} - \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

9. Simplified 55.5%

   \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \color{blue}{\left(\frac{1}{t \cdot \left(k \cdot k\right)} - \frac{0.16666666666666666}{t}\right)}\right) \]

10. Final simplification 55.5%

    \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{k \cdot k} \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} - \frac{0.16666666666666666}{t}\right)\right) \]

Alternative 14: 55.9% accurate, 24.8× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ 1.0 (* t (* k k))) (* l (/ l (* k k))))))

C

double code(double t, double l, double k) {
	return 2.0 * ((1.0 / (t * (k * k))) * (l * (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 * ((1.0d0 / (t * (k * k))) * (l * (l / (k * k))))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * ((1.0 / (t * (k * k))) * (l * (l / (k * k))));
}

Python

def code(t, l, k):
	return 2.0 * ((1.0 / (t * (k * k))) * (l * (l / (k * k))))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(1.0 / Float64(t * Float64(k * k))) * Float64(l * Float64(l / Float64(k * k)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * ((1.0 / (t * (k * k))) * (l * (l / (k * k))));
end

Wolfram

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

Derivation

1. Initial program 54.0%

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

   1. associate-/r* 54.1%

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

   2. associate-*l* 48.1%

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

   3. sqr-neg 48.1%

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

   4. associate-*l* 54.1%

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

   5. *-commutative 54.1%

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

   6. sqr-neg 54.1%

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

   7. associate-*l/ 54.1%

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

   8. associate-*r/ 53.4%

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

   9. associate-/r/ 53.3%

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

3. Simplified 53.3%

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

4. Taylor expanded in k around inf 61.1%

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

   1. times-frac 61.3%

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

   2. unpow2 61.3%

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

   3. unpow2 61.3%

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

6. Simplified 61.3%

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

7. Taylor expanded in k around 0 54.0%

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

   1. *-commutative 54.0%

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

   2. unpow2 54.0%

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

9. Simplified 54.0%

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

10. Taylor expanded in l around 0 54.0%

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

    1. *-rgt-identity 54.0%

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

    2. associate-*r/ 53.6%

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

    3. unpow2 53.6%

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

    4. unpow2 53.6%

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

    5. associate-*l* 53.5%

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

    6. unpow2 53.5%

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

    7. associate-*r/ 53.5%

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

    8. *-rgt-identity 53.5%

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

    9. unpow2 53.5%

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

12. Simplified 53.5%

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

13. Final simplification 53.5%

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

Alternative 15: 53.7% accurate, 24.8× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ (* l l) (* k k)) (/ 1.0 (* k (* t k))))))

C

double code(double t, double l, double k) {
	return 2.0 * (((l * l) / (k * k)) * (1.0 / (k * (t * 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 * (((l * l) / (k * k)) * (1.0d0 / (k * (t * k))))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 * (((l * l) / (k * k)) * (1.0 / (k * (t * k))));
}

Python

def code(t, l, k):
	return 2.0 * (((l * l) / (k * k)) * (1.0 / (k * (t * k))))

Julia

function code(t, l, k)
	return Float64(2.0 * Float64(Float64(Float64(l * l) / Float64(k * k)) * Float64(1.0 / Float64(k * Float64(t * k)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 * (((l * l) / (k * k)) * (1.0 / (k * (t * k))));
end

Wolfram

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

Derivation

1. Initial program 54.0%

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

   1. associate-/r* 54.1%

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

   2. associate-*l* 48.1%

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

   3. sqr-neg 48.1%

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

   4. associate-*l* 54.1%

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

   5. *-commutative 54.1%

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

   6. sqr-neg 54.1%

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

   7. associate-*l/ 54.1%

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

   8. associate-*r/ 53.4%

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

   9. associate-/r/ 53.3%

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

3. Simplified 53.3%

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

4. Taylor expanded in k around inf 61.1%

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

   1. times-frac 61.3%

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

   2. unpow2 61.3%

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

   3. unpow2 61.3%

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

6. Simplified 61.3%

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

7. Taylor expanded in k around 0 54.0%

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

   1. *-commutative 54.0%

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

   2. unpow2 54.0%

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

9. Simplified 54.0%

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

10. Taylor expanded in t around 0 54.0%

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

    1. unpow2 54.0%

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

    2. associate-*l* 54.0%

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

12. Simplified 54.0%

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

13. Final simplification 54.0%

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

Alternative 16: 56.0% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

   1. associate-/r* 54.1%

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

   2. associate-*l* 48.1%

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

   3. sqr-neg 48.1%

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

   4. associate-*l* 54.1%

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

   5. *-commutative 54.1%

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

   6. sqr-neg 54.1%

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

   7. associate-*l/ 54.1%

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

   8. associate-*r/ 53.4%

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

   9. associate-/r/ 53.3%

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

3. Simplified 53.3%

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

4. Taylor expanded in k around inf 61.1%

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

   1. times-frac 61.3%

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

   2. unpow2 61.3%

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

   3. unpow2 61.3%

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

6. Simplified 61.3%

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

7. Taylor expanded in k around 0 54.0%

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

   1. *-commutative 54.0%

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

   2. unpow2 54.0%

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

9. Simplified 54.0%

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

    1. un-div-inv 54.0%

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

    2. times-frac 53.5%

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

11. Applied egg-rr 53.5%

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

12. Final simplification 53.5%

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

Reproduce

herbie shell --seed 2023293 
(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))))