Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.2% → 87.2%

Time: 15.2s

Alternatives: 10

Speedup: 32.4×

• 27.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 87.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq 30500000:\\ \;\;\;\;\frac{{\left(\frac{\frac{\ell}{k}}{t}\right)}^{2}}{t}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{+166}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot t}{\frac{\ell}{k}}}{\ell} \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= k 30500000.0)
     (/ (pow (/ (/ l k) t) 2.0) t)
     (if (<= k 1.1e+166)
       (/ 2.0 (* (/ (/ (* k t) (/ l k)) l) t_1))
       (/ 2.0 (* t_1 (* t (* (/ k l) (/ k l)))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (k <= 30500000.0) {
		tmp = pow(((l / k) / t), 2.0) / t;
	} else if (k <= 1.1e+166) {
		tmp = 2.0 / ((((k * t) / (l / k)) / l) * t_1);
	} else {
		tmp = 2.0 / (t_1 * (t * ((k / l) * (k / l))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) * tan(k)
    if (k <= 30500000.0d0) then
        tmp = (((l / k) / t) ** 2.0d0) / t
    else if (k <= 1.1d+166) then
        tmp = 2.0d0 / ((((k * t) / (l / k)) / l) * t_1)
    else
        tmp = 2.0d0 / (t_1 * (t * ((k / l) * (k / l))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (k <= 30500000.0) {
		tmp = Math.pow(((l / k) / t), 2.0) / t;
	} else if (k <= 1.1e+166) {
		tmp = 2.0 / ((((k * t) / (l / k)) / l) * t_1);
	} else {
		tmp = 2.0 / (t_1 * (t * ((k / l) * (k / l))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = math.sin(k) * math.tan(k)
	tmp = 0
	if k <= 30500000.0:
		tmp = math.pow(((l / k) / t), 2.0) / t
	elif k <= 1.1e+166:
		tmp = 2.0 / ((((k * t) / (l / k)) / l) * t_1)
	else:
		tmp = 2.0 / (t_1 * (t * ((k / l) * (k / l))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (k <= 30500000.0)
		tmp = Float64((Float64(Float64(l / k) / t) ^ 2.0) / t);
	elseif (k <= 1.1e+166)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * t) / Float64(l / k)) / l) * t_1));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(t * Float64(Float64(k / l) * Float64(k / l)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = sin(k) * tan(k);
	tmp = 0.0;
	if (k <= 30500000.0)
		tmp = (((l / k) / t) ^ 2.0) / t;
	elseif (k <= 1.1e+166)
		tmp = 2.0 / ((((k * t) / (l / k)) / l) * t_1);
	else
		tmp = 2.0 / (t_1 * (t * ((k / l) * (k / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 30500000.0], N[(N[Power[N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision], If[LessEqual[k, 1.1e+166], N[(2.0 / N[(N[(N[(N[(k * t), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(t * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.05e7

   1. Initial program 58.1%

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

      1. associate-*l* 58.1%

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

      2. associate-/l/ 58.1%

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

      3. *-commutative 58.1%

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

      4. associate-*r/ 58.2%

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

      5. associate-/l* 57.6%

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

      6. associate-/r/ 54.6%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in k around 0 55.8%

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

      1. associate-/r* 54.7%

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

      2. unpow2 54.7%

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

      3. unpow2 54.7%

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

      4. times-frac 65.6%

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

   6. Simplified 65.6%

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

      1. unpow3 65.6%

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

      2. times-frac 74.2%

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

   8. Applied egg-rr 74.2%

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

   9. Taylor expanded in l around 0 55.8%

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

       1. associate-/r* 54.7%

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

       2. unpow2 54.7%

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

       3. unpow2 54.7%

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

       4. times-frac 65.6%

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

       5. cube-mult 65.6%

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

       6. times-frac 74.2%

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

       7. associate-/l/ 74.3%

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

       8. associate-/r* 75.4%

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

       9. associate-/l/ 78.8%

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

       10. associate-*r/ 78.3%

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

       11. unpow2 78.3%

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

       12. associate-/l/ 76.8%

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

   11. Simplified 76.8%

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

   if 3.05e7 < k < 1.1e166

   1. Initial program 46.5%

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

      1. *-commutative 46.5%

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

      2. associate-*l* 46.5%

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

      3. associate-*r* 46.5%

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

      4. +-commutative 46.5%

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

      5. associate-+r+ 46.5%

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

      6. metadata-eval 46.5%

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

   3. Simplified 46.5%

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

   4. Taylor expanded in k around inf 63.5%

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

      1. *-commutative 63.5%

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

      2. unpow2 63.5%

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

      3. times-frac 63.9%

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

      4. unpow2 63.9%

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

   6. Simplified 63.9%

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

      1. associate-*l/ 77.7%

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

      2. associate-/l* 77.7%

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

   8. Applied egg-rr 77.7%

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

      1. associate-*r/ 83.1%

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

   10. Applied egg-rr 83.1%

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

   if 1.1e166 < k

   1. Initial program 51.8%

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

      1. *-commutative 51.8%

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

      2. associate-*l* 51.8%

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

      3. associate-*r* 51.8%

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

      4. +-commutative 51.8%

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

      5. associate-+r+ 51.8%

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

      6. metadata-eval 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in k around inf 55.1%

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

      1. *-commutative 55.1%

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

      2. unpow2 55.1%

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

      3. times-frac 58.9%

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

      4. unpow2 58.9%

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

   6. Simplified 58.9%

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

   7. Taylor expanded in t around 0 55.1%

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

      1. unpow2 55.1%

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

      2. associate-*l/ 55.1%

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

      3. unpow2 55.1%

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

      4. times-frac 96.7%

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

      5. *-commutative 96.7%

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

   9. Simplified 96.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 30500000:\\ \;\;\;\;\frac{{\left(\frac{\frac{\ell}{k}}{t}\right)}^{2}}{t}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{+166}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot t}{\frac{\ell}{k}}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternative 2: 78.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 18000000:\\ \;\;\;\;\frac{{\left(\frac{\frac{\ell}{k}}{t}\right)}^{2}}{t}\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{+131}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell} \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 18000000.0)
   (/ (pow (/ (/ l k) t) 2.0) t)
   (if (<= k 8.5e+131)
     (* (* l l) (/ 2.0 (* (tan k) (* (* k k) (* t (sin k))))))
     (/ 2.0 (* (/ (* t (/ k (/ l k))) l) (* k k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 18000000.0) {
		tmp = pow(((l / k) / t), 2.0) / t;
	} else if (k <= 8.5e+131) {
		tmp = (l * l) * (2.0 / (tan(k) * ((k * k) * (t * sin(k)))));
	} else {
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 18000000.0d0) then
        tmp = (((l / k) / t) ** 2.0d0) / t
    else if (k <= 8.5d+131) then
        tmp = (l * l) * (2.0d0 / (tan(k) * ((k * k) * (t * sin(k)))))
    else
        tmp = 2.0d0 / (((t * (k / (l / k))) / l) * (k * k))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 18000000.0) {
		tmp = Math.pow(((l / k) / t), 2.0) / t;
	} else if (k <= 8.5e+131) {
		tmp = (l * l) * (2.0 / (Math.tan(k) * ((k * k) * (t * Math.sin(k)))));
	} else {
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 18000000.0:
		tmp = math.pow(((l / k) / t), 2.0) / t
	elif k <= 8.5e+131:
		tmp = (l * l) * (2.0 / (math.tan(k) * ((k * k) * (t * math.sin(k)))))
	else:
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 18000000.0)
		tmp = Float64((Float64(Float64(l / k) / t) ^ 2.0) / t);
	elseif (k <= 8.5e+131)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * Float64(Float64(k * k) * Float64(t * sin(k))))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(k / Float64(l / k))) / l) * Float64(k * k)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 18000000.0)
		tmp = (((l / k) / t) ^ 2.0) / t;
	elseif (k <= 8.5e+131)
		tmp = (l * l) * (2.0 / (tan(k) * ((k * k) * (t * sin(k)))));
	else
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 18000000.0], N[(N[Power[N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision], If[LessEqual[k, 8.5e+131], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.8e7

   1. Initial program 58.1%

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

      1. associate-*l* 58.1%

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

      2. associate-/l/ 58.1%

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

      3. *-commutative 58.1%

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

      4. associate-*r/ 58.2%

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

      5. associate-/l* 57.6%

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

      6. associate-/r/ 54.6%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in k around 0 55.8%

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

      1. associate-/r* 54.7%

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

      2. unpow2 54.7%

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

      3. unpow2 54.7%

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

      4. times-frac 65.6%

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

   6. Simplified 65.6%

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

      1. unpow3 65.6%

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

      2. times-frac 74.2%

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

   8. Applied egg-rr 74.2%

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

   9. Taylor expanded in l around 0 55.8%

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

       1. associate-/r* 54.7%

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

       2. unpow2 54.7%

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

       3. unpow2 54.7%

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

       4. times-frac 65.6%

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

       5. cube-mult 65.6%

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

       6. times-frac 74.2%

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

       7. associate-/l/ 74.3%

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

       8. associate-/r* 75.4%

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

       9. associate-/l/ 78.8%

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

       10. associate-*r/ 78.3%

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

       11. unpow2 78.3%

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

       12. associate-/l/ 76.8%

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

   11. Simplified 76.8%

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

   if 1.8e7 < k < 8.50000000000000063e131

   1. Initial program 46.1%

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

      1. associate-/l/ 46.1%

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

      2. associate-*l/ 46.2%

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

      3. associate-*l/ 46.2%

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

      4. associate-/r/ 46.9%

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

      5. *-commutative 46.9%

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

      6. associate-/l/ 46.9%

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

      7. associate-*r* 46.9%

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

      8. *-commutative 46.9%

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

      9. associate-*r* 46.9%

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

      10. *-commutative 46.9%

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

   3. Simplified 46.9%

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

   4. Taylor expanded in k around inf 68.4%

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

      1. unpow2 68.4%

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

      2. *-commutative 68.4%

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

   6. Simplified 68.4%

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

   if 8.50000000000000063e131 < k

   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. *-commutative 51.5%

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

      2. associate-*l* 51.5%

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

      3. associate-*r* 51.5%

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

      4. +-commutative 51.5%

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

      5. associate-+r+ 51.5%

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

      6. metadata-eval 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in k around inf 52.0%

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

      1. *-commutative 52.0%

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

      2. unpow2 52.0%

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

      3. times-frac 58.2%

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

      4. unpow2 58.2%

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

   6. Simplified 58.2%

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

   7. Taylor expanded in k around 0 58.2%

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

      1. unpow2 58.2%

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

   9. Simplified 58.2%

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

       1. associate-*l/ 61.3%

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

       2. associate-/l* 86.2%

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

   11. Applied egg-rr 58.6%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 18000000:\\ \;\;\;\;\frac{{\left(\frac{\frac{\ell}{k}}{t}\right)}^{2}}{t}\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{+131}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell} \cdot \left(k \cdot k\right)}\\ \end{array} \]

Alternative 3: 85.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 18000000:\\ \;\;\;\;\frac{{\left(\frac{\frac{\ell}{k}}{t}\right)}^{2}}{t}\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{+120}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 18000000.0)
   (/ (pow (/ (/ l k) t) 2.0) t)
   (if (<= k 4.8e+120)
     (* (* l l) (/ 2.0 (* (tan k) (* (* k k) (* t (sin k))))))
     (/ 2.0 (* (* (sin k) (tan k)) (* t (* (/ k l) (/ k l))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 18000000.0) {
		tmp = pow(((l / k) / t), 2.0) / t;
	} else if (k <= 4.8e+120) {
		tmp = (l * l) * (2.0 / (tan(k) * ((k * k) * (t * sin(k)))));
	} else {
		tmp = 2.0 / ((sin(k) * tan(k)) * (t * ((k / l) * (k / l))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 18000000.0d0) then
        tmp = (((l / k) / t) ** 2.0d0) / t
    else if (k <= 4.8d+120) then
        tmp = (l * l) * (2.0d0 / (tan(k) * ((k * k) * (t * sin(k)))))
    else
        tmp = 2.0d0 / ((sin(k) * tan(k)) * (t * ((k / l) * (k / l))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 18000000.0) {
		tmp = Math.pow(((l / k) / t), 2.0) / t;
	} else if (k <= 4.8e+120) {
		tmp = (l * l) * (2.0 / (Math.tan(k) * ((k * k) * (t * Math.sin(k)))));
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.tan(k)) * (t * ((k / l) * (k / l))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 18000000.0:
		tmp = math.pow(((l / k) / t), 2.0) / t
	elif k <= 4.8e+120:
		tmp = (l * l) * (2.0 / (math.tan(k) * ((k * k) * (t * math.sin(k)))))
	else:
		tmp = 2.0 / ((math.sin(k) * math.tan(k)) * (t * ((k / l) * (k / l))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 18000000.0)
		tmp = Float64((Float64(Float64(l / k) / t) ^ 2.0) / t);
	elseif (k <= 4.8e+120)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * Float64(Float64(k * k) * Float64(t * sin(k))))));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * tan(k)) * Float64(t * Float64(Float64(k / l) * Float64(k / l)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 18000000.0)
		tmp = (((l / k) / t) ^ 2.0) / t;
	elseif (k <= 4.8e+120)
		tmp = (l * l) * (2.0 / (tan(k) * ((k * k) * (t * sin(k)))));
	else
		tmp = 2.0 / ((sin(k) * tan(k)) * (t * ((k / l) * (k / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 18000000.0], N[(N[Power[N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision], If[LessEqual[k, 4.8e+120], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.8e7

   1. Initial program 58.1%

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

      1. associate-*l* 58.1%

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

      2. associate-/l/ 58.1%

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

      3. *-commutative 58.1%

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

      4. associate-*r/ 58.2%

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

      5. associate-/l* 57.6%

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

      6. associate-/r/ 54.6%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in k around 0 55.8%

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

      1. associate-/r* 54.7%

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

      2. unpow2 54.7%

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

      3. unpow2 54.7%

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

      4. times-frac 65.6%

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

   6. Simplified 65.6%

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

      1. unpow3 65.6%

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

      2. times-frac 74.2%

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

   8. Applied egg-rr 74.2%

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

   9. Taylor expanded in l around 0 55.8%

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

       1. associate-/r* 54.7%

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

       2. unpow2 54.7%

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

       3. unpow2 54.7%

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

       4. times-frac 65.6%

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

       5. cube-mult 65.6%

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

       6. times-frac 74.2%

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

       7. associate-/l/ 74.3%

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

       8. associate-/r* 75.4%

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

       9. associate-/l/ 78.8%

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

       10. associate-*r/ 78.3%

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

       11. unpow2 78.3%

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

       12. associate-/l/ 76.8%

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

   11. Simplified 76.8%

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

   if 1.8e7 < k < 4.80000000000000002e120

   1. Initial program 51.1%

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

      1. associate-/l/ 51.1%

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

      2. associate-*l/ 51.1%

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

      3. associate-*l/ 51.2%

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

      4. associate-/r/ 52.0%

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

      5. *-commutative 52.0%

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

      6. associate-/l/ 52.0%

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

      7. associate-*r* 52.0%

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

      8. *-commutative 52.0%

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

      9. associate-*r* 52.0%

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

      10. *-commutative 52.0%

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

   3. Simplified 52.0%

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

   4. Taylor expanded in k around inf 68.3%

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

      1. unpow2 68.3%

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

      2. *-commutative 68.3%

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

   6. Simplified 68.3%

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

   if 4.80000000000000002e120 < k

   1. Initial program 47.5%

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

      1. *-commutative 47.5%

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

      2. associate-*l* 47.5%

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

      3. associate-*r* 47.5%

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

      4. +-commutative 47.5%

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

      5. associate-+r+ 47.5%

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

      6. metadata-eval 47.5%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in k around inf 53.3%

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

      1. *-commutative 53.3%

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

      2. unpow2 53.3%

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

      3. times-frac 59.0%

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

      4. unpow2 59.0%

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

   6. Simplified 59.0%

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

   7. Taylor expanded in t around 0 53.3%

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

      1. unpow2 53.3%

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

      2. associate-*l/ 55.8%

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

      3. unpow2 55.8%

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

      4. times-frac 92.2%

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

      5. *-commutative 92.2%

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

   9. Simplified 92.2%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 18000000:\\ \;\;\;\;\frac{{\left(\frac{\frac{\ell}{k}}{t}\right)}^{2}}{t}\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{+120}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternative 4: 86.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq 60000000:\\ \;\;\;\;\frac{{\left(\frac{\frac{\ell}{k}}{t}\right)}^{2}}{t}\\ \mathbf{elif}\;k \leq 10^{+156}:\\ \;\;\;\;\frac{2}{t_1 \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= k 60000000.0)
     (/ (pow (/ (/ l k) t) 2.0) t)
     (if (<= k 1e+156)
       (/ 2.0 (* t_1 (/ (* t (/ k (/ l k))) l)))
       (/ 2.0 (* t_1 (* t (* (/ k l) (/ k l)))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (k <= 60000000.0) {
		tmp = pow(((l / k) / t), 2.0) / t;
	} else if (k <= 1e+156) {
		tmp = 2.0 / (t_1 * ((t * (k / (l / k))) / l));
	} else {
		tmp = 2.0 / (t_1 * (t * ((k / l) * (k / l))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) * tan(k)
    if (k <= 60000000.0d0) then
        tmp = (((l / k) / t) ** 2.0d0) / t
    else if (k <= 1d+156) then
        tmp = 2.0d0 / (t_1 * ((t * (k / (l / k))) / l))
    else
        tmp = 2.0d0 / (t_1 * (t * ((k / l) * (k / l))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (k <= 60000000.0) {
		tmp = Math.pow(((l / k) / t), 2.0) / t;
	} else if (k <= 1e+156) {
		tmp = 2.0 / (t_1 * ((t * (k / (l / k))) / l));
	} else {
		tmp = 2.0 / (t_1 * (t * ((k / l) * (k / l))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = math.sin(k) * math.tan(k)
	tmp = 0
	if k <= 60000000.0:
		tmp = math.pow(((l / k) / t), 2.0) / t
	elif k <= 1e+156:
		tmp = 2.0 / (t_1 * ((t * (k / (l / k))) / l))
	else:
		tmp = 2.0 / (t_1 * (t * ((k / l) * (k / l))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (k <= 60000000.0)
		tmp = Float64((Float64(Float64(l / k) / t) ^ 2.0) / t);
	elseif (k <= 1e+156)
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(t * Float64(k / Float64(l / k))) / l)));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(t * Float64(Float64(k / l) * Float64(k / l)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = sin(k) * tan(k);
	tmp = 0.0;
	if (k <= 60000000.0)
		tmp = (((l / k) / t) ^ 2.0) / t;
	elseif (k <= 1e+156)
		tmp = 2.0 / (t_1 * ((t * (k / (l / k))) / l));
	else
		tmp = 2.0 / (t_1 * (t * ((k / l) * (k / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 60000000.0], N[(N[Power[N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision], If[LessEqual[k, 1e+156], N[(2.0 / N[(t$95$1 * N[(N[(t * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(t * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 6e7

   1. Initial program 58.1%

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

      1. associate-*l* 58.1%

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

      2. associate-/l/ 58.1%

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

      3. *-commutative 58.1%

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

      4. associate-*r/ 58.2%

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

      5. associate-/l* 57.6%

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

      6. associate-/r/ 54.6%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in k around 0 55.8%

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

      1. associate-/r* 54.7%

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

      2. unpow2 54.7%

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

      3. unpow2 54.7%

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

      4. times-frac 65.6%

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

   6. Simplified 65.6%

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

      1. unpow3 65.6%

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

      2. times-frac 74.2%

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

   8. Applied egg-rr 74.2%

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

   9. Taylor expanded in l around 0 55.8%

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

       1. associate-/r* 54.7%

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

       2. unpow2 54.7%

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

       3. unpow2 54.7%

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

       4. times-frac 65.6%

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

       5. cube-mult 65.6%

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

       6. times-frac 74.2%

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

       7. associate-/l/ 74.3%

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

       8. associate-/r* 75.4%

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

       9. associate-/l/ 78.8%

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

       10. associate-*r/ 78.3%

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

       11. unpow2 78.3%

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

       12. associate-/l/ 76.8%

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

   11. Simplified 76.8%

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

   if 6e7 < k < 9.9999999999999998e155

   1. Initial program 47.9%

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

      1. *-commutative 47.9%

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

      2. associate-*l* 47.9%

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

      3. associate-*r* 47.9%

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

      4. +-commutative 47.9%

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

      5. associate-+r+ 47.9%

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

      6. metadata-eval 47.9%

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

   3. Simplified 47.9%

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

   4. Taylor expanded in k around inf 65.3%

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

      1. *-commutative 65.3%

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

      2. unpow2 65.3%

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

      3. times-frac 65.7%

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

      4. unpow2 65.7%

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

   6. Simplified 65.7%

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

      1. associate-*l/ 79.9%

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

      2. associate-/l* 79.9%

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

   8. Applied egg-rr 79.9%

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

   if 9.9999999999999998e155 < k

   1. Initial program 50.1%

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

      1. *-commutative 50.1%

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

      2. associate-*l* 50.1%

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

      3. associate-*r* 50.1%

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

      4. +-commutative 50.1%

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

      5. associate-+r+ 50.1%

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

      6. metadata-eval 50.1%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in k around inf 53.6%

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

      1. *-commutative 53.6%

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

      2. unpow2 53.6%

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

      3. times-frac 57.3%

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

      4. unpow2 57.3%

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

   6. Simplified 57.3%

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

   7. Taylor expanded in t around 0 53.6%

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

      1. unpow2 53.6%

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

      2. associate-*l/ 53.6%

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

      3. unpow2 53.6%

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

      4. times-frac 93.8%

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

      5. *-commutative 93.8%

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

   9. Simplified 93.8%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 60000000:\\ \;\;\;\;\frac{{\left(\frac{\frac{\ell}{k}}{t}\right)}^{2}}{t}\\ \mathbf{elif}\;k \leq 10^{+156}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternative 5: 87.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq 17000000:\\ \;\;\;\;\frac{{\left(\frac{\frac{\ell}{k}}{t}\right)}^{2}}{t}\\ \mathbf{elif}\;k \leq 2.55 \cdot 10^{+173}:\\ \;\;\;\;\frac{2}{t_1 \cdot \frac{k \cdot \frac{k \cdot t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= k 17000000.0)
     (/ (pow (/ (/ l k) t) 2.0) t)
     (if (<= k 2.55e+173)
       (/ 2.0 (* t_1 (/ (* k (/ (* k t) l)) l)))
       (/ 2.0 (* t_1 (* t (* (/ k l) (/ k l)))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (k <= 17000000.0) {
		tmp = pow(((l / k) / t), 2.0) / t;
	} else if (k <= 2.55e+173) {
		tmp = 2.0 / (t_1 * ((k * ((k * t) / l)) / l));
	} else {
		tmp = 2.0 / (t_1 * (t * ((k / l) * (k / l))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) * tan(k)
    if (k <= 17000000.0d0) then
        tmp = (((l / k) / t) ** 2.0d0) / t
    else if (k <= 2.55d+173) then
        tmp = 2.0d0 / (t_1 * ((k * ((k * t) / l)) / l))
    else
        tmp = 2.0d0 / (t_1 * (t * ((k / l) * (k / l))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (k <= 17000000.0) {
		tmp = Math.pow(((l / k) / t), 2.0) / t;
	} else if (k <= 2.55e+173) {
		tmp = 2.0 / (t_1 * ((k * ((k * t) / l)) / l));
	} else {
		tmp = 2.0 / (t_1 * (t * ((k / l) * (k / l))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = math.sin(k) * math.tan(k)
	tmp = 0
	if k <= 17000000.0:
		tmp = math.pow(((l / k) / t), 2.0) / t
	elif k <= 2.55e+173:
		tmp = 2.0 / (t_1 * ((k * ((k * t) / l)) / l))
	else:
		tmp = 2.0 / (t_1 * (t * ((k / l) * (k / l))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (k <= 17000000.0)
		tmp = Float64((Float64(Float64(l / k) / t) ^ 2.0) / t);
	elseif (k <= 2.55e+173)
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(k * Float64(Float64(k * t) / l)) / l)));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(t * Float64(Float64(k / l) * Float64(k / l)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = sin(k) * tan(k);
	tmp = 0.0;
	if (k <= 17000000.0)
		tmp = (((l / k) / t) ^ 2.0) / t;
	elseif (k <= 2.55e+173)
		tmp = 2.0 / (t_1 * ((k * ((k * t) / l)) / l));
	else
		tmp = 2.0 / (t_1 * (t * ((k / l) * (k / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 17000000.0], N[(N[Power[N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision], If[LessEqual[k, 2.55e+173], N[(2.0 / N[(t$95$1 * N[(N[(k * N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(t * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.7e7

   1. Initial program 58.1%

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

      1. associate-*l* 58.1%

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

      2. associate-/l/ 58.1%

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

      3. *-commutative 58.1%

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

      4. associate-*r/ 58.2%

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

      5. associate-/l* 57.6%

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

      6. associate-/r/ 54.6%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in k around 0 55.8%

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

      1. associate-/r* 54.7%

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

      2. unpow2 54.7%

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

      3. unpow2 54.7%

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

      4. times-frac 65.6%

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

   6. Simplified 65.6%

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

      1. unpow3 65.6%

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

      2. times-frac 74.2%

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

   8. Applied egg-rr 74.2%

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

   9. Taylor expanded in l around 0 55.8%

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

       1. associate-/r* 54.7%

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

       2. unpow2 54.7%

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

       3. unpow2 54.7%

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

       4. times-frac 65.6%

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

       5. cube-mult 65.6%

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

       6. times-frac 74.2%

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

       7. associate-/l/ 74.3%

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

       8. associate-/r* 75.4%

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

       9. associate-/l/ 78.8%

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

       10. associate-*r/ 78.3%

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

       11. unpow2 78.3%

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

       12. associate-/l/ 76.8%

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

   11. Simplified 76.8%

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

   if 1.7e7 < k < 2.54999999999999975e173

   1. Initial program 46.5%

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

      1. *-commutative 46.5%

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

      2. associate-*l* 46.5%

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

      3. associate-*r* 46.5%

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

      4. +-commutative 46.5%

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

      5. associate-+r+ 46.5%

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

      6. metadata-eval 46.5%

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

   3. Simplified 46.5%

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

   4. Taylor expanded in k around inf 63.5%

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

      1. *-commutative 63.5%

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

      2. unpow2 63.5%

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

      3. times-frac 63.9%

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

      4. unpow2 63.9%

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

   6. Simplified 63.9%

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

      1. associate-*l/ 77.7%

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

      2. associate-/l* 77.7%

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

   8. Applied egg-rr 77.7%

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

      1. associate-*r/ 83.1%

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

   10. Applied egg-rr 83.1%

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

       1. associate-/r/ 83.1%

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

   12. Applied egg-rr 83.1%

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

   if 2.54999999999999975e173 < k

   1. Initial program 51.8%

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

      1. *-commutative 51.8%

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

      2. associate-*l* 51.8%

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

      3. associate-*r* 51.8%

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

      4. +-commutative 51.8%

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

      5. associate-+r+ 51.8%

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

      6. metadata-eval 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in k around inf 55.1%

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

      1. *-commutative 55.1%

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

      2. unpow2 55.1%

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

      3. times-frac 58.9%

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

      4. unpow2 58.9%

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

   6. Simplified 58.9%

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

   7. Taylor expanded in t around 0 55.1%

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

      1. unpow2 55.1%

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

      2. associate-*l/ 55.1%

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

      3. unpow2 55.1%

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

      4. times-frac 96.7%

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

      5. *-commutative 96.7%

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

   9. Simplified 96.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 17000000:\\ \;\;\;\;\frac{{\left(\frac{\frac{\ell}{k}}{t}\right)}^{2}}{t}\\ \mathbf{elif}\;k \leq 2.55 \cdot 10^{+173}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \frac{k \cdot \frac{k \cdot t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternative 6: 74.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-110} \lor \neg \left(t \leq 3.2 \cdot 10^{-131}\right):\\ \;\;\;\;\frac{{\left(\frac{\frac{\ell}{k}}{t}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell} \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.35e-110) (not (<= t 3.2e-131)))
   (/ (pow (/ (/ l k) t) 2.0) t)
   (/ 2.0 (* (/ (* t (/ k (/ l k))) l) (* k k)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.35e-110) || !(t <= 3.2e-131)) {
		tmp = pow(((l / k) / t), 2.0) / t;
	} else {
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-1.35d-110)) .or. (.not. (t <= 3.2d-131))) then
        tmp = (((l / k) / t) ** 2.0d0) / t
    else
        tmp = 2.0d0 / (((t * (k / (l / k))) / l) * (k * k))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.35e-110) || !(t <= 3.2e-131)) {
		tmp = Math.pow(((l / k) / t), 2.0) / t;
	} else {
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if (t <= -1.35e-110) or not (t <= 3.2e-131):
		tmp = math.pow(((l / k) / t), 2.0) / t
	else:
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.35e-110) || !(t <= 3.2e-131))
		tmp = Float64((Float64(Float64(l / k) / t) ^ 2.0) / t);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(k / Float64(l / k))) / l) * Float64(k * k)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.35e-110) || ~((t <= 3.2e-131)))
		tmp = (((l / k) / t) ^ 2.0) / t;
	else
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -1.35e-110], N[Not[LessEqual[t, 3.2e-131]], $MachinePrecision]], N[(N[Power[N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision], N[(2.0 / N[(N[(N[(t * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.3499999999999999e-110 or 3.2e-131 < t

   1. Initial program 65.1%

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

      1. associate-*l* 65.1%

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

      2. associate-/l/ 65.1%

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

      3. *-commutative 65.1%

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

      4. associate-*r/ 65.2%

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

      5. associate-/l* 64.5%

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

      6. associate-/r/ 61.7%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in k around 0 58.2%

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

      1. associate-/r* 57.1%

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

      2. unpow2 57.1%

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

      3. unpow2 57.1%

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

      4. times-frac 66.5%

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

   6. Simplified 66.5%

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

      1. unpow3 66.5%

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

      2. times-frac 74.6%

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

   8. Applied egg-rr 74.6%

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

   9. Taylor expanded in l around 0 58.2%

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

       1. associate-/r* 57.1%

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

       2. unpow2 57.1%

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

       3. unpow2 57.1%

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

       4. times-frac 66.5%

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

       5. cube-mult 66.5%

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

       6. times-frac 74.6%

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

       7. associate-/l/ 74.6%

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

       8. associate-/r* 75.8%

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

       9. associate-/l/ 79.7%

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

       10. associate-*r/ 78.7%

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

       11. unpow2 78.7%

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

       12. associate-/l/ 77.2%

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

   11. Simplified 77.2%

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

   if -1.3499999999999999e-110 < t < 3.2e-131

   1. Initial program 29.0%

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

      1. *-commutative 29.0%

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

      2. associate-*l* 29.0%

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

      3. associate-*r* 29.0%

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

      4. +-commutative 29.0%

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

      5. associate-+r+ 29.0%

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

      6. metadata-eval 29.0%

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

   3. Simplified 29.0%

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

   4. Taylor expanded in k around inf 64.0%

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

      1. *-commutative 64.0%

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

      2. unpow2 64.0%

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

      3. times-frac 79.3%

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

      4. unpow2 79.3%

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

   6. Simplified 79.3%

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

   7. Taylor expanded in k around 0 70.3%

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

      1. unpow2 70.3%

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

   9. Simplified 70.3%

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

       1. associate-*l/ 81.0%

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

       2. associate-/l* 89.9%

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

   11. Applied egg-rr 70.7%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-110} \lor \neg \left(t \leq 3.2 \cdot 10^{-131}\right):\\ \;\;\;\;\frac{{\left(\frac{\frac{\ell}{k}}{t}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell} \cdot \left(k \cdot k\right)}\\ \end{array} \]

Alternative 7: 70.2% accurate, 22.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -2 \cdot 10^{-177}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t \cdot t} \cdot t_1\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-150}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot t_1}{t \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (* k t))))
   (if (<= t -2e-177)
     (* (/ (/ l k) (* t t)) t_1)
     (if (<= t 3.9e-150)
       (/ 2.0 (* (* k k) (* (/ t l) (* k (/ k l)))))
       (/ (* (/ l k) t_1) (* t t))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = l / (k * t);
	double tmp;
	if (t <= -2e-177) {
		tmp = ((l / k) / (t * t)) * t_1;
	} else if (t <= 3.9e-150) {
		tmp = 2.0 / ((k * k) * ((t / l) * (k * (k / l))));
	} else {
		tmp = ((l / k) * t_1) / (t * t);
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l / (k * t)
    if (t <= (-2d-177)) then
        tmp = ((l / k) / (t * t)) * t_1
    else if (t <= 3.9d-150) then
        tmp = 2.0d0 / ((k * k) * ((t / l) * (k * (k / l))))
    else
        tmp = ((l / k) * t_1) / (t * t)
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = l / (k * t);
	double tmp;
	if (t <= -2e-177) {
		tmp = ((l / k) / (t * t)) * t_1;
	} else if (t <= 3.9e-150) {
		tmp = 2.0 / ((k * k) * ((t / l) * (k * (k / l))));
	} else {
		tmp = ((l / k) * t_1) / (t * t);
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = l / (k * t)
	tmp = 0
	if t <= -2e-177:
		tmp = ((l / k) / (t * t)) * t_1
	elif t <= 3.9e-150:
		tmp = 2.0 / ((k * k) * ((t / l) * (k * (k / l))))
	else:
		tmp = ((l / k) * t_1) / (t * t)
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(l / Float64(k * t))
	tmp = 0.0
	if (t <= -2e-177)
		tmp = Float64(Float64(Float64(l / k) / Float64(t * t)) * t_1);
	elseif (t <= 3.9e-150)
		tmp = Float64(2.0 / Float64(Float64(k * k) * Float64(Float64(t / l) * Float64(k * Float64(k / l)))));
	else
		tmp = Float64(Float64(Float64(l / k) * t_1) / Float64(t * t));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = l / (k * t);
	tmp = 0.0;
	if (t <= -2e-177)
		tmp = ((l / k) / (t * t)) * t_1;
	elseif (t <= 3.9e-150)
		tmp = 2.0 / ((k * k) * ((t / l) * (k * (k / l))));
	else
		tmp = ((l / k) * t_1) / (t * t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2e-177], N[(N[(N[(l / k), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t, 3.9e-150], N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.9999999999999999e-177

   1. Initial program 59.6%

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

      1. associate-*l* 59.6%

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

      2. associate-/l/ 59.6%

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

      3. *-commutative 59.6%

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

      4. associate-*r/ 58.9%

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

      5. associate-/l* 59.6%

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

      6. associate-/r/ 55.6%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in k around 0 53.2%

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

      1. associate-/r* 51.2%

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

      2. unpow2 51.2%

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

      3. unpow2 51.2%

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

      4. times-frac 61.1%

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

   6. Simplified 61.1%

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

      1. unpow3 61.0%

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

      2. times-frac 73.3%

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

   8. Applied egg-rr 73.3%

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

   9. Taylor expanded in l around 0 73.3%

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

   if -1.9999999999999999e-177 < t < 3.9000000000000002e-150

   1. Initial program 27.1%

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

      1. *-commutative 27.1%

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

      2. associate-*l* 27.1%

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

      3. associate-*r* 27.1%

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

      4. +-commutative 27.1%

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

      5. associate-+r+ 27.1%

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

      6. metadata-eval 27.1%

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

   3. Simplified 27.1%

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

   4. Taylor expanded in k around inf 62.7%

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

      1. *-commutative 62.7%

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

      2. unpow2 62.7%

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

      3. times-frac 80.8%

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

      4. unpow2 80.8%

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

   6. Simplified 80.8%

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

   7. Taylor expanded in k around 0 71.9%

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

      1. unpow2 71.9%

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

   9. Simplified 71.9%

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

   10. Taylor expanded in k around 0 71.9%

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

       1. unpow2 71.9%

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

       2. associate-*l/ 71.9%

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

       3. *-commutative 71.9%

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

   12. Simplified 71.9%

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

   if 3.9000000000000002e-150 < t

   1. Initial program 68.8%

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

      1. associate-*l* 68.8%

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

      2. associate-/l/ 68.7%

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

      3. *-commutative 68.7%

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

      4. associate-*r/ 69.8%

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

      5. associate-/l* 67.6%

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

      6. associate-/r/ 66.5%

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

   3. Simplified 75.4%

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

   4. Taylor expanded in k around 0 62.1%

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

      1. associate-/r* 62.2%

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

      2. unpow2 62.2%

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

      3. unpow2 62.2%

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

      4. times-frac 70.2%

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

   6. Simplified 70.2%

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

      1. unpow3 70.2%

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

      2. times-frac 73.7%

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

   8. Applied egg-rr 73.7%

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

      1. associate-*l/ 77.7%

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

      2. associate-/l/ 77.7%

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

   10. Applied egg-rr 77.7%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-177}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-150}:\\ \;\;\;\;\frac{2}{\left(k \cdot k\right) \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}}{t \cdot t}\\ \end{array} \]

Alternative 8: 71.1% accurate, 22.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -2.05 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t \cdot t} \cdot t_1\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{-150}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot t_1}{t \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (* k t))))
   (if (<= t -2.05e-110)
     (* (/ (/ l k) (* t t)) t_1)
     (if (<= t 4.7e-150)
       (/ 2.0 (* (/ (* t (/ k (/ l k))) l) (* k k)))
       (/ (* (/ l k) t_1) (* t t))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = l / (k * t);
	double tmp;
	if (t <= -2.05e-110) {
		tmp = ((l / k) / (t * t)) * t_1;
	} else if (t <= 4.7e-150) {
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	} else {
		tmp = ((l / k) * t_1) / (t * t);
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l / (k * t)
    if (t <= (-2.05d-110)) then
        tmp = ((l / k) / (t * t)) * t_1
    else if (t <= 4.7d-150) then
        tmp = 2.0d0 / (((t * (k / (l / k))) / l) * (k * k))
    else
        tmp = ((l / k) * t_1) / (t * t)
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = l / (k * t);
	double tmp;
	if (t <= -2.05e-110) {
		tmp = ((l / k) / (t * t)) * t_1;
	} else if (t <= 4.7e-150) {
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	} else {
		tmp = ((l / k) * t_1) / (t * t);
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = l / (k * t)
	tmp = 0
	if t <= -2.05e-110:
		tmp = ((l / k) / (t * t)) * t_1
	elif t <= 4.7e-150:
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k))
	else:
		tmp = ((l / k) * t_1) / (t * t)
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(l / Float64(k * t))
	tmp = 0.0
	if (t <= -2.05e-110)
		tmp = Float64(Float64(Float64(l / k) / Float64(t * t)) * t_1);
	elseif (t <= 4.7e-150)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(k / Float64(l / k))) / l) * Float64(k * k)));
	else
		tmp = Float64(Float64(Float64(l / k) * t_1) / Float64(t * t));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = l / (k * t);
	tmp = 0.0;
	if (t <= -2.05e-110)
		tmp = ((l / k) / (t * t)) * t_1;
	elseif (t <= 4.7e-150)
		tmp = 2.0 / (((t * (k / (l / k))) / l) * (k * k));
	else
		tmp = ((l / k) * t_1) / (t * t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.05e-110], N[(N[(N[(l / k), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t, 4.7e-150], N[(2.0 / N[(N[(N[(t * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.04999999999999991e-110

   1. Initial program 61.0%

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

      1. associate-*l* 61.0%

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

      2. associate-/l/ 61.0%

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

      3. *-commutative 61.0%

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

      4. associate-*r/ 60.2%

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

      5. associate-/l* 61.0%

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

      6. associate-/r/ 56.6%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in k around 0 53.9%

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

      1. associate-/r* 51.9%

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

      2. unpow2 51.9%

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

      3. unpow2 51.9%

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

      4. times-frac 62.5%

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

   6. Simplified 62.5%

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

      1. unpow3 62.4%

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

      2. times-frac 74.8%

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

   8. Applied egg-rr 74.8%

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

   9. Taylor expanded in l around 0 74.8%

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

   if -2.04999999999999991e-110 < t < 4.6999999999999999e-150

   1. Initial program 29.5%

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

      1. *-commutative 29.5%

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

      2. associate-*l* 29.5%

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

      3. associate-*r* 29.5%

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

      4. +-commutative 29.5%

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

      5. associate-+r+ 29.5%

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

      6. metadata-eval 29.5%

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

   3. Simplified 29.5%

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

   4. Taylor expanded in k around inf 63.4%

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

      1. *-commutative 63.4%

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

      2. unpow2 63.4%

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

      3. times-frac 79.0%

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

      4. unpow2 79.0%

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

   6. Simplified 79.0%

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

   7. Taylor expanded in k around 0 69.9%

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

      1. unpow2 69.9%

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

   9. Simplified 69.9%

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

       1. associate-*l/ 80.7%

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

       2. associate-/l* 89.7%

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

   11. Applied egg-rr 70.2%

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

   if 4.6999999999999999e-150 < t

   1. Initial program 68.8%

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

      1. associate-*l* 68.8%

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

      2. associate-/l/ 68.7%

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

      3. *-commutative 68.7%

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

      4. associate-*r/ 69.8%

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

      5. associate-/l* 67.6%

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

      6. associate-/r/ 66.5%

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

   3. Simplified 75.4%

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

   4. Taylor expanded in k around 0 62.1%

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

      1. associate-/r* 62.2%

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

      2. unpow2 62.2%

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

      3. unpow2 62.2%

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

      4. times-frac 70.2%

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

   6. Simplified 70.2%

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

      1. unpow3 70.2%

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

      2. times-frac 73.7%

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

   8. Applied egg-rr 73.7%

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

      1. associate-*l/ 77.7%

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

      2. associate-/l/ 77.7%

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

   10. Applied egg-rr 77.7%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.05 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{-150}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k}{\frac{\ell}{k}}}{\ell} \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}}{t \cdot t}\\ \end{array} \]

Alternative 9: 65.3% accurate, 32.4× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \frac{\frac{\ell}{k}}{t \cdot t} \cdot \frac{\ell}{k \cdot t} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (* (/ (/ l k) (* t t)) (/ l (* k t))))

C

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

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((l / k) / (t * t)) * (l / (k * t))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return ((l / k) / (t * t)) * (l / (k * t));
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(N[(l / k), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.8%

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

   1. associate-*l* 55.8%

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

   2. associate-/l/ 55.8%

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

   3. *-commutative 55.8%

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

   4. associate-*r/ 55.9%

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

   5. associate-/l* 55.4%

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

   6. associate-/r/ 53.3%

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

3. Simplified 59.9%

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

4. Taylor expanded in k around 0 53.0%

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

   1. associate-/r* 52.2%

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

   2. unpow2 52.2%

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

   3. unpow2 52.2%

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

   4. times-frac 62.0%

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

6. Simplified 62.0%

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

   1. unpow3 62.0%

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

   2. times-frac 68.5%

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

8. Applied egg-rr 68.5%

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

9. Taylor expanded in l around 0 68.5%

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

10. Final simplification 68.5%

    \[\leadsto \frac{\frac{\ell}{k}}{t \cdot t} \cdot \frac{\ell}{k \cdot t} \]

Alternative 10: 65.5% accurate, 32.4× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \frac{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}}{t \cdot t} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (/ (* (/ l k) (/ l (* k t))) (* t t)))

C

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

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((l / k) * (l / (k * t))) / (t * t)
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return ((l / k) * (l / (k * t))) / (t * t);
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.8%

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

   1. associate-*l* 55.8%

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

   2. associate-/l/ 55.8%

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

   3. *-commutative 55.8%

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

   4. associate-*r/ 55.9%

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

   5. associate-/l* 55.4%

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

   6. associate-/r/ 53.3%

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

3. Simplified 59.9%

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

4. Taylor expanded in k around 0 53.0%

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

   1. associate-/r* 52.2%

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

   2. unpow2 52.2%

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

   3. unpow2 52.2%

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

   4. times-frac 62.0%

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

6. Simplified 62.0%

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

   1. unpow3 62.0%

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

   2. times-frac 68.5%

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

8. Applied egg-rr 68.5%

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

   1. associate-*l/ 69.6%

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

   2. associate-/l/ 69.6%

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

10. Applied egg-rr 69.6%

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

11. Final simplification 69.6%

    \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}}{t \cdot t} \]

Reproduce

herbie shell --seed 2023240 
(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))))