Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.1% → 94.6%

Time: 24.8s

Alternatives: 21

Speedup: 28.1×

• 35.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

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: 34.1% 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: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;\ell \leq 7 \cdot 10^{+169}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}}{\frac{t_1}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\left(\frac{\ell}{k}\right)}^{2}}{t \cdot t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= l 7e+169)
     (* 2.0 (/ (* (/ l k) (/ l (* k t))) (/ t_1 (cos k))))
     (* 2.0 (/ (* (cos k) (pow (/ l k) 2.0)) (* t t_1))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (l <= 7e+169) {
		tmp = 2.0 * (((l / k) * (l / (k * t))) / (t_1 / cos(k)));
	} else {
		tmp = 2.0 * ((cos(k) * pow((l / k), 2.0)) / (t * t_1));
	}
	return tmp;
}

Fortran

NOTE: l 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) ** 2.0d0
    if (l <= 7d+169) then
        tmp = 2.0d0 * (((l / k) * (l / (k * t))) / (t_1 / cos(k)))
    else
        tmp = 2.0d0 * ((cos(k) * ((l / k) ** 2.0d0)) / (t * t_1))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (l <= 7e+169) {
		tmp = 2.0 * (((l / k) * (l / (k * t))) / (t_1 / Math.cos(k)));
	} else {
		tmp = 2.0 * ((Math.cos(k) * Math.pow((l / k), 2.0)) / (t * t_1));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if l <= 7e+169:
		tmp = 2.0 * (((l / k) * (l / (k * t))) / (t_1 / math.cos(k)))
	else:
		tmp = 2.0 * ((math.cos(k) * math.pow((l / k), 2.0)) / (t * t_1))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (l <= 7e+169)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / Float64(k * t))) / Float64(t_1 / cos(k))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * (Float64(l / k) ^ 2.0)) / Float64(t * t_1)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (l <= 7e+169)
		tmp = 2.0 * (((l / k) * (l / (k * t))) / (t_1 / cos(k)));
	else
		tmp = 2.0 * ((cos(k) * ((l / k) ^ 2.0)) / (t * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, 7e+169], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 7.00000000000000038e169

   1. Initial program 34.6%

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

      1. associate-*l* 34.6%

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

      2. associate-*l* 34.6%

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

      3. associate-/r* 34.6%

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

      4. associate-/r/ 34.6%

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

      5. *-commutative 34.6%

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

      6. times-frac 34.8%

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

      7. +-commutative 34.8%

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

      8. associate--l+ 45.3%

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

      9. metadata-eval 45.3%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 45.3%

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

      11. times-frac 48.9%

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

   3. Simplified 48.9%

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

   4. Taylor expanded in t around 0 86.7%

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

      1. unpow2 86.7%

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

   6. Simplified 86.7%

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

   7. Taylor expanded in k around inf 82.1%

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

      1. associate-/l* 82.1%

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

      2. associate-*r* 79.5%

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

      3. associate-/l* 78.5%

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

      4. associate-/l* 78.5%

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

      5. unpow2 78.5%

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

      6. associate-/l* 82.9%

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

      7. *-commutative 82.9%

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

      8. times-frac 84.2%

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

   9. Simplified 89.5%

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

       1. associate-*r/ 89.8%

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

   11. Applied egg-rr 89.8%

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

       1. *-un-lft-identity 89.8%

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

       2. associate-/l* 89.8%

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

       3. frac-times 82.8%

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

       4. *-commutative 82.8%

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

       5. associate-*r* 85.4%

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

       6. times-frac 96.5%

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

   13. Applied egg-rr 96.5%

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

   if 7.00000000000000038e169 < l

   1. Initial program 35.6%

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

      1. associate-*l* 35.6%

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

      2. associate-*l* 35.6%

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

      3. associate-/r* 35.6%

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

      4. associate-/r/ 35.6%

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

      5. *-commutative 35.6%

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

      6. times-frac 35.6%

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

      7. +-commutative 35.6%

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

      8. associate--l+ 35.6%

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

      9. metadata-eval 35.6%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 35.6%

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

      11. times-frac 35.6%

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

   3. Simplified 35.6%

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

   4. Taylor expanded in t around 0 61.4%

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

      1. unpow2 61.4%

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

   6. Simplified 61.4%

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

   7. Taylor expanded in k around inf 61.4%

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

      1. associate-/l* 61.4%

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

      2. associate-*r* 61.4%

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

      3. associate-/l* 61.8%

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

      4. associate-/l* 61.8%

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

      5. unpow2 61.8%

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

      6. associate-/l* 74.6%

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

      7. *-commutative 74.6%

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

      8. times-frac 74.5%

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

   9. Simplified 80.8%

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

   10. Taylor expanded in l around 0 61.4%

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

       1. associate-/r* 61.8%

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

       2. unpow2 61.8%

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

       3. unpow2 61.8%

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

       4. times-frac 96.7%

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

       5. unpow2 96.7%

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

   12. Simplified 96.7%

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

       1. frac-times 96.7%

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

   14. Applied egg-rr 96.7%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7 \cdot 10^{+169}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}}{\frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\left(\frac{\ell}{k}\right)}^{2}}{t \cdot {\sin k}^{2}}\\ \end{array} \]

Alternative 2: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;\ell \leq 2.1 \cdot 10^{+144}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}}{\frac{t_1}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{t_1}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= l 2.1e+144)
     (* 2.0 (/ (* (/ l k) (/ l (* k t))) (/ t_1 (cos k))))
     (* 2.0 (* (/ (pow (/ l k) 2.0) t) (/ (cos k) t_1))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (l <= 2.1e+144) {
		tmp = 2.0 * (((l / k) * (l / (k * t))) / (t_1 / cos(k)));
	} else {
		tmp = 2.0 * ((pow((l / k), 2.0) / t) * (cos(k) / t_1));
	}
	return tmp;
}

Fortran

NOTE: l 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) ** 2.0d0
    if (l <= 2.1d+144) then
        tmp = 2.0d0 * (((l / k) * (l / (k * t))) / (t_1 / cos(k)))
    else
        tmp = 2.0d0 * ((((l / k) ** 2.0d0) / t) * (cos(k) / t_1))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (l <= 2.1e+144) {
		tmp = 2.0 * (((l / k) * (l / (k * t))) / (t_1 / Math.cos(k)));
	} else {
		tmp = 2.0 * ((Math.pow((l / k), 2.0) / t) * (Math.cos(k) / t_1));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if l <= 2.1e+144:
		tmp = 2.0 * (((l / k) * (l / (k * t))) / (t_1 / math.cos(k)))
	else:
		tmp = 2.0 * ((math.pow((l / k), 2.0) / t) * (math.cos(k) / t_1))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (l <= 2.1e+144)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / Float64(k * t))) / Float64(t_1 / cos(k))));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k) ^ 2.0) / t) * Float64(cos(k) / t_1)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (l <= 2.1e+144)
		tmp = 2.0 * (((l / k) * (l / (k * t))) / (t_1 / cos(k)));
	else
		tmp = 2.0 * ((((l / k) ^ 2.0) / t) * (cos(k) / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, 2.1e+144], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 2.09999999999999996e144

   1. Initial program 34.6%

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

      1. associate-*l* 34.6%

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

      2. associate-*l* 34.6%

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

      3. associate-/r* 34.6%

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

      4. associate-/r/ 34.6%

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

      5. *-commutative 34.6%

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

      6. times-frac 34.9%

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

      7. +-commutative 34.9%

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

      8. associate--l+ 45.6%

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

      9. metadata-eval 45.6%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 45.6%

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

      11. times-frac 49.3%

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

   3. Simplified 49.3%

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

   4. Taylor expanded in t around 0 87.2%

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

      1. unpow2 87.2%

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

   6. Simplified 87.2%

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

   7. Taylor expanded in k around inf 82.5%

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

      1. associate-/l* 82.4%

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

      2. associate-*r* 79.8%

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

      3. associate-/l* 78.8%

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

      4. associate-/l* 78.8%

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

      5. unpow2 78.8%

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

      6. associate-/l* 82.8%

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

      7. *-commutative 82.8%

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

      8. times-frac 84.3%

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

   9. Simplified 89.6%

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

       1. associate-*r/ 90.0%

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

   11. Applied egg-rr 90.0%

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

       1. *-un-lft-identity 90.0%

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

       2. associate-/l* 90.0%

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

       3. frac-times 83.2%

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

       4. *-commutative 83.2%

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

       5. associate-*r* 85.9%

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

       6. times-frac 96.4%

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

   13. Applied egg-rr 96.4%

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

   if 2.09999999999999996e144 < l

   1. Initial program 35.2%

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

      1. associate-*l* 35.2%

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

      2. associate-*l* 35.2%

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

      3. associate-/r* 35.2%

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

      4. associate-/r/ 35.2%

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

      5. *-commutative 35.2%

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

      6. times-frac 35.4%

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

      7. +-commutative 35.4%

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

      8. associate--l+ 35.4%

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

      9. metadata-eval 35.4%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 35.4%

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

      11. times-frac 35.4%

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

   3. Simplified 35.4%

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

   4. Taylor expanded in t around 0 62.4%

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

      1. unpow2 62.4%

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

   6. Simplified 62.4%

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

   7. Taylor expanded in k around inf 62.4%

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

      1. associate-/l* 62.4%

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

      2. associate-*r* 62.4%

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

      3. associate-/l* 62.6%

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

      4. associate-/l* 62.6%

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

      5. unpow2 62.6%

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

      6. associate-/l* 76.0%

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

      7. *-commutative 76.0%

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

      8. times-frac 76.0%

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

   9. Simplified 81.4%

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

   10. Taylor expanded in l around 0 62.4%

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

       1. associate-/r* 62.7%

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

       2. unpow2 62.7%

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

       3. unpow2 62.7%

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

       4. times-frac 97.2%

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

       5. unpow2 97.2%

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

   12. Simplified 97.2%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{+144}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}}{\frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 3: 87.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e+262)
   (* (/ (/ 2.0 k) (* k t)) (* (/ l (sin k)) (/ l (tan k))))
   (* 2.0 (* (/ (cos k) (pow (sin k) 2.0)) (* (/ l t) (/ l (* k k)))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e+262) {
		tmp = ((2.0 / k) / (k * t)) * ((l / sin(k)) * (l / tan(k)));
	} else {
		tmp = 2.0 * ((cos(k) / pow(sin(k), 2.0)) * ((l / t) * (l / (k * k))));
	}
	return tmp;
}

Fortran

NOTE: l 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 ((l * l) <= 5d+262) then
        tmp = ((2.0d0 / k) / (k * t)) * ((l / sin(k)) * (l / tan(k)))
    else
        tmp = 2.0d0 * ((cos(k) / (sin(k) ** 2.0d0)) * ((l / t) * (l / (k * k))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e+262) {
		tmp = ((2.0 / k) / (k * t)) * ((l / Math.sin(k)) * (l / Math.tan(k)));
	} else {
		tmp = 2.0 * ((Math.cos(k) / Math.pow(Math.sin(k), 2.0)) * ((l / t) * (l / (k * k))));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if (l * l) <= 5e+262:
		tmp = ((2.0 / k) / (k * t)) * ((l / math.sin(k)) * (l / math.tan(k)))
	else:
		tmp = 2.0 * ((math.cos(k) / math.pow(math.sin(k), 2.0)) * ((l / t) * (l / (k * k))))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e+262)
		tmp = Float64(Float64(Float64(2.0 / k) / Float64(k * t)) * Float64(Float64(l / sin(k)) * Float64(l / tan(k))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / (sin(k) ^ 2.0)) * Float64(Float64(l / t) * Float64(l / Float64(k * k)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 5e+262)
		tmp = ((2.0 / k) / (k * t)) * ((l / sin(k)) * (l / tan(k)));
	else
		tmp = 2.0 * ((cos(k) / (sin(k) ^ 2.0)) * ((l / t) * (l / (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e+262], N[(N[(N[(2.0 / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.00000000000000008e262

   1. Initial program 35.5%

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

      1. associate-*l* 35.4%

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

      2. associate-*l* 35.5%

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

      3. associate-/r* 35.4%

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

      4. associate-/r/ 35.4%

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

      5. *-commutative 35.4%

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

      6. times-frac 35.7%

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

      7. +-commutative 35.7%

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

      8. associate--l+ 48.0%

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

      9. metadata-eval 48.0%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 48.0%

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

      11. times-frac 52.3%

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

   3. Simplified 52.3%

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

   4. Taylor expanded in t around 0 91.0%

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

      1. unpow2 91.0%

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

   6. Simplified 91.0%

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

   7. Taylor expanded in k around 0 91.0%

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

      1. unpow2 91.0%

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

      2. associate-*r* 93.9%

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

      3. associate-/r* 94.1%

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

   9. Simplified 94.1%

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

   if 5.00000000000000008e262 < (*.f64 l l)

   1. Initial program 32.5%

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

      1. associate-*l* 32.5%

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

      2. associate-*l* 32.5%

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

      3. associate-/r* 32.5%

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

      4. associate-/r/ 32.5%

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

      5. *-commutative 32.5%

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

      6. times-frac 32.6%

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

      7. +-commutative 32.6%

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

      8. associate--l+ 32.6%

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

      9. metadata-eval 32.6%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 32.6%

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

      11. times-frac 32.6%

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

   3. Simplified 32.6%

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

   4. Taylor expanded in t around 0 62.0%

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

      1. unpow2 62.0%

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

   6. Simplified 62.0%

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

   7. Taylor expanded in k around inf 63.4%

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

      1. associate-/l* 63.4%

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

      2. associate-*r* 63.4%

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

      3. associate-/l* 63.5%

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

      4. associate-/l* 63.5%

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

      5. unpow2 63.5%

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

      6. associate-/l* 75.7%

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

      7. *-commutative 75.7%

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

      8. times-frac 75.6%

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

   9. Simplified 80.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \]

Alternative 4: 87.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e+262)
   (* (/ (/ 2.0 k) (* k t)) (* (/ l (sin k)) (/ l (tan k))))
   (* 2.0 (/ (* (cos k) (* (/ l t) (/ l (* k k)))) (pow (sin k) 2.0)))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e+262) {
		tmp = ((2.0 / k) / (k * t)) * ((l / sin(k)) * (l / tan(k)));
	} else {
		tmp = 2.0 * ((cos(k) * ((l / t) * (l / (k * k)))) / pow(sin(k), 2.0));
	}
	return tmp;
}

Fortran

NOTE: l 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 ((l * l) <= 5d+262) then
        tmp = ((2.0d0 / k) / (k * t)) * ((l / sin(k)) * (l / tan(k)))
    else
        tmp = 2.0d0 * ((cos(k) * ((l / t) * (l / (k * k)))) / (sin(k) ** 2.0d0))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e+262) {
		tmp = ((2.0 / k) / (k * t)) * ((l / Math.sin(k)) * (l / Math.tan(k)));
	} else {
		tmp = 2.0 * ((Math.cos(k) * ((l / t) * (l / (k * k)))) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if (l * l) <= 5e+262:
		tmp = ((2.0 / k) / (k * t)) * ((l / math.sin(k)) * (l / math.tan(k)))
	else:
		tmp = 2.0 * ((math.cos(k) * ((l / t) * (l / (k * k)))) / math.pow(math.sin(k), 2.0))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e+262)
		tmp = Float64(Float64(Float64(2.0 / k) / Float64(k * t)) * Float64(Float64(l / sin(k)) * Float64(l / tan(k))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(Float64(l / t) * Float64(l / Float64(k * k)))) / (sin(k) ^ 2.0)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 5e+262)
		tmp = ((2.0 / k) / (k * t)) * ((l / sin(k)) * (l / tan(k)));
	else
		tmp = 2.0 * ((cos(k) * ((l / t) * (l / (k * k)))) / (sin(k) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e+262], N[(N[(N[(2.0 / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.00000000000000008e262

   1. Initial program 35.5%

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

      1. associate-*l* 35.4%

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

      2. associate-*l* 35.5%

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

      3. associate-/r* 35.4%

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

      4. associate-/r/ 35.4%

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

      5. *-commutative 35.4%

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

      6. times-frac 35.7%

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

      7. +-commutative 35.7%

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

      8. associate--l+ 48.0%

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

      9. metadata-eval 48.0%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 48.0%

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

      11. times-frac 52.3%

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

   3. Simplified 52.3%

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

   4. Taylor expanded in t around 0 91.0%

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

      1. unpow2 91.0%

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

   6. Simplified 91.0%

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

   7. Taylor expanded in k around 0 91.0%

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

      1. unpow2 91.0%

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

      2. associate-*r* 93.9%

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

      3. associate-/r* 94.1%

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

   9. Simplified 94.1%

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

   if 5.00000000000000008e262 < (*.f64 l l)

   1. Initial program 32.5%

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

      1. associate-*l* 32.5%

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

      2. associate-*l* 32.5%

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

      3. associate-/r* 32.5%

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

      4. associate-/r/ 32.5%

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

      5. *-commutative 32.5%

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

      6. times-frac 32.6%

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

      7. +-commutative 32.6%

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

      8. associate--l+ 32.6%

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

      9. metadata-eval 32.6%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 32.6%

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

      11. times-frac 32.6%

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

   3. Simplified 32.6%

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

   4. Taylor expanded in t around 0 62.0%

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

      1. unpow2 62.0%

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

   6. Simplified 62.0%

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

   7. Taylor expanded in k around inf 63.4%

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

      1. associate-/l* 63.4%

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

      2. associate-*r* 63.4%

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

      3. associate-/l* 63.5%

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

      4. associate-/l* 63.5%

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

      5. unpow2 63.5%

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

      6. associate-/l* 75.7%

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

      7. *-commutative 75.7%

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

      8. times-frac 75.6%

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

   9. Simplified 80.2%

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

       1. associate-*r/ 80.2%

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

   11. Applied egg-rr 80.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)}{{\sin k}^{2}}\\ \end{array} \]

Alternative 5: 94.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.4 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}}{\frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{0.5 - \frac{\cos \left(k + k\right)}{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= l 7.4e+161)
   (* 2.0 (/ (* (/ l k) (/ l (* k t))) (/ (pow (sin k) 2.0) (cos k))))
   (*
    2.0
    (* (/ (pow (/ l k) 2.0) t) (/ (cos k) (- 0.5 (/ (cos (+ k k)) 2.0)))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (l <= 7.4e+161) {
		tmp = 2.0 * (((l / k) * (l / (k * t))) / (pow(sin(k), 2.0) / cos(k)));
	} else {
		tmp = 2.0 * ((pow((l / k), 2.0) / t) * (cos(k) / (0.5 - (cos((k + k)) / 2.0))));
	}
	return tmp;
}

Fortran

NOTE: l 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 (l <= 7.4d+161) then
        tmp = 2.0d0 * (((l / k) * (l / (k * t))) / ((sin(k) ** 2.0d0) / cos(k)))
    else
        tmp = 2.0d0 * ((((l / k) ** 2.0d0) / t) * (cos(k) / (0.5d0 - (cos((k + k)) / 2.0d0))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 7.4e+161) {
		tmp = 2.0 * (((l / k) * (l / (k * t))) / (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	} else {
		tmp = 2.0 * ((Math.pow((l / k), 2.0) / t) * (Math.cos(k) / (0.5 - (Math.cos((k + k)) / 2.0))));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if l <= 7.4e+161:
		tmp = 2.0 * (((l / k) * (l / (k * t))) / (math.pow(math.sin(k), 2.0) / math.cos(k)))
	else:
		tmp = 2.0 * ((math.pow((l / k), 2.0) / t) * (math.cos(k) / (0.5 - (math.cos((k + k)) / 2.0))))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (l <= 7.4e+161)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / Float64(k * t))) / Float64((sin(k) ^ 2.0) / cos(k))));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k) ^ 2.0) / t) * Float64(cos(k) / Float64(0.5 - Float64(cos(Float64(k + k)) / 2.0)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 7.4e+161)
		tmp = 2.0 * (((l / k) * (l / (k * t))) / ((sin(k) ^ 2.0) / cos(k)));
	else
		tmp = 2.0 * ((((l / k) ^ 2.0) / t) * (cos(k) / (0.5 - (cos((k + k)) / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[l, 7.4e+161], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.39999999999999958e161

   1. Initial program 34.4%

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

      1. associate-*l* 34.4%

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

      2. associate-*l* 34.4%

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

      3. associate-/r* 34.4%

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

      4. associate-/r/ 34.4%

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

      5. *-commutative 34.4%

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

      6. times-frac 34.7%

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

      7. +-commutative 34.7%

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

      8. associate--l+ 45.2%

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

      9. metadata-eval 45.2%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 45.2%

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

      11. times-frac 48.9%

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

   3. Simplified 48.9%

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

   4. Taylor expanded in t around 0 87.0%

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

      1. unpow2 87.0%

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

   6. Simplified 87.0%

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

   7. Taylor expanded in k around inf 82.3%

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

      1. associate-/l* 82.3%

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

      2. associate-*r* 79.7%

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

      3. associate-/l* 78.7%

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

      4. associate-/l* 78.7%

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

      5. unpow2 78.7%

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

      6. associate-/l* 82.7%

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

      7. *-commutative 82.7%

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

      8. times-frac 84.1%

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

   9. Simplified 89.4%

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

       1. associate-*r/ 89.7%

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

   11. Applied egg-rr 89.7%

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

       1. *-un-lft-identity 89.7%

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

       2. associate-/l* 89.7%

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

       3. frac-times 83.1%

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

       4. *-commutative 83.1%

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

       5. associate-*r* 85.7%

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

       6. times-frac 96.5%

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

   13. Applied egg-rr 96.5%

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

   if 7.39999999999999958e161 < l

   1. Initial program 36.5%

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

      1. associate-*l* 36.5%

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

      2. associate-*l* 36.5%

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

      3. associate-/r* 36.5%

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

      4. associate-/r/ 36.5%

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

      5. *-commutative 36.5%

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

      6. times-frac 36.5%

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

      7. +-commutative 36.5%

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

      8. associate--l+ 36.5%

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

      9. metadata-eval 36.5%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 36.5%

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

      11. times-frac 36.5%

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

   3. Simplified 36.5%

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

   4. Taylor expanded in t around 0 60.7%

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

      1. unpow2 60.7%

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

   6. Simplified 60.7%

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

   7. Taylor expanded in k around inf 60.7%

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

      1. associate-/l* 60.7%

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

      2. associate-*r* 60.7%

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

      3. associate-/l* 61.1%

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

      4. associate-/l* 61.1%

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

      5. unpow2 61.1%

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

      6. associate-/l* 76.1%

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

      7. *-commutative 76.1%

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

      8. times-frac 76.1%

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

   9. Simplified 82.0%

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

   10. Taylor expanded in l around 0 60.7%

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

       1. associate-/r* 61.1%

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

       2. unpow2 61.1%

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

       3. unpow2 61.1%

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

       4. times-frac 96.9%

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

       5. unpow2 96.9%

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

   12. Simplified 96.9%

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

       1. unpow2 82.0%

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

       2. sin-mult 82.2%

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

   14. Applied egg-rr 96.7%

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

       1. div-sub 82.2%

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

       2. +-inverses 82.2%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}}\right) \]

       3. cos-0 82.2%

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

       4. metadata-eval 82.2%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}}\right) \]

   16. Simplified 96.7%

       \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{\color{blue}{0.5 - \frac{\cos \left(k + k\right)}{2}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.4 \cdot 10^{+161}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}}{\frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{0.5 - \frac{\cos \left(k + k\right)}{2}}\right)\\ \end{array} \]

Alternative 6: 93.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ 2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}}{\frac{{\sin k}^{2}}{\cos k}} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* 2.0 (/ (* (/ l k) (/ l (* k t))) (/ (pow (sin k) 2.0) (cos k)))))

C

l = abs(l);
double code(double t, double l, double k) {
	return 2.0 * (((l / k) * (l / (k * t))) / (pow(sin(k), 2.0) / cos(k)));
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (((l / k) * (l / (k * t))) / ((sin(k) ** 2.0d0) / cos(k)))
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	return 2.0 * (((l / k) * (l / (k * t))) / (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
}

Python

l = abs(l)
def code(t, l, k):
	return 2.0 * (((l / k) * (l / (k * t))) / (math.pow(math.sin(k), 2.0) / math.cos(k)))

Julia

l = abs(l)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / Float64(k * t))) / Float64((sin(k) ^ 2.0) / cos(k))))
end

MATLAB

l = abs(l)
function tmp = code(t, l, k)
	tmp = 2.0 * (((l / k) * (l / (k * t))) / ((sin(k) ^ 2.0) / cos(k)));
end

Wolfram

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

Derivation

1. Initial program 34.7%

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

   1. associate-*l* 34.7%

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

   2. associate-*l* 34.7%

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

   3. associate-/r* 34.7%

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

   4. associate-/r/ 34.7%

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

   5. *-commutative 34.7%

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

   6. times-frac 34.9%

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

   7. +-commutative 34.9%

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

   8. associate--l+ 44.1%

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

   9. metadata-eval 44.1%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   10. +-rgt-identity 44.1%

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

   11. times-frac 47.3%

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

3. Simplified 47.3%

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

4. Taylor expanded in t around 0 83.6%

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

   1. unpow2 83.6%

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

6. Simplified 83.6%

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

7. Taylor expanded in k around inf 79.6%

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

   1. associate-/l* 79.6%

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

   2. associate-*r* 77.3%

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

   3. associate-/l* 76.4%

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

   4. associate-/l* 76.5%

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

   5. unpow2 76.5%

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

   6. associate-/l* 81.9%

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

   7. *-commutative 81.9%

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

   8. times-frac 83.1%

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

9. Simplified 88.4%

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

    1. associate-*r/ 88.7%

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

11. Applied egg-rr 88.7%

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

    1. *-un-lft-identity 88.7%

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

    2. associate-/l* 88.7%

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

    3. frac-times 80.2%

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

    4. *-commutative 80.2%

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

    5. associate-*r* 82.9%

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

    6. times-frac 95.4%

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

13. Applied egg-rr 95.4%

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

14. Final simplification 95.4%

    \[\leadsto 2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}}{\frac{{\sin k}^{2}}{\cos k}} \]

Alternative 7: 86.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.45 \cdot 10^{+159}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{0.5 - \frac{\cos \left(k + k\right)}{2}} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= l 1.45e+159)
   (* (/ (/ 2.0 k) (* k t)) (* (/ l (sin k)) (/ l (tan k))))
   (*
    2.0
    (* (/ (cos k) (- 0.5 (/ (cos (+ k k)) 2.0))) (* (/ l t) (/ l (* k k)))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.45e+159) {
		tmp = ((2.0 / k) / (k * t)) * ((l / sin(k)) * (l / tan(k)));
	} else {
		tmp = 2.0 * ((cos(k) / (0.5 - (cos((k + k)) / 2.0))) * ((l / t) * (l / (k * k))));
	}
	return tmp;
}

Fortran

NOTE: l 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 (l <= 1.45d+159) then
        tmp = ((2.0d0 / k) / (k * t)) * ((l / sin(k)) * (l / tan(k)))
    else
        tmp = 2.0d0 * ((cos(k) / (0.5d0 - (cos((k + k)) / 2.0d0))) * ((l / t) * (l / (k * k))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.45e+159) {
		tmp = ((2.0 / k) / (k * t)) * ((l / Math.sin(k)) * (l / Math.tan(k)));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (0.5 - (Math.cos((k + k)) / 2.0))) * ((l / t) * (l / (k * k))));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if l <= 1.45e+159:
		tmp = ((2.0 / k) / (k * t)) * ((l / math.sin(k)) * (l / math.tan(k)))
	else:
		tmp = 2.0 * ((math.cos(k) / (0.5 - (math.cos((k + k)) / 2.0))) * ((l / t) * (l / (k * k))))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (l <= 1.45e+159)
		tmp = Float64(Float64(Float64(2.0 / k) / Float64(k * t)) * Float64(Float64(l / sin(k)) * Float64(l / tan(k))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(0.5 - Float64(cos(Float64(k + k)) / 2.0))) * Float64(Float64(l / t) * Float64(l / Float64(k * k)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 1.45e+159)
		tmp = ((2.0 / k) / (k * t)) * ((l / sin(k)) * (l / tan(k)));
	else
		tmp = 2.0 * ((cos(k) / (0.5 - (cos((k + k)) / 2.0))) * ((l / t) * (l / (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[l, 1.45e+159], N[(N[(N[(2.0 / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(0.5 - N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.45000000000000007e159

   1. Initial program 34.4%

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

      1. associate-*l* 34.4%

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

      2. associate-*l* 34.4%

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

      3. associate-/r* 34.4%

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

      4. associate-/r/ 34.4%

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

      5. *-commutative 34.4%

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

      6. times-frac 34.7%

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

      7. +-commutative 34.7%

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

      8. associate--l+ 45.2%

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

      9. metadata-eval 45.2%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 45.2%

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

      11. times-frac 48.9%

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

   3. Simplified 48.9%

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

   4. Taylor expanded in t around 0 87.0%

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

      1. unpow2 87.0%

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

   6. Simplified 87.0%

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

   7. Taylor expanded in k around 0 87.0%

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

      1. unpow2 87.0%

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

      2. associate-*r* 90.4%

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

      3. associate-/r* 90.6%

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

   9. Simplified 90.6%

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

   if 1.45000000000000007e159 < l

   1. Initial program 36.5%

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

      1. associate-*l* 36.5%

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

      2. associate-*l* 36.5%

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

      3. associate-/r* 36.5%

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

      4. associate-/r/ 36.5%

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

      5. *-commutative 36.5%

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

      6. times-frac 36.5%

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

      7. +-commutative 36.5%

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

      8. associate--l+ 36.5%

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

      9. metadata-eval 36.5%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 36.5%

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

      11. times-frac 36.5%

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

   3. Simplified 36.5%

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

   4. Taylor expanded in t around 0 60.7%

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

      1. unpow2 60.7%

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

   6. Simplified 60.7%

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

   7. Taylor expanded in k around inf 60.7%

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

      1. associate-/l* 60.7%

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

      2. associate-*r* 60.7%

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

      3. associate-/l* 61.1%

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

      4. associate-/l* 61.1%

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

      5. unpow2 61.1%

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

      6. associate-/l* 76.1%

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

      7. *-commutative 76.1%

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

      8. times-frac 76.1%

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

   9. Simplified 82.0%

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

       1. unpow2 82.0%

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

       2. sin-mult 82.2%

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

   11. Applied egg-rr 82.2%

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

       1. div-sub 82.2%

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

       2. +-inverses 82.2%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}}\right) \]

       3. cos-0 82.2%

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

       4. metadata-eval 82.2%

          \[\leadsto 2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}}\right) \]

   13. Simplified 82.2%

       \[\leadsto 2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{\color{blue}{0.5 - \frac{\cos \left(k + k\right)}{2}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.45 \cdot 10^{+159}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{0.5 - \frac{\cos \left(k + k\right)}{2}} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \]

Alternative 8: 76.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \frac{\mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k \cdot t}, 0.3333333333333333 \cdot \frac{\ell}{\frac{t}{\ell}}\right)}{k \cdot k}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{2}{t \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 6.2e-9)
   (*
    2.0
    (/
     (fma (/ l k) (/ l (* k t)) (* 0.3333333333333333 (/ l (/ t l))))
     (* k k)))
   (if (<= k 1.5e+154)
     (* (* (/ l (sin k)) (/ l (tan k))) (/ 2.0 (* t (* k k))))
     (* (/ (pow (/ l k) 2.0) t) -0.3333333333333333))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.2e-9) {
		tmp = 2.0 * (fma((l / k), (l / (k * t)), (0.3333333333333333 * (l / (t / l)))) / (k * k));
	} else if (k <= 1.5e+154) {
		tmp = ((l / sin(k)) * (l / tan(k))) * (2.0 / (t * (k * k)));
	} else {
		tmp = (pow((l / k), 2.0) / t) * -0.3333333333333333;
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (k <= 6.2e-9)
		tmp = Float64(2.0 * Float64(fma(Float64(l / k), Float64(l / Float64(k * t)), Float64(0.3333333333333333 * Float64(l / Float64(t / l)))) / Float64(k * k)));
	elseif (k <= 1.5e+154)
		tmp = Float64(Float64(Float64(l / sin(k)) * Float64(l / tan(k))) * Float64(2.0 / Float64(t * Float64(k * k))));
	else
		tmp = Float64(Float64((Float64(l / k) ^ 2.0) / t) * -0.3333333333333333);
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 6.2e-9], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.5e+154], N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 6.2000000000000001e-9

   1. Initial program 32.8%

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

      1. associate-*l* 32.8%

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

      2. associate-*l* 32.8%

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

      3. associate-/r* 32.8%

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

      4. associate-/r/ 32.8%

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

      5. *-commutative 32.8%

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

      6. times-frac 33.4%

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

      7. +-commutative 33.4%

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

      8. associate--l+ 40.7%

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

      9. metadata-eval 40.7%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 40.7%

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

      11. times-frac 45.1%

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

   3. Simplified 45.1%

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

   4. Taylor expanded in t around 0 79.1%

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

      1. unpow2 79.1%

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

      2. times-frac 79.5%

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

      3. unpow2 79.5%

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

      4. *-commutative 79.5%

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

      5. associate-/r* 76.0%

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

   6. Simplified 76.0%

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

   7. Taylor expanded in k around 0 70.4%

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

      1. unpow2 70.4%

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

   9. Simplified 70.4%

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

   10. Taylor expanded in k around 0 73.8%

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

       1. unpow2 73.8%

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

       2. unpow2 73.8%

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

       3. associate-*r* 73.8%

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

       4. unpow2 73.8%

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

   12. Simplified 73.8%

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

       1. associate-*l/ 73.8%

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

       2. *-un-lft-identity 73.8%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}}}{k \cdot k} \]

       3. times-frac 81.2%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}}{k \cdot k} \]

       4. fma-def 81.2%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k \cdot t}, 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)}}{k \cdot k} \]

       5. associate-/l* 83.6%

          \[\leadsto 2 \cdot \frac{\mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k \cdot t}, 0.3333333333333333 \cdot \color{blue}{\frac{\ell}{\frac{t}{\ell}}}\right)}{k \cdot k} \]

   14. Applied egg-rr 83.6%

       \[\leadsto 2 \cdot \color{blue}{\frac{\mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k \cdot t}, 0.3333333333333333 \cdot \frac{\ell}{\frac{t}{\ell}}\right)}{k \cdot k}} \]

   if 6.2000000000000001e-9 < k < 1.50000000000000013e154

   1. Initial program 34.8%

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

      1. associate-*l* 34.7%

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

      2. associate-*l* 34.7%

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

      3. associate-/r* 34.7%

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

      4. associate-/r/ 34.7%

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

      5. *-commutative 34.7%

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

      6. times-frac 33.4%

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

      7. +-commutative 33.4%

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

      8. associate--l+ 53.2%

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

      9. metadata-eval 53.2%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 53.2%

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

      11. times-frac 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in t around 0 86.0%

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

      1. unpow2 86.0%

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

   6. Simplified 86.0%

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

   if 1.50000000000000013e154 < k

   1. Initial program 44.4%

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

      1. associate-*l* 44.4%

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

      2. associate-*l* 44.4%

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

      3. associate-/r* 44.4%

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

      4. associate-/r/ 44.4%

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

      5. *-commutative 44.4%

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

      6. times-frac 44.6%

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

      7. +-commutative 44.6%

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

      8. associate--l+ 52.9%

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

      9. metadata-eval 52.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 52.9%

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

      11. times-frac 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in t around 0 75.5%

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

      1. unpow2 75.5%

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

   6. Simplified 75.5%

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

   7. Taylor expanded in k around 0 75.5%

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

      1. *-commutative 75.5%

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

      2. fma-def 75.5%

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

      3. unpow2 75.5%

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

      4. unpow2 75.5%

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

      5. unpow2 75.5%

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

      6. times-frac 75.5%

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

   9. Simplified 75.5%

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

   10. Taylor expanded in k around inf 75.5%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   11. Step-by-step derivation

       1. associate-/r* 75.5%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} \]

       2. unpow2 75.5%

          \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t} \]

       3. unpow2 75.5%

          \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} \]

       4. times-frac 79.9%

          \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \]

       5. unpow2 79.9%

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

   12. Simplified 79.9%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \frac{\mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k \cdot t}, 0.3333333333333333 \cdot \frac{\ell}{\frac{t}{\ell}}\right)}{k \cdot k}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{2}{t \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333\\ \end{array} \]

Alternative 9: 83.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{+182}:\\ \;\;\;\;\frac{\ell}{\frac{\sin k}{\frac{\ell}{\tan k}}} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 2.6e+182)
   (* (/ l (/ (sin k) (/ l (tan k)))) (/ 2.0 (* k (* k t))))
   (* (/ (pow (/ l k) 2.0) t) -0.3333333333333333)))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.6e+182) {
		tmp = (l / (sin(k) / (l / tan(k)))) * (2.0 / (k * (k * t)));
	} else {
		tmp = (pow((l / k), 2.0) / t) * -0.3333333333333333;
	}
	return tmp;
}

Fortran

NOTE: l 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 <= 2.6d+182) then
        tmp = (l / (sin(k) / (l / tan(k)))) * (2.0d0 / (k * (k * t)))
    else
        tmp = (((l / k) ** 2.0d0) / t) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.6e+182) {
		tmp = (l / (Math.sin(k) / (l / Math.tan(k)))) * (2.0 / (k * (k * t)));
	} else {
		tmp = (Math.pow((l / k), 2.0) / t) * -0.3333333333333333;
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if k <= 2.6e+182:
		tmp = (l / (math.sin(k) / (l / math.tan(k)))) * (2.0 / (k * (k * t)))
	else:
		tmp = (math.pow((l / k), 2.0) / t) * -0.3333333333333333
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (k <= 2.6e+182)
		tmp = Float64(Float64(l / Float64(sin(k) / Float64(l / tan(k)))) * Float64(2.0 / Float64(k * Float64(k * t))));
	else
		tmp = Float64(Float64((Float64(l / k) ^ 2.0) / t) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.6e+182)
		tmp = (l / (sin(k) / (l / tan(k)))) * (2.0 / (k * (k * t)));
	else
		tmp = (((l / k) ^ 2.0) / t) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 2.6e+182], N[(N[(l / N[(N[Sin[k], $MachinePrecision] / N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.6e182

   1. Initial program 32.7%

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

      1. associate-*l* 32.7%

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

      2. associate-*l* 32.7%

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

      3. associate-/r* 32.7%

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

      4. associate-/r/ 32.7%

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

      5. *-commutative 32.7%

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

      6. times-frac 32.9%

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

      7. +-commutative 32.9%

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

      8. associate--l+ 42.4%

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

      9. metadata-eval 42.4%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 42.4%

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

      11. times-frac 46.1%

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

   3. Simplified 46.1%

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

   4. Taylor expanded in t around 0 84.5%

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

      1. unpow2 84.5%

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

   6. Simplified 84.5%

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

      1. associate-*l/ 84.5%

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

      2. associate-*l* 87.9%

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

   8. Applied egg-rr 87.9%

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

      1. associate-*l/ 87.8%

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

      2. *-commutative 87.8%

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

      3. associate-*l/ 87.4%

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

      4. associate-/l* 87.9%

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

   10. Simplified 87.9%

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

   if 2.6e182 < k

   1. Initial program 49.9%

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

      1. associate-*l* 49.9%

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

      2. associate-*l* 49.9%

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

      3. associate-/r* 49.9%

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

      4. associate-/r/ 49.9%

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

      5. *-commutative 49.9%

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

      6. times-frac 50.1%

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

      7. +-commutative 50.1%

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

      8. associate--l+ 56.8%

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

      9. metadata-eval 56.8%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 56.8%

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

      11. times-frac 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in t around 0 77.1%

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

      1. unpow2 77.1%

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

   6. Simplified 77.1%

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

   7. Taylor expanded in k around 0 77.1%

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

      1. *-commutative 77.1%

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

      2. fma-def 77.1%

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

      3. unpow2 77.1%

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

      4. unpow2 77.1%

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

      5. unpow2 77.1%

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

      6. times-frac 77.1%

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

   9. Simplified 77.1%

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

   10. Taylor expanded in k around inf 77.1%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   11. Step-by-step derivation

       1. associate-/r* 77.1%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} \]

       2. unpow2 77.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t} \]

       3. unpow2 77.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} \]

       4. times-frac 81.9%

          \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \]

       5. unpow2 81.9%

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

   12. Simplified 81.9%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{+182}:\\ \;\;\;\;\frac{\ell}{\frac{\sin k}{\frac{\ell}{\tan k}}} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333\\ \end{array} \]

Alternative 10: 83.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{+182}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 2.7e+182)
   (* (/ (/ 2.0 k) (* k t)) (* (/ l (sin k)) (/ l (tan k))))
   (* (/ (pow (/ l k) 2.0) t) -0.3333333333333333)))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e+182) {
		tmp = ((2.0 / k) / (k * t)) * ((l / sin(k)) * (l / tan(k)));
	} else {
		tmp = (pow((l / k), 2.0) / t) * -0.3333333333333333;
	}
	return tmp;
}

Fortran

NOTE: l 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 <= 2.7d+182) then
        tmp = ((2.0d0 / k) / (k * t)) * ((l / sin(k)) * (l / tan(k)))
    else
        tmp = (((l / k) ** 2.0d0) / t) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e+182) {
		tmp = ((2.0 / k) / (k * t)) * ((l / Math.sin(k)) * (l / Math.tan(k)));
	} else {
		tmp = (Math.pow((l / k), 2.0) / t) * -0.3333333333333333;
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if k <= 2.7e+182:
		tmp = ((2.0 / k) / (k * t)) * ((l / math.sin(k)) * (l / math.tan(k)))
	else:
		tmp = (math.pow((l / k), 2.0) / t) * -0.3333333333333333
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (k <= 2.7e+182)
		tmp = Float64(Float64(Float64(2.0 / k) / Float64(k * t)) * Float64(Float64(l / sin(k)) * Float64(l / tan(k))));
	else
		tmp = Float64(Float64((Float64(l / k) ^ 2.0) / t) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.7e+182)
		tmp = ((2.0 / k) / (k * t)) * ((l / sin(k)) * (l / tan(k)));
	else
		tmp = (((l / k) ^ 2.0) / t) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 2.7e+182], N[(N[(N[(2.0 / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.7000000000000003e182

   1. Initial program 32.7%

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

      1. associate-*l* 32.7%

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

      2. associate-*l* 32.7%

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

      3. associate-/r* 32.7%

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

      4. associate-/r/ 32.7%

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

      5. *-commutative 32.7%

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

      6. times-frac 32.9%

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

      7. +-commutative 32.9%

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

      8. associate--l+ 42.4%

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

      9. metadata-eval 42.4%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 42.4%

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

      11. times-frac 46.1%

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

   3. Simplified 46.1%

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

   4. Taylor expanded in t around 0 84.5%

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

      1. unpow2 84.5%

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

   6. Simplified 84.5%

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

   7. Taylor expanded in k around 0 84.5%

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

      1. unpow2 84.5%

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

      2. associate-*r* 87.8%

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

      3. associate-/r* 88.1%

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

   9. Simplified 88.1%

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

   if 2.7000000000000003e182 < k

   1. Initial program 49.9%

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

      1. associate-*l* 49.9%

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

      2. associate-*l* 49.9%

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

      3. associate-/r* 49.9%

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

      4. associate-/r/ 49.9%

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

      5. *-commutative 49.9%

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

      6. times-frac 50.1%

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

      7. +-commutative 50.1%

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

      8. associate--l+ 56.8%

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

      9. metadata-eval 56.8%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 56.8%

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

      11. times-frac 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in t around 0 77.1%

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

      1. unpow2 77.1%

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

   6. Simplified 77.1%

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

   7. Taylor expanded in k around 0 77.1%

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

      1. *-commutative 77.1%

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

      2. fma-def 77.1%

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

      3. unpow2 77.1%

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

      4. unpow2 77.1%

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

      5. unpow2 77.1%

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

      6. times-frac 77.1%

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

   9. Simplified 77.1%

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

   10. Taylor expanded in k around inf 77.1%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   11. Step-by-step derivation

       1. associate-/r* 77.1%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} \]

       2. unpow2 77.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t} \]

       3. unpow2 77.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} \]

       4. times-frac 81.9%

          \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \]

       5. unpow2 81.9%

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

   12. Simplified 81.9%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{+182}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333\\ \end{array} \]

Alternative 11: 72.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \frac{\mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k \cdot t}, 0.3333333333333333 \cdot \frac{\ell}{\frac{t}{\ell}}\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)}{\frac{t}{{\left(\frac{\ell}{k}\right)}^{2}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 6.2e-9)
   (*
    2.0
    (/
     (fma (/ l k) (/ l (* k t)) (* 0.3333333333333333 (/ l (/ t l))))
     (* k k)))
   (/
    (* 2.0 (+ -0.16666666666666666 (/ 1.0 (* k k))))
    (/ t (pow (/ l k) 2.0)))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.2e-9) {
		tmp = 2.0 * (fma((l / k), (l / (k * t)), (0.3333333333333333 * (l / (t / l)))) / (k * k));
	} else {
		tmp = (2.0 * (-0.16666666666666666 + (1.0 / (k * k)))) / (t / pow((l / k), 2.0));
	}
	return tmp;
}

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (k <= 6.2e-9)
		tmp = Float64(2.0 * Float64(fma(Float64(l / k), Float64(l / Float64(k * t)), Float64(0.3333333333333333 * Float64(l / Float64(t / l)))) / Float64(k * k)));
	else
		tmp = Float64(Float64(2.0 * Float64(-0.16666666666666666 + Float64(1.0 / Float64(k * k)))) / Float64(t / (Float64(l / k) ^ 2.0)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 6.2e-9], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(-0.16666666666666666 + N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t / N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.2000000000000001e-9

   1. Initial program 32.8%

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

      1. associate-*l* 32.8%

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

      2. associate-*l* 32.8%

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

      3. associate-/r* 32.8%

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

      4. associate-/r/ 32.8%

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

      5. *-commutative 32.8%

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

      6. times-frac 33.4%

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

      7. +-commutative 33.4%

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

      8. associate--l+ 40.7%

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

      9. metadata-eval 40.7%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 40.7%

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

      11. times-frac 45.1%

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

   3. Simplified 45.1%

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

   4. Taylor expanded in t around 0 79.1%

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

      1. unpow2 79.1%

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

      2. times-frac 79.5%

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

      3. unpow2 79.5%

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

      4. *-commutative 79.5%

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

      5. associate-/r* 76.0%

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

   6. Simplified 76.0%

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

   7. Taylor expanded in k around 0 70.4%

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

      1. unpow2 70.4%

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

   9. Simplified 70.4%

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

   10. Taylor expanded in k around 0 73.8%

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

       1. unpow2 73.8%

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

       2. unpow2 73.8%

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

       3. associate-*r* 73.8%

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

       4. unpow2 73.8%

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

   12. Simplified 73.8%

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

       1. associate-*l/ 73.8%

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

       2. *-un-lft-identity 73.8%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}}}{k \cdot k} \]

       3. times-frac 81.2%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}}{k \cdot k} \]

       4. fma-def 81.2%

          \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k \cdot t}, 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)}}{k \cdot k} \]

       5. associate-/l* 83.6%

          \[\leadsto 2 \cdot \frac{\mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k \cdot t}, 0.3333333333333333 \cdot \color{blue}{\frac{\ell}{\frac{t}{\ell}}}\right)}{k \cdot k} \]

   14. Applied egg-rr 83.6%

       \[\leadsto 2 \cdot \color{blue}{\frac{\mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k \cdot t}, 0.3333333333333333 \cdot \frac{\ell}{\frac{t}{\ell}}\right)}{k \cdot k}} \]

   if 6.2000000000000001e-9 < k

   1. Initial program 39.6%

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

      1. associate-*l* 39.6%

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

      2. associate-*l* 39.6%

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

      3. associate-/r* 39.6%

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

      4. associate-/r/ 39.6%

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

      5. *-commutative 39.6%

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

      6. times-frac 39.1%

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

      7. +-commutative 39.1%

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

      8. associate--l+ 53.1%

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

      9. metadata-eval 53.1%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 53.1%

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

      11. times-frac 53.1%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in t around 0 80.7%

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

      1. unpow2 80.7%

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

   6. Simplified 80.7%

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

   7. Taylor expanded in k around 0 65.0%

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

      1. *-commutative 65.0%

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

      2. fma-def 65.0%

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

      3. unpow2 65.0%

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

      4. unpow2 65.0%

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

      5. unpow2 65.0%

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

      6. times-frac 70.6%

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

   9. Simplified 70.6%

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

   10. Taylor expanded in l around 0 72.0%

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

       1. associate-/l* 72.0%

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

       2. associate-*r/ 72.0%

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

       3. sub-neg 72.0%

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

       4. unpow2 72.0%

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

       5. associate-/l/ 72.0%

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

       6. metadata-eval 72.0%

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

       7. +-commutative 72.0%

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

       8. associate-/l/ 72.0%

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

       9. *-commutative 72.0%

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

       10. associate-/l* 72.1%

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

       11. unpow2 72.1%

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

       12. unpow2 72.1%

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

       13. times-frac 74.3%

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

       14. unpow2 74.3%

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

   12. Simplified 74.3%

       \[\leadsto \color{blue}{\frac{2 \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)}{\frac{t}{{\left(\frac{\ell}{k}\right)}^{2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \frac{\mathsf{fma}\left(\frac{\ell}{k}, \frac{\ell}{k \cdot t}, 0.3333333333333333 \cdot \frac{\ell}{\frac{t}{\ell}}\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)}{\frac{t}{{\left(\frac{\ell}{k}\right)}^{2}}}\\ \end{array} \]

Alternative 12: 73.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;k \leq 4.6 \cdot 10^{+90}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.75e-94)
   (* (/ 2.0 (* k (* k t))) (* (/ l k) (/ l k)))
   (if (<= k 4.6e+90)
     (* 2.0 (* (* (/ l t) (/ l (* k k))) (/ (cos k) (* k k))))
     (* (/ (pow (/ l k) 2.0) t) -0.3333333333333333))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.75e-94) {
		tmp = (2.0 / (k * (k * t))) * ((l / k) * (l / k));
	} else if (k <= 4.6e+90) {
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (cos(k) / (k * k)));
	} else {
		tmp = (pow((l / k), 2.0) / t) * -0.3333333333333333;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.75d-94) then
        tmp = (2.0d0 / (k * (k * t))) * ((l / k) * (l / k))
    else if (k <= 4.6d+90) then
        tmp = 2.0d0 * (((l / t) * (l / (k * k))) * (cos(k) / (k * k)))
    else
        tmp = (((l / k) ** 2.0d0) / t) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.75e-94) {
		tmp = (2.0 / (k * (k * t))) * ((l / k) * (l / k));
	} else if (k <= 4.6e+90) {
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (Math.cos(k) / (k * k)));
	} else {
		tmp = (Math.pow((l / k), 2.0) / t) * -0.3333333333333333;
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if k <= 1.75e-94:
		tmp = (2.0 / (k * (k * t))) * ((l / k) * (l / k))
	elif k <= 4.6e+90:
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (math.cos(k) / (k * k)))
	else:
		tmp = (math.pow((l / k), 2.0) / t) * -0.3333333333333333
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.75e-94)
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k * t))) * Float64(Float64(l / k) * Float64(l / k)));
	elseif (k <= 4.6e+90)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / t) * Float64(l / Float64(k * k))) * Float64(cos(k) / Float64(k * k))));
	else
		tmp = Float64(Float64((Float64(l / k) ^ 2.0) / t) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.75e-94)
		tmp = (2.0 / (k * (k * t))) * ((l / k) * (l / k));
	elseif (k <= 4.6e+90)
		tmp = 2.0 * (((l / t) * (l / (k * k))) * (cos(k) / (k * k)));
	else
		tmp = (((l / k) ^ 2.0) / t) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.75e-94], N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.6e+90], N[(2.0 * N[(N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.74999999999999999e-94

   1. Initial program 35.0%

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

      1. associate-*l* 35.0%

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

      2. associate-*l* 35.0%

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

      3. associate-/r* 35.0%

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

      4. associate-/r/ 35.0%

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

      5. *-commutative 35.0%

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

      6. times-frac 35.6%

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

      7. +-commutative 35.6%

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

      8. associate--l+ 42.1%

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

      9. metadata-eval 42.1%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 42.1%

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

      11. times-frac 47.2%

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

   3. Simplified 47.2%

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

   4. Taylor expanded in t around 0 83.8%

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

      1. unpow2 83.8%

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

   6. Simplified 83.8%

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

   7. Taylor expanded in k around 0 70.7%

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

      1. unpow2 70.7%

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

      2. unpow2 70.7%

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

      3. times-frac 78.9%

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

   9. Simplified 78.9%

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

   10. Taylor expanded in k around 0 78.9%

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

       1. unpow2 78.9%

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

       2. associate-*r* 80.0%

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

   12. Simplified 80.0%

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

   if 1.74999999999999999e-94 < k < 4.6e90

   1. Initial program 24.2%

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

      1. associate-*l* 24.2%

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

      2. associate-*l* 24.2%

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

      3. associate-/r* 24.2%

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

      4. associate-/r/ 24.2%

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

      5. *-commutative 24.2%

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

      6. times-frac 24.2%

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

      7. +-commutative 24.2%

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

      8. associate--l+ 45.8%

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

      9. metadata-eval 45.8%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 45.8%

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

      11. times-frac 45.8%

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

   3. Simplified 45.8%

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

   4. Taylor expanded in t around 0 93.3%

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

      1. unpow2 93.3%

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

      2. times-frac 93.6%

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

      3. unpow2 93.6%

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

      4. *-commutative 93.6%

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

      5. associate-/r* 91.4%

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

   6. Simplified 91.4%

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

   7. Taylor expanded in k around 0 88.0%

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

      1. unpow2 88.0%

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

      2. *-commutative 88.0%

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

      3. times-frac 90.1%

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

      4. unpow2 90.1%

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

   9. Simplified 90.1%

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

   if 4.6e90 < k

   1. Initial program 44.0%

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

      1. associate-*l* 44.0%

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

      2. associate-*l* 44.0%

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

      3. associate-/r* 44.0%

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

      4. associate-/r/ 44.0%

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

      5. *-commutative 44.0%

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

      6. times-frac 43.1%

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

      7. +-commutative 43.1%

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

      8. associate--l+ 49.3%

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

      9. metadata-eval 49.3%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 49.3%

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

      11. times-frac 49.3%

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

   3. Simplified 49.3%

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

   4. Taylor expanded in t around 0 73.5%

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

      1. unpow2 73.5%

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

   6. Simplified 73.5%

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

   7. Taylor expanded in k around 0 65.8%

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

      1. *-commutative 65.8%

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

      2. fma-def 65.8%

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

      3. unpow2 65.8%

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

      4. unpow2 65.8%

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

      5. unpow2 65.8%

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

      6. times-frac 70.0%

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

   9. Simplified 70.0%

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

   10. Taylor expanded in k around inf 70.0%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   11. Step-by-step derivation

       1. associate-/r* 70.1%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} \]

       2. unpow2 70.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t} \]

       3. unpow2 70.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} \]

       4. times-frac 73.5%

          \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \]

       5. unpow2 73.5%

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

   12. Simplified 73.5%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;k \leq 4.6 \cdot 10^{+90}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333\\ \end{array} \]

Alternative 13: 73.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;k \leq 4.6 \cdot 10^{+90}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 2.7e-94)
   (* (/ 2.0 (* k (* k t))) (* (/ l k) (/ l k)))
   (if (<= k 4.6e+90)
     (* 2.0 (/ (* (cos k) (* (/ l t) (/ l (* k k)))) (* k k)))
     (* (/ (pow (/ l k) 2.0) t) -0.3333333333333333))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e-94) {
		tmp = (2.0 / (k * (k * t))) * ((l / k) * (l / k));
	} else if (k <= 4.6e+90) {
		tmp = 2.0 * ((cos(k) * ((l / t) * (l / (k * k)))) / (k * k));
	} else {
		tmp = (pow((l / k), 2.0) / t) * -0.3333333333333333;
	}
	return tmp;
}

Fortran

NOTE: l 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 <= 2.7d-94) then
        tmp = (2.0d0 / (k * (k * t))) * ((l / k) * (l / k))
    else if (k <= 4.6d+90) then
        tmp = 2.0d0 * ((cos(k) * ((l / t) * (l / (k * k)))) / (k * k))
    else
        tmp = (((l / k) ** 2.0d0) / t) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.7e-94) {
		tmp = (2.0 / (k * (k * t))) * ((l / k) * (l / k));
	} else if (k <= 4.6e+90) {
		tmp = 2.0 * ((Math.cos(k) * ((l / t) * (l / (k * k)))) / (k * k));
	} else {
		tmp = (Math.pow((l / k), 2.0) / t) * -0.3333333333333333;
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if k <= 2.7e-94:
		tmp = (2.0 / (k * (k * t))) * ((l / k) * (l / k))
	elif k <= 4.6e+90:
		tmp = 2.0 * ((math.cos(k) * ((l / t) * (l / (k * k)))) / (k * k))
	else:
		tmp = (math.pow((l / k), 2.0) / t) * -0.3333333333333333
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (k <= 2.7e-94)
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k * t))) * Float64(Float64(l / k) * Float64(l / k)));
	elseif (k <= 4.6e+90)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(Float64(l / t) * Float64(l / Float64(k * k)))) / Float64(k * k)));
	else
		tmp = Float64(Float64((Float64(l / k) ^ 2.0) / t) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.7e-94)
		tmp = (2.0 / (k * (k * t))) * ((l / k) * (l / k));
	elseif (k <= 4.6e+90)
		tmp = 2.0 * ((cos(k) * ((l / t) * (l / (k * k)))) / (k * k));
	else
		tmp = (((l / k) ^ 2.0) / t) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 2.7e-94], N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.6e+90], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.7000000000000001e-94

   1. Initial program 35.0%

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

      1. associate-*l* 35.0%

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

      2. associate-*l* 35.0%

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

      3. associate-/r* 35.0%

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

      4. associate-/r/ 35.0%

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

      5. *-commutative 35.0%

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

      6. times-frac 35.6%

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

      7. +-commutative 35.6%

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

      8. associate--l+ 42.1%

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

      9. metadata-eval 42.1%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 42.1%

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

      11. times-frac 47.2%

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

   3. Simplified 47.2%

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

   4. Taylor expanded in t around 0 83.8%

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

      1. unpow2 83.8%

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

   6. Simplified 83.8%

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

   7. Taylor expanded in k around 0 70.7%

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

      1. unpow2 70.7%

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

      2. unpow2 70.7%

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

      3. times-frac 78.9%

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

   9. Simplified 78.9%

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

   10. Taylor expanded in k around 0 78.9%

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

       1. unpow2 78.9%

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

       2. associate-*r* 80.0%

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

   12. Simplified 80.0%

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

   if 2.7000000000000001e-94 < k < 4.6e90

   1. Initial program 24.2%

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

      1. associate-*l* 24.2%

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

      2. associate-*l* 24.2%

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

      3. associate-/r* 24.2%

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

      4. associate-/r/ 24.2%

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

      5. *-commutative 24.2%

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

      6. times-frac 24.2%

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

      7. +-commutative 24.2%

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

      8. associate--l+ 45.8%

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

      9. metadata-eval 45.8%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 45.8%

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

      11. times-frac 45.8%

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

   3. Simplified 45.8%

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

   4. Taylor expanded in t around 0 93.5%

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

      1. unpow2 93.5%

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

   6. Simplified 93.5%

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

   7. Taylor expanded in k around inf 93.3%

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

      1. associate-/l* 93.3%

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

      2. associate-*r* 91.0%

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

      3. associate-/l* 89.1%

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

      4. associate-/l* 89.1%

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

      5. unpow2 89.1%

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

      6. associate-/l* 91.2%

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

      7. *-commutative 91.2%

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

      8. times-frac 93.6%

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

   9. Simplified 97.6%

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

       1. associate-*r/ 97.6%

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

   11. Applied egg-rr 97.6%

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

   12. Taylor expanded in k around 0 90.1%

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

       1. unpow2 90.1%

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

   14. Simplified 90.1%

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

   if 4.6e90 < k

   1. Initial program 44.0%

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

      1. associate-*l* 44.0%

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

      2. associate-*l* 44.0%

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

      3. associate-/r* 44.0%

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

      4. associate-/r/ 44.0%

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

      5. *-commutative 44.0%

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

      6. times-frac 43.1%

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

      7. +-commutative 43.1%

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

      8. associate--l+ 49.3%

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

      9. metadata-eval 49.3%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 49.3%

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

      11. times-frac 49.3%

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

   3. Simplified 49.3%

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

   4. Taylor expanded in t around 0 73.5%

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

      1. unpow2 73.5%

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

   6. Simplified 73.5%

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

   7. Taylor expanded in k around 0 65.8%

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

      1. *-commutative 65.8%

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

      2. fma-def 65.8%

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

      3. unpow2 65.8%

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

      4. unpow2 65.8%

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

      5. unpow2 65.8%

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

      6. times-frac 70.0%

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

   9. Simplified 70.0%

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

   10. Taylor expanded in k around inf 70.0%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   11. Step-by-step derivation

       1. associate-/r* 70.1%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} \]

       2. unpow2 70.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t} \]

       3. unpow2 70.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} \]

       4. times-frac 73.5%

          \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \]

       5. unpow2 73.5%

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

   12. Simplified 73.5%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;k \leq 4.6 \cdot 10^{+90}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333\\ \end{array} \]

Alternative 14: 73.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := {\left(\frac{\ell}{k}\right)}^{2}\\ \mathbf{if}\;k \leq 1.85 \cdot 10^{-94}:\\ \;\;\;\;2 \cdot \frac{t_1 - \ell \cdot \left(\ell \cdot -0.3333333333333333\right)}{k \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 4.6 \cdot 10^{+90}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ l k) 2.0)))
   (if (<= k 1.85e-94)
     (* 2.0 (/ (- t_1 (* l (* l -0.3333333333333333))) (* k (* k t))))
     (if (<= k 4.6e+90)
       (* 2.0 (/ (* (cos k) (* (/ l t) (/ l (* k k)))) (* k k)))
       (* (/ t_1 t) -0.3333333333333333)))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = pow((l / k), 2.0);
	double tmp;
	if (k <= 1.85e-94) {
		tmp = 2.0 * ((t_1 - (l * (l * -0.3333333333333333))) / (k * (k * t)));
	} else if (k <= 4.6e+90) {
		tmp = 2.0 * ((cos(k) * ((l / t) * (l / (k * k)))) / (k * k));
	} else {
		tmp = (t_1 / t) * -0.3333333333333333;
	}
	return tmp;
}

Fortran

NOTE: l 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) ** 2.0d0
    if (k <= 1.85d-94) then
        tmp = 2.0d0 * ((t_1 - (l * (l * (-0.3333333333333333d0)))) / (k * (k * t)))
    else if (k <= 4.6d+90) then
        tmp = 2.0d0 * ((cos(k) * ((l / t) * (l / (k * k)))) / (k * k))
    else
        tmp = (t_1 / t) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow((l / k), 2.0);
	double tmp;
	if (k <= 1.85e-94) {
		tmp = 2.0 * ((t_1 - (l * (l * -0.3333333333333333))) / (k * (k * t)));
	} else if (k <= 4.6e+90) {
		tmp = 2.0 * ((Math.cos(k) * ((l / t) * (l / (k * k)))) / (k * k));
	} else {
		tmp = (t_1 / t) * -0.3333333333333333;
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	t_1 = math.pow((l / k), 2.0)
	tmp = 0
	if k <= 1.85e-94:
		tmp = 2.0 * ((t_1 - (l * (l * -0.3333333333333333))) / (k * (k * t)))
	elif k <= 4.6e+90:
		tmp = 2.0 * ((math.cos(k) * ((l / t) * (l / (k * k)))) / (k * k))
	else:
		tmp = (t_1 / t) * -0.3333333333333333
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64(l / k) ^ 2.0
	tmp = 0.0
	if (k <= 1.85e-94)
		tmp = Float64(2.0 * Float64(Float64(t_1 - Float64(l * Float64(l * -0.3333333333333333))) / Float64(k * Float64(k * t))));
	elseif (k <= 4.6e+90)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * Float64(Float64(l / t) * Float64(l / Float64(k * k)))) / Float64(k * k)));
	else
		tmp = Float64(Float64(t_1 / t) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	t_1 = (l / k) ^ 2.0;
	tmp = 0.0;
	if (k <= 1.85e-94)
		tmp = 2.0 * ((t_1 - (l * (l * -0.3333333333333333))) / (k * (k * t)));
	elseif (k <= 4.6e+90)
		tmp = 2.0 * ((cos(k) * ((l / t) * (l / (k * k)))) / (k * k));
	else
		tmp = (t_1 / t) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 1.85e-94], N[(2.0 * N[(N[(t$95$1 - N[(l * N[(l * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.6e+90], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.8499999999999999e-94

   1. Initial program 35.0%

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

      1. associate-*l* 35.0%

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

      2. associate-*l* 35.0%

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

      3. associate-/r* 35.0%

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

      4. associate-/r/ 35.0%

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

      5. *-commutative 35.0%

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

      6. times-frac 35.6%

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

      7. +-commutative 35.6%

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

      8. associate--l+ 42.1%

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

      9. metadata-eval 42.1%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 42.1%

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

      11. times-frac 47.2%

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

   3. Simplified 47.2%

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

   4. Taylor expanded in t around 0 77.4%

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

      1. unpow2 77.4%

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

      2. times-frac 77.8%

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

      3. unpow2 77.8%

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

      4. *-commutative 77.8%

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

      5. associate-/r* 74.4%

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

   6. Simplified 74.4%

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

   7. Taylor expanded in k around 0 68.0%

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

      1. unpow2 68.0%

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

   9. Simplified 68.0%

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

   10. Taylor expanded in k around 0 71.2%

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

       1. unpow2 71.2%

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

       2. unpow2 71.2%

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

       3. associate-*r* 71.3%

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

       4. unpow2 71.3%

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

   12. Simplified 71.3%

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

   13. Taylor expanded in t around -inf 71.0%

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

       1. associate-*r/ 71.0%

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

       2. mul-1-neg 71.0%

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

       3. unpow2 71.0%

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

       4. unpow2 71.0%

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

       5. times-frac 76.7%

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

       6. unpow2 76.7%

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

       7. neg-mul-1 76.7%

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

       8. unsub-neg 76.7%

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

       9. *-commutative 76.7%

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

       10. unpow2 76.7%

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

       11. associate-*l* 76.7%

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

       12. unpow2 76.7%

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

       13. associate-*r* 78.0%

           \[\leadsto 2 \cdot \frac{-\left(\ell \cdot \left(\ell \cdot -0.3333333333333333\right) - {\left(\frac{\ell}{k}\right)}^{2}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   15. Simplified 78.0%

       \[\leadsto 2 \cdot \color{blue}{\frac{-\left(\ell \cdot \left(\ell \cdot -0.3333333333333333\right) - {\left(\frac{\ell}{k}\right)}^{2}\right)}{k \cdot \left(k \cdot t\right)}} \]

   if 1.8499999999999999e-94 < k < 4.6e90

   1. Initial program 24.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 24.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 24.2%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 24.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 24.2%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 24.2%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 24.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 24.2%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 45.8%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 45.8%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 45.8%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 45.8%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 45.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 93.5%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 93.5%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 93.5%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   7. Taylor expanded in k around inf 93.3%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   8. Step-by-step derivation

      1. associate-/l* 93.3%

         \[\leadsto 2 \cdot \color{blue}{\frac{\cos k}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{{\ell}^{2}}}} \]

      2. associate-*r* 91.0%

         \[\leadsto 2 \cdot \frac{\cos k}{\frac{\color{blue}{\left({k}^{2} \cdot {\sin k}^{2}\right) \cdot t}}{{\ell}^{2}}} \]

      3. associate-/l* 89.1%

         \[\leadsto 2 \cdot \frac{\cos k}{\color{blue}{\frac{{k}^{2} \cdot {\sin k}^{2}}{\frac{{\ell}^{2}}{t}}}} \]

      4. associate-/l* 89.1%

         \[\leadsto 2 \cdot \color{blue}{\frac{\cos k \cdot \frac{{\ell}^{2}}{t}}{{k}^{2} \cdot {\sin k}^{2}}} \]

      5. unpow2 89.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}}{{k}^{2} \cdot {\sin k}^{2}} \]

      6. associate-/l* 91.2%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{{k}^{2} \cdot {\sin k}^{2}} \]

      7. *-commutative 91.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}} \cdot \cos k}}{{k}^{2} \cdot {\sin k}^{2}} \]

      8. times-frac 93.6%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   9. Simplified 97.6%

      \[\leadsto \color{blue}{2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 97.6%

          \[\leadsto 2 \cdot \color{blue}{\frac{\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \cos k}{{\sin k}^{2}}} \]

   11. Applied egg-rr 97.6%

       \[\leadsto 2 \cdot \color{blue}{\frac{\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \cos k}{{\sin k}^{2}}} \]

   12. Taylor expanded in k around 0 90.1%

       \[\leadsto 2 \cdot \frac{\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \cos k}{\color{blue}{{k}^{2}}} \]
   13. Step-by-step derivation

       1. unpow2 90.1%

          \[\leadsto 2 \cdot \frac{\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \cos k}{\color{blue}{k \cdot k}} \]

   14. Simplified 90.1%

       \[\leadsto 2 \cdot \frac{\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \cos k}{\color{blue}{k \cdot k}} \]

   if 4.6e90 < k

   1. Initial program 44.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 44.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 44.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 44.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 44.0%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 44.0%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 43.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 43.1%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 49.3%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 49.3%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 49.3%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 49.3%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 49.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 73.5%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 73.5%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 73.5%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   7. Taylor expanded in k around 0 65.8%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(-0.16666666666666666 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 65.8%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{{\ell}^{2} \cdot -0.16666666666666666} + \frac{{\ell}^{2}}{{k}^{2}}\right) \]

      2. fma-def 65.8%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right)} \]

      3. unpow2 65.8%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right) \]

      4. unpow2 65.8%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \]

      5. unpow2 65.8%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \]

      6. times-frac 70.0%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}\right) \]

   9. Simplified 70.0%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \]

   10. Taylor expanded in k around inf 70.0%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   11. Step-by-step derivation

       1. associate-/r* 70.1%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} \]

       2. unpow2 70.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t} \]

       3. unpow2 70.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} \]

       4. times-frac 73.5%

          \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \]

       5. unpow2 73.5%

          \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{t} \]

   12. Simplified 73.5%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{-94}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} - \ell \cdot \left(\ell \cdot -0.3333333333333333\right)}{k \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 4.6 \cdot 10^{+90}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333\\ \end{array} \]

Alternative 15: 71.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{1}{k \cdot k}\\ \mathbf{if}\;k \leq 4.2 \cdot 10^{-10}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(-0.16666666666666666 + t_1\right)}{\frac{t}{{\left(\frac{\ell}{k}\right)}^{2}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ 1.0 (* k k))))
   (if (<= k 4.2e-10)
     (*
      2.0
      (*
       t_1
       (+ (* (/ l k) (/ l (* k t))) (* 0.3333333333333333 (/ (* l l) t)))))
     (/ (* 2.0 (+ -0.16666666666666666 t_1)) (/ t (pow (/ l k) 2.0))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = 1.0 / (k * k);
	double tmp;
	if (k <= 4.2e-10) {
		tmp = 2.0 * (t_1 * (((l / k) * (l / (k * t))) + (0.3333333333333333 * ((l * l) / t))));
	} else {
		tmp = (2.0 * (-0.16666666666666666 + t_1)) / (t / pow((l / k), 2.0));
	}
	return tmp;
}

Fortran

NOTE: l 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 = 1.0d0 / (k * k)
    if (k <= 4.2d-10) then
        tmp = 2.0d0 * (t_1 * (((l / k) * (l / (k * t))) + (0.3333333333333333d0 * ((l * l) / t))))
    else
        tmp = (2.0d0 * ((-0.16666666666666666d0) + t_1)) / (t / ((l / k) ** 2.0d0))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = 1.0 / (k * k);
	double tmp;
	if (k <= 4.2e-10) {
		tmp = 2.0 * (t_1 * (((l / k) * (l / (k * t))) + (0.3333333333333333 * ((l * l) / t))));
	} else {
		tmp = (2.0 * (-0.16666666666666666 + t_1)) / (t / Math.pow((l / k), 2.0));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	t_1 = 1.0 / (k * k)
	tmp = 0
	if k <= 4.2e-10:
		tmp = 2.0 * (t_1 * (((l / k) * (l / (k * t))) + (0.3333333333333333 * ((l * l) / t))))
	else:
		tmp = (2.0 * (-0.16666666666666666 + t_1)) / (t / math.pow((l / k), 2.0))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64(1.0 / Float64(k * k))
	tmp = 0.0
	if (k <= 4.2e-10)
		tmp = Float64(2.0 * Float64(t_1 * Float64(Float64(Float64(l / k) * Float64(l / Float64(k * t))) + Float64(0.3333333333333333 * Float64(Float64(l * l) / t)))));
	else
		tmp = Float64(Float64(2.0 * Float64(-0.16666666666666666 + t_1)) / Float64(t / (Float64(l / k) ^ 2.0)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	t_1 = 1.0 / (k * k);
	tmp = 0.0;
	if (k <= 4.2e-10)
		tmp = 2.0 * (t_1 * (((l / k) * (l / (k * t))) + (0.3333333333333333 * ((l * l) / t))));
	else
		tmp = (2.0 * (-0.16666666666666666 + t_1)) / (t / ((l / k) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 4.2e-10], N[(2.0 * N[(t$95$1 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(-0.16666666666666666 + t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t / N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.2e-10

   1. Initial program 32.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 32.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 32.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 32.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 32.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 32.8%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 33.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 33.4%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 40.7%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 40.7%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 40.7%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 45.1%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 45.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 79.1%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. unpow2 79.1%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 79.5%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 79.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2} \cdot t}\right) \]

      4. *-commutative 79.5%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

      5. associate-/r* 76.0%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}}\right) \]

   6. Simplified 76.0%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 70.4%

      \[\leadsto 2 \cdot \left(\color{blue}{\frac{1}{{k}^{2}}} \cdot \frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}\right) \]
   8. Step-by-step derivation

      1. unpow2 70.4%

         \[\leadsto 2 \cdot \left(\frac{1}{\color{blue}{k \cdot k}} \cdot \frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}\right) \]

   9. Simplified 70.4%

      \[\leadsto 2 \cdot \left(\color{blue}{\frac{1}{k \cdot k}} \cdot \frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}\right) \]

   10. Taylor expanded in k around 0 73.8%

       \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
   11. Step-by-step derivation

       1. unpow2 73.8%

          \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

       2. unpow2 73.8%

          \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

       3. associate-*r* 73.8%

          \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

       4. unpow2 73.8%

          \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + 0.3333333333333333 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right)\right) \]

   12. Simplified 73.8%

       \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)}\right) \]
   13. Step-by-step derivation

       1. times-frac 80.8%

          \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)\right) \]

   14. Applied egg-rr 80.8%

       \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)\right) \]

   if 4.2e-10 < k

   1. Initial program 39.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 39.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 39.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 39.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 39.6%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 39.6%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 39.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 39.1%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 53.1%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 53.1%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 53.1%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 53.1%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 53.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 80.7%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 80.7%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 80.7%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   7. Taylor expanded in k around 0 65.0%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(-0.16666666666666666 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 65.0%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{{\ell}^{2} \cdot -0.16666666666666666} + \frac{{\ell}^{2}}{{k}^{2}}\right) \]

      2. fma-def 65.0%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right)} \]

      3. unpow2 65.0%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right) \]

      4. unpow2 65.0%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \]

      5. unpow2 65.0%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \]

      6. times-frac 70.6%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}\right) \]

   9. Simplified 70.6%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \]

   10. Taylor expanded in l around 0 72.0%

       \[\leadsto \color{blue}{2 \cdot \frac{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right) \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]
   11. Step-by-step derivation

       1. associate-/l* 72.0%

          \[\leadsto 2 \cdot \color{blue}{\frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]

       2. associate-*r/ 72.0%

          \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{1}{{k}^{2}} - 0.16666666666666666\right)}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}}} \]

       3. sub-neg 72.0%

          \[\leadsto \frac{2 \cdot \color{blue}{\left(\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)\right)}}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \]

       4. unpow2 72.0%

          \[\leadsto \frac{2 \cdot \left(\frac{1}{\color{blue}{k \cdot k}} + \left(-0.16666666666666666\right)\right)}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \]

       5. associate-/l/ 72.0%

          \[\leadsto \frac{2 \cdot \left(\color{blue}{\frac{\frac{1}{k}}{k}} + \left(-0.16666666666666666\right)\right)}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \]

       6. metadata-eval 72.0%

          \[\leadsto \frac{2 \cdot \left(\frac{\frac{1}{k}}{k} + \color{blue}{-0.16666666666666666}\right)}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \]

       7. +-commutative 72.0%

          \[\leadsto \frac{2 \cdot \color{blue}{\left(-0.16666666666666666 + \frac{\frac{1}{k}}{k}\right)}}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \]

       8. associate-/l/ 72.0%

          \[\leadsto \frac{2 \cdot \left(-0.16666666666666666 + \color{blue}{\frac{1}{k \cdot k}}\right)}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \]

       9. *-commutative 72.0%

          \[\leadsto \frac{2 \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}}} \]

       10. associate-/l* 72.1%

           \[\leadsto \frac{2 \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)}{\color{blue}{\frac{t}{\frac{{\ell}^{2}}{{k}^{2}}}}} \]

       11. unpow2 72.1%

           \[\leadsto \frac{2 \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)}{\frac{t}{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}} \]

       12. unpow2 72.1%

           \[\leadsto \frac{2 \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)}{\frac{t}{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}} \]

       13. times-frac 74.3%

           \[\leadsto \frac{2 \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)}{\frac{t}{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}} \]

       14. unpow2 74.3%

           \[\leadsto \frac{2 \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)}{\frac{t}{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}} \]

   12. Simplified 74.3%

       \[\leadsto \color{blue}{\frac{2 \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)}{\frac{t}{{\left(\frac{\ell}{k}\right)}^{2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-10}:\\ \;\;\;\;2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)}{\frac{t}{{\left(\frac{\ell}{k}\right)}^{2}}}\\ \end{array} \]

Alternative 16: 71.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{+20}:\\ \;\;\;\;2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 6.8e+20)
   (*
    2.0
    (*
     (/ 1.0 (* k k))
     (+ (* (/ l k) (/ l (* k t))) (* 0.3333333333333333 (/ (* l l) t)))))
   (* (/ (pow (/ l k) 2.0) t) -0.3333333333333333)))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.8e+20) {
		tmp = 2.0 * ((1.0 / (k * k)) * (((l / k) * (l / (k * t))) + (0.3333333333333333 * ((l * l) / t))));
	} else {
		tmp = (pow((l / k), 2.0) / t) * -0.3333333333333333;
	}
	return tmp;
}

Fortran

NOTE: l 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 <= 6.8d+20) then
        tmp = 2.0d0 * ((1.0d0 / (k * k)) * (((l / k) * (l / (k * t))) + (0.3333333333333333d0 * ((l * l) / t))))
    else
        tmp = (((l / k) ** 2.0d0) / t) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.8e+20) {
		tmp = 2.0 * ((1.0 / (k * k)) * (((l / k) * (l / (k * t))) + (0.3333333333333333 * ((l * l) / t))));
	} else {
		tmp = (Math.pow((l / k), 2.0) / t) * -0.3333333333333333;
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if k <= 6.8e+20:
		tmp = 2.0 * ((1.0 / (k * k)) * (((l / k) * (l / (k * t))) + (0.3333333333333333 * ((l * l) / t))))
	else:
		tmp = (math.pow((l / k), 2.0) / t) * -0.3333333333333333
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (k <= 6.8e+20)
		tmp = Float64(2.0 * Float64(Float64(1.0 / Float64(k * k)) * Float64(Float64(Float64(l / k) * Float64(l / Float64(k * t))) + Float64(0.3333333333333333 * Float64(Float64(l * l) / t)))));
	else
		tmp = Float64(Float64((Float64(l / k) ^ 2.0) / t) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6.8e+20)
		tmp = 2.0 * ((1.0 / (k * k)) * (((l / k) * (l / (k * t))) + (0.3333333333333333 * ((l * l) / t))));
	else
		tmp = (((l / k) ^ 2.0) / t) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 6.8e+20], N[(2.0 * N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.8e20

   1. Initial program 33.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 33.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 33.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 33.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 33.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 33.4%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 33.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 33.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 41.6%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 41.6%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 41.6%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 46.0%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 46.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 79.5%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. unpow2 79.5%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 79.8%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 79.8%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2} \cdot t}\right) \]

      4. *-commutative 79.8%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

      5. associate-/r* 76.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}}\right) \]

   6. Simplified 76.4%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 70.5%

      \[\leadsto 2 \cdot \left(\color{blue}{\frac{1}{{k}^{2}}} \cdot \frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}\right) \]
   8. Step-by-step derivation

      1. unpow2 70.5%

         \[\leadsto 2 \cdot \left(\frac{1}{\color{blue}{k \cdot k}} \cdot \frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}\right) \]

   9. Simplified 70.5%

      \[\leadsto 2 \cdot \left(\color{blue}{\frac{1}{k \cdot k}} \cdot \frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}\right) \]

   10. Taylor expanded in k around 0 73.8%

       \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
   11. Step-by-step derivation

       1. unpow2 73.8%

          \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

       2. unpow2 73.8%

          \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

       3. associate-*r* 73.8%

          \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

       4. unpow2 73.8%

          \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + 0.3333333333333333 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right)\right) \]

   12. Simplified 73.8%

       \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)}\right) \]
   13. Step-by-step derivation

       1. times-frac 80.7%

          \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)\right) \]

   14. Applied egg-rr 80.7%

       \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)\right) \]

   if 6.8e20 < k

   1. Initial program 38.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 38.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 38.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 38.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 38.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 38.4%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 37.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 37.8%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 51.0%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 51.0%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 51.0%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 51.0%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 51.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 79.8%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 79.8%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 79.8%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   7. Taylor expanded in k around 0 65.8%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(-0.16666666666666666 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 65.8%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{{\ell}^{2} \cdot -0.16666666666666666} + \frac{{\ell}^{2}}{{k}^{2}}\right) \]

      2. fma-def 65.8%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right)} \]

      3. unpow2 65.8%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right) \]

      4. unpow2 65.8%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \]

      5. unpow2 65.8%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \]

      6. times-frac 71.7%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}\right) \]

   9. Simplified 71.7%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \]

   10. Taylor expanded in k around inf 71.7%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   11. Step-by-step derivation

       1. associate-/r* 71.7%

          \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} \]

       2. unpow2 71.7%

          \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t} \]

       3. unpow2 71.7%

          \[\leadsto -0.3333333333333333 \cdot \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} \]

       4. times-frac 74.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \]

       5. unpow2 74.1%

          \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{t} \]

   12. Simplified 74.1%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{+20}:\\ \;\;\;\;2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.3333333333333333\\ \end{array} \]

Alternative 17: 71.2% accurate, 15.6× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{1}{k \cdot k}\\ \mathbf{if}\;k \leq 3.5 \cdot 10^{-15}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666 + t_1}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ 1.0 (* k k))))
   (if (<= k 3.5e-15)
     (*
      2.0
      (*
       t_1
       (+ (* (/ l k) (/ l (* k t))) (* 0.3333333333333333 (/ (* l l) t)))))
     (* 2.0 (* (/ (+ -0.16666666666666666 t_1) (* k k)) (* l (/ l t)))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double t_1 = 1.0 / (k * k);
	double tmp;
	if (k <= 3.5e-15) {
		tmp = 2.0 * (t_1 * (((l / k) * (l / (k * t))) + (0.3333333333333333 * ((l * l) / t))));
	} else {
		tmp = 2.0 * (((-0.16666666666666666 + t_1) / (k * k)) * (l * (l / t)));
	}
	return tmp;
}

Fortran

NOTE: l 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 = 1.0d0 / (k * k)
    if (k <= 3.5d-15) then
        tmp = 2.0d0 * (t_1 * (((l / k) * (l / (k * t))) + (0.3333333333333333d0 * ((l * l) / t))))
    else
        tmp = 2.0d0 * ((((-0.16666666666666666d0) + t_1) / (k * k)) * (l * (l / t)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double t_1 = 1.0 / (k * k);
	double tmp;
	if (k <= 3.5e-15) {
		tmp = 2.0 * (t_1 * (((l / k) * (l / (k * t))) + (0.3333333333333333 * ((l * l) / t))));
	} else {
		tmp = 2.0 * (((-0.16666666666666666 + t_1) / (k * k)) * (l * (l / t)));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	t_1 = 1.0 / (k * k)
	tmp = 0
	if k <= 3.5e-15:
		tmp = 2.0 * (t_1 * (((l / k) * (l / (k * t))) + (0.3333333333333333 * ((l * l) / t))))
	else:
		tmp = 2.0 * (((-0.16666666666666666 + t_1) / (k * k)) * (l * (l / t)))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	t_1 = Float64(1.0 / Float64(k * k))
	tmp = 0.0
	if (k <= 3.5e-15)
		tmp = Float64(2.0 * Float64(t_1 * Float64(Float64(Float64(l / k) * Float64(l / Float64(k * t))) + Float64(0.3333333333333333 * Float64(Float64(l * l) / t)))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(-0.16666666666666666 + t_1) / Float64(k * k)) * Float64(l * Float64(l / t))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	t_1 = 1.0 / (k * k);
	tmp = 0.0;
	if (k <= 3.5e-15)
		tmp = 2.0 * (t_1 * (((l / k) * (l / (k * t))) + (0.3333333333333333 * ((l * l) / t))));
	else
		tmp = 2.0 * (((-0.16666666666666666 + t_1) / (k * k)) * (l * (l / t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 3.5e-15], N[(2.0 * N[(t$95$1 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(-0.16666666666666666 + t$95$1), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.5000000000000001e-15

   1. Initial program 33.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 33.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 33.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 33.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 33.0%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 33.0%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 33.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 33.5%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 40.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 40.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 40.9%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 45.4%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 45.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 79.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. unpow2 79.0%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 79.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 79.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2} \cdot t}\right) \]

      4. *-commutative 79.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

      5. associate-/r* 75.9%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}}\right) \]

   6. Simplified 75.9%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 70.2%

      \[\leadsto 2 \cdot \left(\color{blue}{\frac{1}{{k}^{2}}} \cdot \frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}\right) \]
   8. Step-by-step derivation

      1. unpow2 70.2%

         \[\leadsto 2 \cdot \left(\frac{1}{\color{blue}{k \cdot k}} \cdot \frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}\right) \]

   9. Simplified 70.2%

      \[\leadsto 2 \cdot \left(\color{blue}{\frac{1}{k \cdot k}} \cdot \frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}\right) \]

   10. Taylor expanded in k around 0 73.6%

       \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
   11. Step-by-step derivation

       1. unpow2 73.6%

          \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

       2. unpow2 73.6%

          \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

       3. associate-*r* 73.6%

          \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

       4. unpow2 73.6%

          \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + 0.3333333333333333 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right)\right) \]

   12. Simplified 73.6%

       \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)}\right) \]
   13. Step-by-step derivation

       1. times-frac 80.7%

          \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)\right) \]

   14. Applied egg-rr 80.7%

       \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)\right) \]

   if 3.5000000000000001e-15 < k

   1. Initial program 39.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 39.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 39.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 39.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 39.1%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 39.1%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 38.5%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 38.5%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 52.3%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 52.3%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 52.3%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 52.3%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 81.0%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 81.0%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   7. Taylor expanded in k around 0 65.5%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(-0.16666666666666666 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 65.5%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{{\ell}^{2} \cdot -0.16666666666666666} + \frac{{\ell}^{2}}{{k}^{2}}\right) \]

      2. fma-def 65.5%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right)} \]

      3. unpow2 65.5%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right) \]

      4. unpow2 65.5%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \]

      5. unpow2 65.5%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \]

      6. times-frac 71.0%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}\right) \]

   9. Simplified 71.0%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \]

   10. Taylor expanded in l around 0 72.4%

       \[\leadsto \color{blue}{2 \cdot \frac{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right) \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]
   11. Step-by-step derivation

       1. times-frac 72.4%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t}\right)} \]

       2. sub-neg 72.4%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t}\right) \]

       3. unpow2 72.4%

          \[\leadsto 2 \cdot \left(\frac{\frac{1}{\color{blue}{k \cdot k}} + \left(-0.16666666666666666\right)}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t}\right) \]

       4. metadata-eval 72.4%

          \[\leadsto 2 \cdot \left(\frac{\frac{1}{k \cdot k} + \color{blue}{-0.16666666666666666}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t}\right) \]

       5. unpow2 72.4%

          \[\leadsto 2 \cdot \left(\frac{\frac{1}{k \cdot k} + -0.16666666666666666}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{t}\right) \]

       6. unpow2 72.4%

          \[\leadsto 2 \cdot \left(\frac{\frac{1}{k \cdot k} + -0.16666666666666666}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right) \]

       7. associate-*r/ 72.5%

          \[\leadsto 2 \cdot \left(\frac{\frac{1}{k \cdot k} + -0.16666666666666666}{k \cdot k} \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)}\right) \]

   12. Simplified 72.5%

       \[\leadsto \color{blue}{2 \cdot \left(\frac{\frac{1}{k \cdot k} + -0.16666666666666666}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-15}:\\ \;\;\;\;2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666 + \frac{1}{k \cdot k}}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)\\ \end{array} \]

Alternative 18: 71.6% accurate, 20.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-51}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666 + \frac{1}{k \cdot k}}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 5e-51)
   (* (/ 2.0 (* k (* k t))) (* (/ l k) (/ l k)))
   (*
    2.0
    (* (/ (+ -0.16666666666666666 (/ 1.0 (* k k))) (* k k)) (* l (/ l t))))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-51) {
		tmp = (2.0 / (k * (k * t))) * ((l / k) * (l / k));
	} else {
		tmp = 2.0 * (((-0.16666666666666666 + (1.0 / (k * k))) / (k * k)) * (l * (l / t)));
	}
	return tmp;
}

Fortran

NOTE: l 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 <= 5d-51) then
        tmp = (2.0d0 / (k * (k * t))) * ((l / k) * (l / k))
    else
        tmp = 2.0d0 * ((((-0.16666666666666666d0) + (1.0d0 / (k * k))) / (k * k)) * (l * (l / t)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-51) {
		tmp = (2.0 / (k * (k * t))) * ((l / k) * (l / k));
	} else {
		tmp = 2.0 * (((-0.16666666666666666 + (1.0 / (k * k))) / (k * k)) * (l * (l / t)));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if k <= 5e-51:
		tmp = (2.0 / (k * (k * t))) * ((l / k) * (l / k))
	else:
		tmp = 2.0 * (((-0.16666666666666666 + (1.0 / (k * k))) / (k * k)) * (l * (l / t)))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (k <= 5e-51)
		tmp = Float64(Float64(2.0 / Float64(k * Float64(k * t))) * Float64(Float64(l / k) * Float64(l / k)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(-0.16666666666666666 + Float64(1.0 / Float64(k * k))) / Float64(k * k)) * Float64(l * Float64(l / t))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5e-51)
		tmp = (2.0 / (k * (k * t))) * ((l / k) * (l / k));
	else
		tmp = 2.0 * (((-0.16666666666666666 + (1.0 / (k * k))) / (k * k)) * (l * (l / t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 5e-51], N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(-0.16666666666666666 + N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.00000000000000004e-51

   1. Initial program 33.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 33.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 33.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 33.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 33.7%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 33.7%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 34.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 34.3%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 41.7%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 41.7%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 41.7%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 46.4%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 46.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 84.1%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 84.1%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 84.1%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   7. Taylor expanded in k around 0 71.7%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 71.7%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \]

      2. unpow2 71.7%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \]

      3. times-frac 79.5%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \]

   9. Simplified 79.5%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \]

   10. Taylor expanded in k around 0 79.5%

       \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]
   11. Step-by-step derivation

       1. unpow2 79.5%

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

       2. associate-*r* 80.5%

          \[\leadsto \frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

   12. Simplified 80.5%

       \[\leadsto \frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

   if 5.00000000000000004e-51 < k

   1. Initial program 36.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 36.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 36.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 36.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 36.6%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 36.6%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 36.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 36.1%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 49.1%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 49.1%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 49.1%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 49.1%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 49.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 82.7%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 82.7%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 82.7%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   7. Taylor expanded in k around 0 61.4%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(-0.16666666666666666 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 61.4%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{{\ell}^{2} \cdot -0.16666666666666666} + \frac{{\ell}^{2}}{{k}^{2}}\right) \]

      2. fma-def 61.4%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right)} \]

      3. unpow2 61.4%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right) \]

      4. unpow2 61.4%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \]

      5. unpow2 61.4%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \]

      6. times-frac 66.0%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}\right) \]

   9. Simplified 66.0%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \]

   10. Taylor expanded in l around 0 75.5%

       \[\leadsto \color{blue}{2 \cdot \frac{\left(\frac{1}{{k}^{2}} - 0.16666666666666666\right) \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]
   11. Step-by-step derivation

       1. times-frac 75.5%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\frac{1}{{k}^{2}} - 0.16666666666666666}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t}\right)} \]

       2. sub-neg 75.5%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\frac{1}{{k}^{2}} + \left(-0.16666666666666666\right)}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t}\right) \]

       3. unpow2 75.5%

          \[\leadsto 2 \cdot \left(\frac{\frac{1}{\color{blue}{k \cdot k}} + \left(-0.16666666666666666\right)}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t}\right) \]

       4. metadata-eval 75.5%

          \[\leadsto 2 \cdot \left(\frac{\frac{1}{k \cdot k} + \color{blue}{-0.16666666666666666}}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t}\right) \]

       5. unpow2 75.5%

          \[\leadsto 2 \cdot \left(\frac{\frac{1}{k \cdot k} + -0.16666666666666666}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{t}\right) \]

       6. unpow2 75.5%

          \[\leadsto 2 \cdot \left(\frac{\frac{1}{k \cdot k} + -0.16666666666666666}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right) \]

       7. associate-*r/ 75.6%

          \[\leadsto 2 \cdot \left(\frac{\frac{1}{k \cdot k} + -0.16666666666666666}{k \cdot k} \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)}\right) \]

   12. Simplified 75.6%

       \[\leadsto \color{blue}{2 \cdot \left(\frac{\frac{1}{k \cdot k} + -0.16666666666666666}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-51}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666 + \frac{1}{k \cdot k}}{k \cdot k} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)\\ \end{array} \]

Alternative 19: 57.2% accurate, 28.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{+20}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{k} \cdot \frac{\ell \cdot \ell}{k \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 6.8e+20)
   (* 2.0 (* 0.3333333333333333 (/ (* l l) (* k (* k t)))))
   (* (/ -0.3333333333333333 k) (/ (* l l) (* k t)))))

C

l = abs(l);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.8e+20) {
		tmp = 2.0 * (0.3333333333333333 * ((l * l) / (k * (k * t))));
	} else {
		tmp = (-0.3333333333333333 / k) * ((l * l) / (k * t));
	}
	return tmp;
}

Fortran

NOTE: l 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 <= 6.8d+20) then
        tmp = 2.0d0 * (0.3333333333333333d0 * ((l * l) / (k * (k * t))))
    else
        tmp = ((-0.3333333333333333d0) / k) * ((l * l) / (k * t))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.8e+20) {
		tmp = 2.0 * (0.3333333333333333 * ((l * l) / (k * (k * t))));
	} else {
		tmp = (-0.3333333333333333 / k) * ((l * l) / (k * t));
	}
	return tmp;
}

Python

l = abs(l)
def code(t, l, k):
	tmp = 0
	if k <= 6.8e+20:
		tmp = 2.0 * (0.3333333333333333 * ((l * l) / (k * (k * t))))
	else:
		tmp = (-0.3333333333333333 / k) * ((l * l) / (k * t))
	return tmp

Julia

l = abs(l)
function code(t, l, k)
	tmp = 0.0
	if (k <= 6.8e+20)
		tmp = Float64(2.0 * Float64(0.3333333333333333 * Float64(Float64(l * l) / Float64(k * Float64(k * t)))));
	else
		tmp = Float64(Float64(-0.3333333333333333 / k) * Float64(Float64(l * l) / Float64(k * t)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6.8e+20)
		tmp = 2.0 * (0.3333333333333333 * ((l * l) / (k * (k * t))));
	else
		tmp = (-0.3333333333333333 / k) * ((l * l) / (k * t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 6.8e+20], N[(2.0 * N[(0.3333333333333333 * N[(N[(l * l), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 / k), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.8e20

   1. Initial program 33.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 33.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 33.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 33.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 33.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 33.4%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 33.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 33.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 41.6%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 41.6%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 41.6%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 46.0%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 46.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 79.5%

      \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   5. Step-by-step derivation

      1. unpow2 79.5%

         \[\leadsto 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

      2. times-frac 79.8%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2} \cdot t}\right)} \]

      3. unpow2 79.8%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2} \cdot t}\right) \]

      4. *-commutative 79.8%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

      5. associate-/r* 76.4%

         \[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}}\right) \]

   6. Simplified 76.4%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}\right)} \]

   7. Taylor expanded in k around 0 70.5%

      \[\leadsto 2 \cdot \left(\color{blue}{\frac{1}{{k}^{2}}} \cdot \frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}\right) \]
   8. Step-by-step derivation

      1. unpow2 70.5%

         \[\leadsto 2 \cdot \left(\frac{1}{\color{blue}{k \cdot k}} \cdot \frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}\right) \]

   9. Simplified 70.5%

      \[\leadsto 2 \cdot \left(\color{blue}{\frac{1}{k \cdot k}} \cdot \frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}\right) \]

   10. Taylor expanded in k around 0 73.8%

       \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}\right) \]
   11. Step-by-step derivation

       1. unpow2 73.8%

          \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

       2. unpow2 73.8%

          \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

       3. associate-*r* 73.8%

          \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} + 0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)\right) \]

       4. unpow2 73.8%

          \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + 0.3333333333333333 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}\right)\right) \]

   12. Simplified 73.8%

       \[\leadsto 2 \cdot \left(\frac{1}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + 0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}\right)}\right) \]

   13. Taylor expanded in k around inf 62.0%

       \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
   14. Step-by-step derivation

       1. *-commutative 62.0%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot 0.3333333333333333\right)} \]

       2. unpow2 62.0%

          \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot t} \cdot 0.3333333333333333\right) \]

       3. unpow2 62.0%

          \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot 0.3333333333333333\right) \]

       4. associate-*r* 59.6%

          \[\leadsto 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot 0.3333333333333333\right) \]

   15. Simplified 59.6%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} \cdot 0.3333333333333333\right)} \]

   if 6.8e20 < k

   1. Initial program 38.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 38.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      2. associate-*l* 38.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      3. associate-/r* 38.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-/r/ 38.4%

         \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. *-commutative 38.4%

         \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      6. times-frac 37.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

      7. +-commutative 37.8%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      8. associate--l+ 51.0%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      9. metadata-eval 51.0%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 51.0%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      11. times-frac 51.0%

          \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Simplified 51.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   4. Taylor expanded in t around 0 79.8%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   5. Step-by-step derivation

      1. unpow2 79.8%

         \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   6. Simplified 79.8%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   7. Taylor expanded in k around 0 65.8%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(-0.16666666666666666 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 65.8%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{{\ell}^{2} \cdot -0.16666666666666666} + \frac{{\ell}^{2}}{{k}^{2}}\right) \]

      2. fma-def 65.8%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right)} \]

      3. unpow2 65.8%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right) \]

      4. unpow2 65.8%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \]

      5. unpow2 65.8%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \]

      6. times-frac 71.7%

         \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}\right) \]

   9. Simplified 71.7%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \]

   10. Taylor expanded in k around inf 71.7%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(-0.16666666666666666 \cdot {\ell}^{2}\right)} \]
   11. Step-by-step derivation

       1. *-commutative 71.7%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left({\ell}^{2} \cdot -0.16666666666666666\right)} \]

       2. unpow2 71.7%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot -0.16666666666666666\right) \]

       3. associate-*l* 71.7%

          \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot -0.16666666666666666\right)\right)} \]

   12. Simplified 71.7%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot -0.16666666666666666\right)\right)} \]

   13. Taylor expanded in k around 0 71.7%

       \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   14. Step-by-step derivation

       1. associate-*r/ 71.7%

          \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

       2. unpow2 71.7%

          \[\leadsto \frac{-0.3333333333333333 \cdot {\ell}^{2}}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

       3. associate-*r* 73.2%

          \[\leadsto \frac{-0.3333333333333333 \cdot {\ell}^{2}}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

       4. times-frac 73.4%

          \[\leadsto \color{blue}{\frac{-0.3333333333333333}{k} \cdot \frac{{\ell}^{2}}{k \cdot t}} \]

       5. unpow2 73.4%

          \[\leadsto \frac{-0.3333333333333333}{k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{k \cdot t} \]

   15. Simplified 73.4%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{k} \cdot \frac{\ell \cdot \ell}{k \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{+20}:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{k} \cdot \frac{\ell \cdot \ell}{k \cdot t}\\ \end{array} \]

Alternative 20: 71.0% accurate, 28.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* (/ 2.0 (* k (* k t))) (* (/ l k) (/ l k))))

C

l = abs(l);
double code(double t, double l, double k) {
	return (2.0 / (k * (k * t))) * ((l / k) * (l / k));
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (2.0d0 / (k * (k * t))) * ((l / k) * (l / k))
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	return (2.0 / (k * (k * t))) * ((l / k) * (l / k));
}

Python

l = abs(l)
def code(t, l, k):
	return (2.0 / (k * (k * t))) * ((l / k) * (l / k))

Julia

l = abs(l)
function code(t, l, k)
	return Float64(Float64(2.0 / Float64(k * Float64(k * t))) * Float64(Float64(l / k) * Float64(l / k)))
end

MATLAB

l = abs(l)
function tmp = code(t, l, k)
	tmp = (2.0 / (k * (k * t))) * ((l / k) * (l / k));
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 34.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. associate-*l* 34.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   3. associate-/r* 34.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   4. associate-/r/ 34.7%

      \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   5. *-commutative 34.7%

      \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   6. times-frac 34.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

   7. +-commutative 34.9%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   8. associate--l+ 44.1%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   9. metadata-eval 44.1%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   10. +-rgt-identity 44.1%

       \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   11. times-frac 47.3%

       \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

3. Simplified 47.3%

   \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

4. Taylor expanded in t around 0 83.6%

   \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
5. Step-by-step derivation

   1. unpow2 83.6%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

6. Simplified 83.6%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

7. Taylor expanded in k around 0 72.0%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 72.0%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \]

   2. unpow2 72.0%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \]

   3. times-frac 77.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \]

9. Simplified 77.3%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \]

10. Taylor expanded in k around 0 77.3%

    \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]
11. Step-by-step derivation

    1. unpow2 77.3%

       \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

    2. associate-*r* 78.0%

       \[\leadsto \frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

12. Simplified 78.0%

    \[\leadsto \frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

13. Final simplification 78.0%

    \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \]

Alternative 21: 33.4% accurate, 38.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \frac{-0.3333333333333333}{k} \cdot \frac{\ell \cdot \ell}{k \cdot t} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* (/ -0.3333333333333333 k) (/ (* l l) (* k t))))

C

l = abs(l);
double code(double t, double l, double k) {
	return (-0.3333333333333333 / k) * ((l * l) / (k * t));
}

Fortran

NOTE: l 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 = ((-0.3333333333333333d0) / k) * ((l * l) / (k * t))
end function

Java

l = Math.abs(l);
public static double code(double t, double l, double k) {
	return (-0.3333333333333333 / k) * ((l * l) / (k * t));
}

Python

l = abs(l)
def code(t, l, k):
	return (-0.3333333333333333 / k) * ((l * l) / (k * t))

Julia

l = abs(l)
function code(t, l, k)
	return Float64(Float64(-0.3333333333333333 / k) * Float64(Float64(l * l) / Float64(k * t)))
end

MATLAB

l = abs(l)
function tmp = code(t, l, k)
	tmp = (-0.3333333333333333 / k) * ((l * l) / (k * t));
end

Wolfram

NOTE: l should be positive before calling this function
code[t_, l_, k_] := N[(N[(-0.3333333333333333 / k), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 34.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. associate-*l* 34.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   3. associate-/r* 34.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   4. associate-/r/ 34.7%

      \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   5. *-commutative 34.7%

      \[\leadsto \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   6. times-frac 34.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]

   7. +-commutative 34.9%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   8. associate--l+ 44.1%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   9. metadata-eval 44.1%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   10. +-rgt-identity 44.1%

       \[\leadsto \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   11. times-frac 47.3%

       \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

3. Simplified 47.3%

   \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

4. Taylor expanded in t around 0 83.6%

   \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
5. Step-by-step derivation

   1. unpow2 83.6%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

6. Simplified 83.6%

   \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

7. Taylor expanded in k around 0 60.3%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(-0.16666666666666666 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}\right)} \]
8. Step-by-step derivation

   1. *-commutative 60.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{{\ell}^{2} \cdot -0.16666666666666666} + \frac{{\ell}^{2}}{{k}^{2}}\right) \]

   2. fma-def 60.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right)} \]

   3. unpow2 60.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, -0.16666666666666666, \frac{{\ell}^{2}}{{k}^{2}}\right) \]

   4. unpow2 60.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \]

   5. unpow2 60.3%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \]

   6. times-frac 65.8%

      \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}\right) \]

9. Simplified 65.8%

   \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \]

10. Taylor expanded in k around inf 37.0%

    \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(-0.16666666666666666 \cdot {\ell}^{2}\right)} \]
11. Step-by-step derivation

    1. *-commutative 37.0%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left({\ell}^{2} \cdot -0.16666666666666666\right)} \]

    2. unpow2 37.0%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot -0.16666666666666666\right) \]

    3. associate-*l* 37.0%

       \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot -0.16666666666666666\right)\right)} \]

12. Simplified 37.0%

    \[\leadsto \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot -0.16666666666666666\right)\right)} \]

13. Taylor expanded in k around 0 37.0%

    \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
14. Step-by-step derivation

    1. associate-*r/ 37.0%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

    2. unpow2 37.0%

       \[\leadsto \frac{-0.3333333333333333 \cdot {\ell}^{2}}{\color{blue}{\left(k \cdot k\right)} \cdot t} \]

    3. associate-*r* 38.2%

       \[\leadsto \frac{-0.3333333333333333 \cdot {\ell}^{2}}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

    4. times-frac 38.3%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{k} \cdot \frac{{\ell}^{2}}{k \cdot t}} \]

    5. unpow2 38.3%

       \[\leadsto \frac{-0.3333333333333333}{k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{k \cdot t} \]

15. Simplified 38.3%

    \[\leadsto \color{blue}{\frac{-0.3333333333333333}{k} \cdot \frac{\ell \cdot \ell}{k \cdot t}} \]

16. Final simplification 38.3%

    \[\leadsto \frac{-0.3333333333333333}{k} \cdot \frac{\ell \cdot \ell}{k \cdot t} \]

Reproduce

herbie shell --seed 2023227 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))