Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.5% → 95.0%

Time: 27.0s

Alternatives: 10

Speedup: 38.3×

• 35.1% 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: 34.5% 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: 95.0% accurate, 1.9× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (tan k))))
   (if (<= k 1e+144)
     (/ (* 2.0 (* (/ (/ l (sin k)) k) (/ t_1 t))) k)
     (* t_1 (* (/ l k) (/ 2.0 (* (sin k) (* k t))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = l / tan(k);
	double tmp;
	if (k <= 1e+144) {
		tmp = (2.0 * (((l / sin(k)) / k) * (t_1 / t))) / k;
	} else {
		tmp = t_1 * ((l / k) * (2.0 / (sin(k) * (k * t))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = l / tan(k)
    if (k <= 1d+144) then
        tmp = (2.0d0 * (((l / sin(k)) / k) * (t_1 / t))) / k
    else
        tmp = t_1 * ((l / k) * (2.0d0 / (sin(k) * (k * t))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = l / Math.tan(k);
	double tmp;
	if (k <= 1e+144) {
		tmp = (2.0 * (((l / Math.sin(k)) / k) * (t_1 / t))) / k;
	} else {
		tmp = t_1 * ((l / k) * (2.0 / (Math.sin(k) * (k * t))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = l / math.tan(k)
	tmp = 0
	if k <= 1e+144:
		tmp = (2.0 * (((l / math.sin(k)) / k) * (t_1 / t))) / k
	else:
		tmp = t_1 * ((l / k) * (2.0 / (math.sin(k) * (k * t))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(l / tan(k))
	tmp = 0.0
	if (k <= 1e+144)
		tmp = Float64(Float64(2.0 * Float64(Float64(Float64(l / sin(k)) / k) * Float64(t_1 / t))) / k);
	else
		tmp = Float64(t_1 * Float64(Float64(l / k) * Float64(2.0 / Float64(sin(k) * Float64(k * t)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = l / tan(k);
	tmp = 0.0;
	if (k <= 1e+144)
		tmp = (2.0 * (((l / sin(k)) / k) * (t_1 / t))) / k;
	else
		tmp = t_1 * ((l / k) * (2.0 / (sin(k) * (k * t))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1.00000000000000002e144

   1. Initial program 30.8%

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

      1. associate-*l* 30.8%

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

      2. associate-*l* 30.8%

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

      3. associate-/r* 31.0%

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

      4. associate-/r/ 30.8%

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

      5. *-commutative 30.8%

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

      6. times-frac 30.4%

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

      7. +-commutative 30.4%

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

      8. associate--l+ 38.4%

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

      9. metadata-eval 38.4%

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

      10. +-rgt-identity 38.4%

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

      11. times-frac 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 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%

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

      2. associate-*l* 87.2%

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

   6. Simplified 87.2%

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

      1. associate-*l/ 87.3%

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

   8. Applied egg-rr 87.3%

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

      1. times-frac 88.9%

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

   10. Applied egg-rr 88.9%

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

       1. associate-*l/ 88.9%

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

       2. times-frac 96.5%

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

   12. Applied egg-rr 96.5%

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

   if 1.00000000000000002e144 < k

   1. Initial program 18.5%

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

      1. associate-*l* 18.5%

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

      2. associate-*l* 18.5%

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

      3. associate-/r* 18.5%

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

      4. associate-/r/ 18.5%

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

      5. *-commutative 18.5%

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

      6. times-frac 18.5%

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

      7. +-commutative 18.5%

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

      8. associate--l+ 27.3%

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

      9. metadata-eval 27.3%

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

      10. +-rgt-identity 27.3%

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

      11. times-frac 27.3%

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

   3. Simplified 27.3%

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

   4. Taylor expanded in t around 0 45.1%

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

      1. unpow2 45.1%

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

      2. associate-*l* 59.7%

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

   6. Simplified 59.7%

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

      1. associate-*l/ 59.7%

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

   8. Applied egg-rr 59.7%

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

      1. div-inv 59.7%

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

   10. Applied egg-rr 59.7%

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

       1. associate-*r/ 59.7%

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

       2. *-rgt-identity 59.7%

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

       3. associate-*l/ 59.7%

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

       4. associate-*r* 77.3%

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

       5. associate-*r* 54.7%

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

   12. Simplified 54.7%

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

   13. Taylor expanded in k around inf 54.7%

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

       1. associate-*r/ 54.7%

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

       2. *-commutative 54.7%

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

       3. associate-*r* 54.7%

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

       4. unpow2 54.7%

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

       5. associate-*r* 77.5%

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

       6. *-commutative 77.5%

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

       7. associate-*l* 77.4%

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

       8. times-frac 94.2%

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

   15. Simplified 94.2%

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

4. Final simplification 96.2%

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

Alternative 2: 85.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 5.7 \cdot 10^{-117}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{2}{t \cdot \left(k \cdot k\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 5.7e-117)
   (* (/ 2.0 k) (/ (* (/ l k) (/ l k)) (* k t)))
   (* (/ l (tan k)) (* (/ l (sin k)) (/ 2.0 (* t (* k k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.7e-117) {
		tmp = (2.0 / k) * (((l / k) * (l / k)) / (k * t));
	} else {
		tmp = (l / tan(k)) * ((l / sin(k)) * (2.0 / (t * (k * k))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.7d-117) then
        tmp = (2.0d0 / k) * (((l / k) * (l / k)) / (k * t))
    else
        tmp = (l / tan(k)) * ((l / sin(k)) * (2.0d0 / (t * (k * k))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.7e-117) {
		tmp = (2.0 / k) * (((l / k) * (l / k)) / (k * t));
	} else {
		tmp = (l / Math.tan(k)) * ((l / Math.sin(k)) * (2.0 / (t * (k * k))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 5.7e-117:
		tmp = (2.0 / k) * (((l / k) * (l / k)) / (k * t))
	else:
		tmp = (l / math.tan(k)) * ((l / math.sin(k)) * (2.0 / (t * (k * k))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 5.7e-117)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(Float64(l / k) * Float64(l / k)) / Float64(k * t)));
	else
		tmp = Float64(Float64(l / tan(k)) * Float64(Float64(l / sin(k)) * Float64(2.0 / Float64(t * Float64(k * k)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5.7e-117)
		tmp = (2.0 / k) * (((l / k) * (l / k)) / (k * t));
	else
		tmp = (l / tan(k)) * ((l / sin(k)) * (2.0 / (t * (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 5.7e-117], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.6999999999999999e-117

   1. Initial program 34.1%

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

      1. associate-*l* 34.1%

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

      2. associate-*l* 34.1%

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

      3. associate-/r* 34.2%

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

      4. associate-/r/ 34.0%

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

      5. *-commutative 34.0%

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

      6. times-frac 33.9%

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

      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+ 38.1%

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

      9. metadata-eval 38.1%

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

      10. +-rgt-identity 38.1%

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

      11. times-frac 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in t around 0 81.8%

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

      1. unpow2 81.8%

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

      2. associate-*l* 86.5%

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

   6. Simplified 86.5%

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

      1. associate-*l/ 86.6%

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

   8. Applied egg-rr 86.6%

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

      1. times-frac 88.7%

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

   10. Applied egg-rr 88.7%

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

   11. Taylor expanded in k around 0 64.9%

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

       1. unpow2 64.9%

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

       2. unpow2 64.9%

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

       3. times-frac 78.1%

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

   13. Simplified 78.1%

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

   if 5.6999999999999999e-117 < k

   1. Initial program 22.7%

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

      1. associate-*l* 22.7%

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

      2. associate-*l* 22.7%

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

      3. associate-/r* 23.0%

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

      4. associate-/r/ 23.0%

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

      5. *-commutative 23.0%

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

      6. times-frac 22.1%

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

      7. +-commutative 22.1%

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

      8. associate--l+ 35.5%

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

      9. metadata-eval 35.5%

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

      10. +-rgt-identity 35.5%

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

      11. times-frac 38.0%

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

   3. Simplified 38.0%

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

   4. Taylor expanded in t around 0 75.1%

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

      1. unpow2 75.1%

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

      2. associate-*l* 79.6%

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

   6. Simplified 79.6%

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

      1. associate-*l/ 79.7%

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

   8. Applied egg-rr 79.7%

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

      1. div-inv 79.6%

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

   10. Applied egg-rr 79.6%

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

       1. associate-*r/ 79.7%

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

       2. *-rgt-identity 79.7%

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

       3. associate-*l/ 79.6%

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

       4. associate-*r* 89.2%

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

       5. associate-*r* 82.3%

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

   12. Simplified 82.3%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.7 \cdot 10^{-117}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{2}{t \cdot \left(k \cdot k\right)}\right)\\ \end{array} \]

Alternative 3: 93.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.2%

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

   1. associate-*l* 29.2%

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

   2. associate-*l* 29.2%

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

   3. associate-/r* 29.3%

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

   4. associate-/r/ 29.2%

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

   5. *-commutative 29.2%

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

   6. times-frac 28.8%

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

   7. +-commutative 28.8%

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

   8. associate--l+ 36.9%

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

   9. metadata-eval 36.9%

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

   10. +-rgt-identity 36.9%

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

   11. times-frac 42.7%

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

3. Simplified 42.7%

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

4. Taylor expanded in t around 0 78.9%

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

   1. unpow2 78.9%

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

   2. associate-*l* 83.5%

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

6. Simplified 83.5%

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

   1. associate-*l/ 83.6%

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

8. Applied egg-rr 83.6%

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

   1. div-inv 83.5%

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

10. Applied egg-rr 83.5%

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

    1. associate-*r/ 83.6%

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

    2. *-rgt-identity 83.6%

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

    3. associate-*l/ 83.5%

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

    4. associate-*r* 89.3%

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

    5. associate-*r* 83.2%

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

12. Simplified 83.2%

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

13. Taylor expanded in k around inf 81.9%

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

    1. associate-*r/ 81.9%

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

    2. *-commutative 81.9%

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

    3. associate-*r* 81.9%

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

    4. unpow2 81.9%

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

    5. associate-*r* 86.5%

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

    6. *-commutative 86.5%

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

    7. associate-*l* 86.5%

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

    8. times-frac 94.5%

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

15. Simplified 94.5%

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

16. Final simplification 94.5%

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

Alternative 4: 73.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k \cdot \sin k}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5e+296)
   (*
    (/ 2.0 k)
    (/ (fma (* l l) -0.16666666666666666 (* (/ l k) (/ l k))) (* k t)))
   (* (/ 2.0 k) (* (/ l (* k t)) (/ l (* k (sin k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 5e+296) {
		tmp = (2.0 / k) * (fma((l * l), -0.16666666666666666, ((l / k) * (l / k))) / (k * t));
	} else {
		tmp = (2.0 / k) * ((l / (k * t)) * (l / (k * sin(k))));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 5e+296)
		tmp = Float64(Float64(2.0 / k) * Float64(fma(Float64(l * l), -0.16666666666666666, Float64(Float64(l / k) * Float64(l / k))) / Float64(k * t)));
	else
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(l / Float64(k * t)) * Float64(l / Float64(k * sin(k)))));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 5e+296], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * -0.16666666666666666 + N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.0000000000000001e296

   1. Initial program 29.5%

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

      1. associate-*l* 29.5%

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

      2. associate-*l* 29.5%

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

      3. associate-/r* 29.7%

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

      4. associate-/r/ 29.5%

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

      5. *-commutative 29.5%

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

      6. times-frac 29.0%

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

      7. +-commutative 29.0%

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

      8. associate--l+ 40.3%

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

      9. metadata-eval 40.3%

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

      10. +-rgt-identity 40.3%

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

      11. times-frac 48.3%

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

   3. Simplified 48.3%

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

   4. Taylor expanded in t around 0 87.4%

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

      1. unpow2 87.4%

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

      2. associate-*l* 92.6%

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

   6. Simplified 92.6%

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

      1. associate-*l/ 92.7%

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

   8. Applied egg-rr 92.7%

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

      1. times-frac 96.1%

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

   10. Applied egg-rr 96.1%

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

   11. Taylor expanded in k around 0 67.7%

       \[\leadsto \frac{2}{k} \cdot \frac{\color{blue}{-0.16666666666666666 \cdot {\ell}^{2} + \frac{{\ell}^{2}}{{k}^{2}}}}{k \cdot t} \]
   12. Step-by-step derivation

       1. *-commutative 67.7%

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

       2. fma-def 67.7%

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

       3. unpow2 67.7%

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

       4. unpow2 67.7%

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

       5. unpow2 67.7%

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

       6. times-frac 82.1%

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

   13. Simplified 82.1%

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

   if 5.0000000000000001e296 < (*.f64 l l)

   1. Initial program 28.4%

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

      1. associate-*l* 28.4%

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

      2. associate-*l* 28.4%

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

      3. associate-/r* 28.4%

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

      4. associate-/r/ 28.4%

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

      5. *-commutative 28.4%

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

      6. times-frac 28.2%

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

      7. +-commutative 28.2%

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

      8. associate--l+ 28.2%

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

      9. metadata-eval 28.2%

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

      10. +-rgt-identity 28.2%

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

      11. times-frac 28.2%

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

   3. Simplified 28.2%

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

   4. Taylor expanded in t around 0 56.8%

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

      1. unpow2 56.8%

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

      2. associate-*l* 59.9%

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

   6. Simplified 59.9%

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

      1. associate-*l/ 59.9%

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

   8. Applied egg-rr 59.9%

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

   9. Taylor expanded in k around 0 57.2%

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

       1. expm1-log1p-u 21.8%

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

       2. expm1-udef 21.8%

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

       3. times-frac 21.8%

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

       4. associate-*l/ 21.8%

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

   11. Applied egg-rr 21.8%

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

       1. expm1-def 21.8%

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

       2. expm1-log1p 57.3%

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

       3. associate-/l/ 57.3%

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

       4. times-frac 58.7%

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

       5. associate-/r* 58.7%

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

   13. Simplified 58.7%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\mathsf{fma}\left(\ell \cdot \ell, -0.16666666666666666, \frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k \cdot \sin k}\right)\\ \end{array} \]

Alternative 5: 73.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k \cdot \sin k}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 5.8e-88)
   (* (/ 2.0 k) (/ (* (/ l k) (/ l k)) (* k t)))
   (* (/ 2.0 k) (* (/ l (* k t)) (/ l (* k (sin k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.8e-88) {
		tmp = (2.0 / k) * (((l / k) * (l / k)) / (k * t));
	} else {
		tmp = (2.0 / k) * ((l / (k * t)) * (l / (k * sin(k))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.8d-88) then
        tmp = (2.0d0 / k) * (((l / k) * (l / k)) / (k * t))
    else
        tmp = (2.0d0 / k) * ((l / (k * t)) * (l / (k * sin(k))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.8e-88) {
		tmp = (2.0 / k) * (((l / k) * (l / k)) / (k * t));
	} else {
		tmp = (2.0 / k) * ((l / (k * t)) * (l / (k * Math.sin(k))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 5.8e-88:
		tmp = (2.0 / k) * (((l / k) * (l / k)) / (k * t))
	else:
		tmp = (2.0 / k) * ((l / (k * t)) * (l / (k * math.sin(k))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 5.8e-88)
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(Float64(l / k) * Float64(l / k)) / Float64(k * t)));
	else
		tmp = Float64(Float64(2.0 / k) * Float64(Float64(l / Float64(k * t)) * Float64(l / Float64(k * sin(k)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5.8e-88)
		tmp = (2.0 / k) * (((l / k) * (l / k)) / (k * t));
	else
		tmp = (2.0 / k) * ((l / (k * t)) * (l / (k * sin(k))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 5.8e-88], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / k), $MachinePrecision] * N[(N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.8000000000000003e-88

   1. Initial program 32.6%

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

      1. associate-*l* 32.6%

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

      2. associate-*l* 32.6%

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

      3. associate-/r* 32.6%

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

      4. associate-/r/ 32.4%

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

      5. *-commutative 32.4%

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

      6. times-frac 32.4%

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

      7. +-commutative 32.4%

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

      8. associate--l+ 37.0%

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

      9. metadata-eval 37.0%

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

      10. +-rgt-identity 37.0%

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

      11. times-frac 46.6%

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

   3. Simplified 46.6%

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

   4. Taylor expanded in t around 0 81.8%

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

      1. unpow2 81.8%

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

      2. associate-*l* 86.2%

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

   6. Simplified 86.2%

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

      1. associate-*l/ 86.3%

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

   8. Applied egg-rr 86.3%

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

      1. times-frac 88.2%

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

   10. Applied egg-rr 88.2%

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

   11. Taylor expanded in k around 0 63.4%

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

       1. unpow2 63.4%

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

       2. unpow2 63.4%

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

       3. times-frac 78.3%

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

   13. Simplified 78.3%

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

   if 5.8000000000000003e-88 < k

   1. Initial program 23.9%

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

      1. associate-*l* 23.9%

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

      2. associate-*l* 23.9%

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

      3. associate-/r* 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 23.3%

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

      7. +-commutative 23.3%

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

      8. associate--l+ 36.9%

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

      9. metadata-eval 36.9%

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

      10. +-rgt-identity 36.9%

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

      11. times-frac 36.9%

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

   3. Simplified 36.9%

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

   4. Taylor expanded in t around 0 74.6%

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

      1. unpow2 74.6%

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

      2. associate-*l* 79.5%

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

   6. Simplified 79.5%

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

      1. associate-*l/ 79.6%

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

   8. Applied egg-rr 79.6%

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

   9. Taylor expanded in k around 0 65.6%

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

       1. expm1-log1p-u 53.8%

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

       2. expm1-udef 50.1%

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

       3. times-frac 50.1%

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

       4. associate-*l/ 50.1%

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

   11. Applied egg-rr 50.1%

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

       1. expm1-def 53.4%

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

       2. expm1-log1p 65.3%

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

       3. associate-/l/ 65.3%

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

       4. times-frac 65.3%

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

       5. associate-/r* 65.3%

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

   13. Simplified 65.3%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k \cdot \sin k}\right)\\ \end{array} \]

Alternative 6: 70.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.2%

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

   1. associate-*l* 29.2%

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

   2. associate-*l* 29.2%

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

   3. associate-/r* 29.3%

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

   4. associate-/r/ 29.2%

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

   5. *-commutative 29.2%

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

   6. times-frac 28.8%

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

   7. +-commutative 28.8%

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

   8. associate--l+ 36.9%

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

   9. metadata-eval 36.9%

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

   10. +-rgt-identity 36.9%

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

   11. times-frac 42.7%

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

3. Simplified 42.7%

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

4. Taylor expanded in t around 0 78.9%

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

   1. unpow2 78.9%

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

   2. associate-*l* 83.5%

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

6. Simplified 83.5%

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

7. Taylor expanded in k around 0 73.7%

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

8. Final simplification 73.7%

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

Alternative 7: 71.1% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.2%

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

   1. associate-*l* 29.2%

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

   2. associate-*l* 29.2%

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

   3. associate-/r* 29.3%

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

   4. associate-/r/ 29.2%

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

   5. *-commutative 29.2%

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

   6. times-frac 28.8%

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

   7. +-commutative 28.8%

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

   8. associate--l+ 36.9%

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

   9. metadata-eval 36.9%

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

   10. +-rgt-identity 36.9%

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

   11. times-frac 42.7%

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

3. Simplified 42.7%

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

4. Taylor expanded in t around 0 78.9%

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

   1. unpow2 78.9%

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

   2. associate-*l* 83.5%

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

6. Simplified 83.5%

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

   1. associate-*l/ 83.6%

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

8. Applied egg-rr 83.6%

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

9. Taylor expanded in k around 0 73.7%

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

10. Final simplification 73.7%

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

Alternative 8: 72.2% accurate, 18.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{+186}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} + 0.16666666666666666 \cdot \left(k \cdot \ell\right)\right)\right)}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 4.2e+186)
   (/
    (* 2.0 (* (/ l k) (+ (/ l k) (* 0.16666666666666666 (* k l)))))
    (* k (* k t)))
   (* -0.3333333333333333 (* (/ l k) (/ l (* k t))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.2e+186) {
		tmp = (2.0 * ((l / k) * ((l / k) + (0.16666666666666666 * (k * l))))) / (k * (k * t));
	} else {
		tmp = -0.3333333333333333 * ((l / k) * (l / (k * t)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 4.2d+186) then
        tmp = (2.0d0 * ((l / k) * ((l / k) + (0.16666666666666666d0 * (k * l))))) / (k * (k * t))
    else
        tmp = (-0.3333333333333333d0) * ((l / k) * (l / (k * t)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 4.2e+186) {
		tmp = (2.0 * ((l / k) * ((l / k) + (0.16666666666666666 * (k * l))))) / (k * (k * t));
	} else {
		tmp = -0.3333333333333333 * ((l / k) * (l / (k * t)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 4.2e+186:
		tmp = (2.0 * ((l / k) * ((l / k) + (0.16666666666666666 * (k * l))))) / (k * (k * t))
	else:
		tmp = -0.3333333333333333 * ((l / k) * (l / (k * t)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 4.2e+186)
		tmp = Float64(Float64(2.0 * Float64(Float64(l / k) * Float64(Float64(l / k) + Float64(0.16666666666666666 * Float64(k * l))))) / Float64(k * Float64(k * t)));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(l / k) * Float64(l / Float64(k * t))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 4.2e+186)
		tmp = (2.0 * ((l / k) * ((l / k) + (0.16666666666666666 * (k * l))))) / (k * (k * t));
	else
		tmp = -0.3333333333333333 * ((l / k) * (l / (k * t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 4.2e+186], N[(N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] + N[(0.16666666666666666 * N[(k * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.2e186

   1. Initial program 30.2%

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

      1. associate-*l* 30.2%

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

      2. associate-*l* 30.2%

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

      3. associate-/r* 30.4%

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

      4. associate-/r/ 30.3%

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

      5. *-commutative 30.3%

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

      6. times-frac 29.8%

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

      7. +-commutative 29.8%

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

      8. associate--l+ 38.0%

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

      9. metadata-eval 38.0%

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

      10. +-rgt-identity 38.0%

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

      11. times-frac 44.5%

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

   3. Simplified 44.5%

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

   4. Taylor expanded in t around 0 82.1%

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

      1. unpow2 82.1%

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

      2. associate-*l* 86.3%

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

   6. Simplified 86.3%

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

      1. associate-*l/ 86.4%

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

   8. Applied egg-rr 86.4%

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

   9. Taylor expanded in k around 0 74.9%

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

   10. Taylor expanded in k around 0 74.2%

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

   if 4.2e186 < k

   1. Initial program 19.8%

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

      1. associate-*l* 19.8%

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

      2. associate-*l* 19.8%

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

      3. associate-/r* 19.8%

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

      4. associate-/r/ 19.8%

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

      5. *-commutative 19.8%

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

      6. times-frac 19.8%

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

      7. +-commutative 19.8%

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

      8. associate--l+ 27.5%

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

      9. metadata-eval 27.5%

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

      10. +-rgt-identity 27.5%

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

      11. times-frac 27.5%

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

   3. Simplified 27.5%

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

   4. Taylor expanded in k around 0 27.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   5. Step-by-step derivation

      1. +-commutative 27.5%

         \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} + -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}} \]

      2. fma-def 27.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{{k}^{4} \cdot t}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right)} \]

      3. unpow2 27.5%

         \[\leadsto \mathsf{fma}\left(2, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]

      4. *-commutative 27.5%

         \[\leadsto \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]

      5. times-frac 27.5%

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]

      6. associate-/l* 27.5%

         \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{{k}^{2} \cdot {t}^{2}}{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}}}\right) \]

      7. associate-/r/ 27.5%

         \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{2}} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)\right)}\right) \]

      8. unpow2 27.5%

         \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{2}} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)\right)\right) \]

      9. unpow2 27.5%

         \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{2}} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)\right)\right) \]

      10. unpow2 27.5%

          \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)\right)\right) \]

      11. unswap-sqr 50.8%

          \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)}} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)\right)\right) \]

      12. distribute-rgt-out 50.8%

          \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\left(t \cdot \left(-0.16666666666666666 + 0.3333333333333333\right)\right)}\right)\right) \]

      13. metadata-eval 50.8%

          \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \left(t \cdot \color{blue}{0.16666666666666666}\right)\right)\right) \]

   6. Simplified 50.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \left(t \cdot 0.16666666666666666\right)\right)\right)} \]

   7. Taylor expanded in k around inf 50.8%

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

      1. unpow2 50.8%

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

      2. unpow2 50.8%

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

      3. associate-*r* 51.7%

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

   9. Simplified 51.7%

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

       1. times-frac 66.7%

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

   11. Applied egg-rr 66.7%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{+186}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} + 0.16666666666666666 \cdot \left(k \cdot \ell\right)\right)\right)}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right)\\ \end{array} \]

Alternative 9: 71.6% accurate, 28.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.2%

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

   1. associate-*l* 29.2%

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

   2. associate-*l* 29.2%

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

   3. associate-/r* 29.3%

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

   4. associate-/r/ 29.2%

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

   5. *-commutative 29.2%

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

   6. times-frac 28.8%

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

   7. +-commutative 28.8%

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

   8. associate--l+ 36.9%

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

   9. metadata-eval 36.9%

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

   10. +-rgt-identity 36.9%

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

   11. times-frac 42.7%

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

3. Simplified 42.7%

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

4. Taylor expanded in t around 0 78.9%

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

   1. unpow2 78.9%

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

   2. associate-*l* 83.5%

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

6. Simplified 83.5%

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

   1. associate-*l/ 83.6%

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

8. Applied egg-rr 83.6%

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

   1. times-frac 86.2%

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

10. Applied egg-rr 86.2%

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

11. Taylor expanded in k around 0 62.2%

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

    1. unpow2 62.2%

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

    2. unpow2 62.2%

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

    3. times-frac 72.5%

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

13. Simplified 72.5%

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

14. Final simplification 72.5%

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

Alternative 10: 34.6% accurate, 38.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ -0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right) \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* -0.3333333333333333 (* (/ l k) (/ l (* k t)))))

C

k = abs(k);
double code(double t, double l, double k) {
	return -0.3333333333333333 * ((l / k) * (l / (k * t)));
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (-0.3333333333333333d0) * ((l / k) * (l / (k * t)))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return -0.3333333333333333 * ((l / k) * (l / (k * t)));
}

Python

k = abs(k)
def code(t, l, k):
	return -0.3333333333333333 * ((l / k) * (l / (k * t)))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(-0.3333333333333333 * Float64(Float64(l / k) * Float64(l / Float64(k * t))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = -0.3333333333333333 * ((l / k) * (l / (k * t)));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(-0.3333333333333333 * N[(N[(l / k), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.2%

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

   1. associate-*l* 29.2%

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

   2. associate-*l* 29.2%

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

   3. associate-/r* 29.3%

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

   4. associate-/r/ 29.2%

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

   5. *-commutative 29.2%

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

   6. times-frac 28.8%

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

   7. +-commutative 28.8%

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

   8. associate--l+ 36.9%

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

   9. metadata-eval 36.9%

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

   10. +-rgt-identity 36.9%

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

   11. times-frac 42.7%

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

3. Simplified 42.7%

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

4. Taylor expanded in k around 0 21.5%

   \[\leadsto \color{blue}{-2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
5. Step-by-step derivation

   1. +-commutative 21.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} + -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}} \]

   2. fma-def 21.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{{k}^{4} \cdot t}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right)} \]

   3. unpow2 21.5%

      \[\leadsto \mathsf{fma}\left(2, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]

   4. *-commutative 21.5%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]

   5. times-frac 22.0%

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]

   6. associate-/l* 23.7%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{{k}^{2} \cdot {t}^{2}}{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}}}\right) \]

   7. associate-/r/ 23.7%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{2}} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)\right)}\right) \]

   8. unpow2 23.7%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{2}} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)\right)\right) \]

   9. unpow2 23.7%

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{2}} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)\right)\right) \]

   10. unpow2 23.7%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)\right)\right) \]

   11. unswap-sqr 29.7%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)}} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)\right)\right) \]

   12. distribute-rgt-out 29.7%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \color{blue}{\left(t \cdot \left(-0.16666666666666666 + 0.3333333333333333\right)\right)}\right)\right) \]

   13. metadata-eval 29.7%

       \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \left(t \cdot \color{blue}{0.16666666666666666}\right)\right)\right) \]

6. Simplified 29.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \left(\frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot \left(k \cdot t\right)} \cdot \left(t \cdot 0.16666666666666666\right)\right)\right)} \]

7. Taylor expanded in k around inf 27.4%

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

   1. unpow2 27.4%

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

   2. unpow2 27.4%

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

   3. associate-*r* 27.7%

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

9. Simplified 27.7%

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

    1. times-frac 29.9%

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

11. Applied egg-rr 29.9%

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

12. Final simplification 29.9%

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

Reproduce

herbie shell --seed 2023278 
(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))))