Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.2% → 93.2%

Time: 34.1s

Alternatives: 8

Speedup: 38.3×

• 34.9% 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: 34.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq 4.3 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{+112}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t_1} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{k}{\frac{\ell}{k \cdot t}} \cdot t_1}{\ell \cdot \cos k}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= k 4.3e-138)
     (/ 2.0 (* (/ (* k (* k t)) l) (* k (/ k l))))
     (if (<= k 5.5e+112)
       (* 2.0 (* (/ (cos k) t_1) (* (/ l t) (/ l (* k k)))))
       (/ 2.0 (/ (* (/ k (/ l (* k t))) t_1) (* l (cos k))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if (k <= 4.3e-138) {
		tmp = 2.0 / (((k * (k * t)) / l) * (k * (k / l)));
	} else if (k <= 5.5e+112) {
		tmp = 2.0 * ((cos(k) / t_1) * ((l / t) * (l / (k * k))));
	} else {
		tmp = 2.0 / (((k / (l / (k * t))) * t_1) / (l * cos(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) :: t_1
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    if (k <= 4.3d-138) then
        tmp = 2.0d0 / (((k * (k * t)) / l) * (k * (k / l)))
    else if (k <= 5.5d+112) then
        tmp = 2.0d0 * ((cos(k) / t_1) * ((l / t) * (l / (k * k))))
    else
        tmp = 2.0d0 / (((k / (l / (k * t))) * t_1) / (l * cos(k)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= 4.3e-138) {
		tmp = 2.0 / (((k * (k * t)) / l) * (k * (k / l)));
	} else if (k <= 5.5e+112) {
		tmp = 2.0 * ((Math.cos(k) / t_1) * ((l / t) * (l / (k * k))));
	} else {
		tmp = 2.0 / (((k / (l / (k * t))) * t_1) / (l * Math.cos(k)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= 4.3e-138:
		tmp = 2.0 / (((k * (k * t)) / l) * (k * (k / l)))
	elif k <= 5.5e+112:
		tmp = 2.0 * ((math.cos(k) / t_1) * ((l / t) * (l / (k * k))))
	else:
		tmp = 2.0 / (((k / (l / (k * t))) * t_1) / (l * math.cos(k)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= 4.3e-138)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k * t)) / l) * Float64(k * Float64(k / l))));
	elseif (k <= 5.5e+112)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / t_1) * Float64(Float64(l / t) * Float64(l / Float64(k * k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / Float64(l / Float64(k * t))) * t_1) / Float64(l * cos(k))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= 4.3e-138)
		tmp = 2.0 / (((k * (k * t)) / l) * (k * (k / l)));
	elseif (k <= 5.5e+112)
		tmp = 2.0 * ((cos(k) / t_1) * ((l / t) * (l / (k * k))));
	else
		tmp = 2.0 / (((k / (l / (k * t))) * t_1) / (l * cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 4.3e-138], N[(2.0 / N[(N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.5e+112], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.3e-138

   1. Initial program 30.3%

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

   2. Taylor expanded in t around 0 73.4%

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

      1. associate-*r* 73.3%

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

      2. unpow2 73.3%

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

      3. associate-*l* 73.3%

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

      4. times-frac 87.2%

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

      5. unpow2 87.2%

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

      6. associate-*l* 90.9%

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

   4. Simplified 90.9%

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

      1. div-inv 90.9%

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

   6. Applied egg-rr 90.9%

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

   7. Taylor expanded in k around 0 75.7%

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

      1. unpow2 75.7%

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

      2. associate-*l/ 78.2%

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

      3. *-commutative 78.2%

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

   9. Simplified 78.2%

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

   if 4.3e-138 < k < 5.50000000000000026e112

   1. Initial program 27.4%

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

      1. associate-/r* 27.4%

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

      2. *-commutative 27.4%

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

      3. associate-/r* 27.6%

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

      4. associate-*r/ 27.7%

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

      5. associate-/l* 27.6%

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

      6. +-commutative 27.6%

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

      7. unpow2 27.6%

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

      8. sqr-neg 27.6%

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

      9. distribute-frac-neg 27.6%

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

      10. distribute-frac-neg 27.6%

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

      11. unpow2 27.6%

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

      12. associate--l+ 49.0%

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

      13. metadata-eval 49.0%

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

      14. +-rgt-identity 49.0%

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

      15. unpow2 49.0%

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

      16. distribute-frac-neg 49.0%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in k around inf 80.6%

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

      1. associate-*r* 80.5%

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

      2. times-frac 79.0%

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

      3. unpow2 79.0%

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

      4. *-commutative 79.0%

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

      5. times-frac 97.9%

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

      6. unpow2 97.9%

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

   6. Simplified 97.9%

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

   if 5.50000000000000026e112 < k

   1. Initial program 41.3%

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

   2. Taylor expanded in t around 0 68.1%

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

      1. associate-*r* 68.1%

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

      2. unpow2 68.1%

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

      3. associate-*l* 68.1%

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

      4. times-frac 70.6%

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

      5. unpow2 70.6%

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

      6. associate-*l* 84.0%

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

   4. Simplified 84.0%

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

      1. associate-*r/ 83.9%

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

      2. associate-/l* 93.7%

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

   6. Applied egg-rr 93.7%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.3 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{+112}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{k}{\frac{\ell}{k \cdot t}} \cdot {\sin k}^{2}}{\ell \cdot \cos k}}\\ \end{array} \]

Alternative 2: 94.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \sqrt[3]{\frac{k}{\frac{\ell}{k \cdot t}}}\\ \mathbf{if}\;k \leq 6 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\left(t_1 \cdot \left(t_1 \cdot t_1\right)\right) \cdot \frac{k}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{k} \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{t}\right)\right) \cdot \frac{\cos k}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (cbrt (/ k (/ l (* k t))))))
   (if (<= k 6e-14)
     (/ 2.0 (* (* t_1 (* t_1 t_1)) (/ k (/ l k))))
     (* (* (/ 2.0 k) (* l (/ (/ l k) t))) (/ (cos k) (pow (sin k) 2.0))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = cbrt((k / (l / (k * t))));
	double tmp;
	if (k <= 6e-14) {
		tmp = 2.0 / ((t_1 * (t_1 * t_1)) * (k / (l / k)));
	} else {
		tmp = ((2.0 / k) * (l * ((l / k) / t))) * (cos(k) / pow(sin(k), 2.0));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.cbrt((k / (l / (k * t))));
	double tmp;
	if (k <= 6e-14) {
		tmp = 2.0 / ((t_1 * (t_1 * t_1)) * (k / (l / k)));
	} else {
		tmp = ((2.0 / k) * (l * ((l / k) / t))) * (Math.cos(k) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	t_1 = cbrt(Float64(k / Float64(l / Float64(k * t))))
	tmp = 0.0
	if (k <= 6e-14)
		tmp = Float64(2.0 / Float64(Float64(t_1 * Float64(t_1 * t_1)) * Float64(k / Float64(l / k))));
	else
		tmp = Float64(Float64(Float64(2.0 / k) * Float64(l * Float64(Float64(l / k) / t))) * Float64(cos(k) / (sin(k) ^ 2.0)));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[k, 6e-14], N[(2.0 / N[(N[(t$95$1 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / k), $MachinePrecision] * N[(l * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.9999999999999997e-14

   1. Initial program 31.6%

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

   2. Taylor expanded in t around 0 74.3%

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

      1. associate-*r* 74.3%

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

      2. unpow2 74.3%

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

      3. associate-*l* 74.3%

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

      4. times-frac 87.1%

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

      5. unpow2 87.1%

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

      6. associate-*l* 90.2%

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

   4. Simplified 90.2%

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

      1. add-cube-cbrt 89.9%

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

      2. associate-/l* 89.9%

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

      3. associate-/l* 89.9%

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

      4. associate-/l* 94.5%

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

   6. Applied egg-rr 94.5%

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

   7. Taylor expanded in k around 0 79.8%

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

      1. unpow2 79.8%

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

      2. associate-/l* 82.3%

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

   9. Simplified 82.3%

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

   if 5.9999999999999997e-14 < k

   1. Initial program 32.3%

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

   2. Taylor expanded in t around 0 72.9%

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

      1. associate-*r* 72.8%

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

      2. unpow2 72.8%

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

      3. associate-*l* 72.8%

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

      4. times-frac 75.7%

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

      5. unpow2 75.7%

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

      6. associate-*l* 84.5%

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

   4. Simplified 84.5%

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

      1. div-inv 84.5%

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

   6. Applied egg-rr 84.5%

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

   7. Taylor expanded in k around inf 72.9%

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

      1. associate-*r* 72.8%

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

      2. unpow2 72.8%

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

      3. associate-*r* 80.3%

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

      4. times-frac 80.3%

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

      5. associate-/l* 78.3%

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

      6. unpow2 78.3%

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

      7. associate-/l* 85.9%

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

      8. associate-/r/ 85.9%

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

      9. *-commutative 85.9%

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

      10. associate-/r* 88.5%

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

   9. Simplified 88.5%

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

   10. Taylor expanded in k around inf 72.9%

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

       1. associate-*r/ 72.9%

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

       2. associate-/l* 72.9%

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

       3. associate-*r* 72.8%

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

       4. unpow2 72.8%

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

       5. associate-*r* 80.3%

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

       6. times-frac 80.3%

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

       7. associate-/l* 78.3%

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

       8. unpow2 78.3%

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

       9. associate-*r/ 85.9%

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

       10. associate-*r/ 85.9%

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

       11. associate-/l* 85.8%

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

       12. associate-/r/ 85.9%

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

   12. Simplified 93.3%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\left(\sqrt[3]{\frac{k}{\frac{\ell}{k \cdot t}}} \cdot \left(\sqrt[3]{\frac{k}{\frac{\ell}{k \cdot t}}} \cdot \sqrt[3]{\frac{k}{\frac{\ell}{k \cdot t}}}\right)\right) \cdot \frac{k}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{k} \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{t}\right)\right) \cdot \frac{\cos k}{{\sin k}^{2}}\\ \end{array} \]

Alternative 3: 92.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+135}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t_1} \cdot \frac{\ell}{k \cdot \frac{k}{\frac{\ell}{t}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \frac{t_1}{\cos k}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= (* l l) 5e+135)
     (* 2.0 (* (/ (cos k) t_1) (/ l (* k (/ k (/ l t))))))
     (/ 2.0 (* (* t (* (/ k l) (/ k l))) (/ t_1 (cos k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if ((l * l) <= 5e+135) {
		tmp = 2.0 * ((cos(k) / t_1) * (l / (k * (k / (l / t)))));
	} else {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (t_1 / cos(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) :: t_1
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    if ((l * l) <= 5d+135) then
        tmp = 2.0d0 * ((cos(k) / t_1) * (l / (k * (k / (l / t)))))
    else
        tmp = 2.0d0 / ((t * ((k / l) * (k / l))) * (t_1 / cos(k)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if ((l * l) <= 5e+135) {
		tmp = 2.0 * ((Math.cos(k) / t_1) * (l / (k * (k / (l / t)))));
	} else {
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (t_1 / Math.cos(k)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if (l * l) <= 5e+135:
		tmp = 2.0 * ((math.cos(k) / t_1) * (l / (k * (k / (l / t)))))
	else:
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (t_1 / math.cos(k)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (Float64(l * l) <= 5e+135)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / t_1) * Float64(l / Float64(k * Float64(k / Float64(l / t))))));
	else
		tmp = Float64(2.0 / Float64(Float64(t * Float64(Float64(k / l) * Float64(k / l))) * Float64(t_1 / cos(k))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if ((l * l) <= 5e+135)
		tmp = 2.0 * ((cos(k) / t_1) * (l / (k * (k / (l / t)))));
	else
		tmp = 2.0 / ((t * ((k / l) * (k / l))) * (t_1 / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 5e+135], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / t$95$1), $MachinePrecision] * N[(l / N[(k * N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.00000000000000029e135

   1. Initial program 30.5%

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

   2. Taylor expanded in t around 0 75.3%

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

      1. associate-*r* 75.3%

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

      2. unpow2 75.3%

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

      3. associate-*l* 75.3%

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

      4. times-frac 88.3%

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

      5. unpow2 88.3%

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

      6. associate-*l* 92.4%

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

   4. Simplified 92.4%

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

      1. div-inv 92.4%

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

   6. Applied egg-rr 92.4%

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

   7. Taylor expanded in k around inf 75.5%

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

      1. associate-*r* 75.4%

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

      2. unpow2 75.4%

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

      3. associate-*r* 79.5%

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

      4. times-frac 79.8%

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

      5. unpow2 79.8%

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

      6. associate-/l* 92.0%

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

      7. associate-*l/ 94.2%

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

      8. *-commutative 94.2%

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

      9. associate-*r* 95.5%

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

      10. associate-/r/ 92.9%

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

   9. Simplified 92.9%

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

   if 5.00000000000000029e135 < (*.f64 l l)

   1. Initial program 33.9%

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

   2. Taylor expanded in t around 0 71.6%

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

      1. associate-*r* 71.5%

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

      2. unpow2 71.5%

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

      3. associate-*l* 71.5%

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

      4. times-frac 76.0%

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

      5. unpow2 76.0%

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

      6. associate-*l* 82.0%

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

   4. Simplified 82.0%

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

      1. div-inv 82.0%

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

   6. Applied egg-rr 82.0%

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

   7. Taylor expanded in k around inf 71.6%

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

      1. associate-*r* 71.5%

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

      2. unpow2 71.5%

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

      3. associate-*r* 76.6%

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

      4. times-frac 76.6%

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

      5. associate-/l* 78.1%

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

      6. unpow2 78.1%

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

      7. associate-/l* 88.2%

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

      8. associate-/r/ 88.2%

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

      9. *-commutative 88.2%

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

      10. associate-/r* 91.2%

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

   9. Simplified 91.2%

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

   10. Taylor expanded in k around 0 71.6%

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

       1. associate-/l* 73.5%

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

       2. associate-/r/ 75.6%

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

       3. unpow2 75.6%

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

       4. unpow2 75.6%

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

       5. times-frac 97.1%

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

   12. Simplified 97.1%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+135}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{\ell}{k \cdot \frac{k}{\frac{\ell}{t}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right) \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \end{array} \]

Alternative 4: 86.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-136}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.6e-136)
   (/ 2.0 (* (/ (* k (* k t)) l) (* k (/ k l))))
   (* 2.0 (* (/ (cos k) (pow (sin k) 2.0)) (* (/ l t) (/ l (* k k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.6e-136) {
		tmp = 2.0 / (((k * (k * t)) / l) * (k * (k / l)));
	} else {
		tmp = 2.0 * ((cos(k) / pow(sin(k), 2.0)) * ((l / t) * (l / (k * k))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.6d-136) then
        tmp = 2.0d0 / (((k * (k * t)) / l) * (k * (k / l)))
    else
        tmp = 2.0d0 * ((cos(k) / (sin(k) ** 2.0d0)) * ((l / t) * (l / (k * k))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.6e-136) {
		tmp = 2.0 / (((k * (k * t)) / l) * (k * (k / l)));
	} else {
		tmp = 2.0 * ((Math.cos(k) / Math.pow(Math.sin(k), 2.0)) * ((l / t) * (l / (k * k))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.6e-136:
		tmp = 2.0 / (((k * (k * t)) / l) * (k * (k / l)))
	else:
		tmp = 2.0 * ((math.cos(k) / math.pow(math.sin(k), 2.0)) * ((l / t) * (l / (k * k))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.6e-136)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k * t)) / l) * Float64(k * Float64(k / l))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / (sin(k) ^ 2.0)) * Float64(Float64(l / t) * Float64(l / Float64(k * k)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.6e-136)
		tmp = 2.0 / (((k * (k * t)) / l) * (k * (k / l)));
	else
		tmp = 2.0 * ((cos(k) / (sin(k) ^ 2.0)) * ((l / t) * (l / (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.6e-136], N[(2.0 / N[(N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.59999999999999996e-136

   1. Initial program 30.3%

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

   2. Taylor expanded in t around 0 73.4%

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

      1. associate-*r* 73.3%

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

      2. unpow2 73.3%

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

      3. associate-*l* 73.3%

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

      4. times-frac 87.2%

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

      5. unpow2 87.2%

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

      6. associate-*l* 90.9%

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

   4. Simplified 90.9%

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

      1. div-inv 90.9%

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

   6. Applied egg-rr 90.9%

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

   7. Taylor expanded in k around 0 75.7%

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

      1. unpow2 75.7%

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

      2. associate-*l/ 78.2%

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

      3. *-commutative 78.2%

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

   9. Simplified 78.2%

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

   if 1.59999999999999996e-136 < k

   1. Initial program 33.9%

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

      1. associate-/r* 33.9%

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

      2. *-commutative 33.9%

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

      3. associate-/r* 34.0%

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

      4. associate-*r/ 34.1%

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

      5. associate-/l* 34.1%

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

      6. +-commutative 34.1%

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

      7. unpow2 34.1%

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

      8. sqr-neg 34.1%

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

      9. distribute-frac-neg 34.1%

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

      10. distribute-frac-neg 34.1%

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

      11. unpow2 34.1%

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

      12. associate--l+ 50.7%

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

      13. metadata-eval 50.7%

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

      14. +-rgt-identity 50.7%

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

      15. unpow2 50.7%

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

      16. distribute-frac-neg 50.7%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in k around inf 74.7%

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

      1. associate-*r* 74.7%

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

      2. times-frac 73.9%

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

      3. unpow2 73.9%

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

      4. *-commutative 73.9%

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

      5. times-frac 84.6%

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

      6. unpow2 84.6%

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

   6. Simplified 84.6%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-136}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\ \end{array} \]

Alternative 5: 90.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-137}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{\ell}{k \cdot \frac{k}{\frac{\ell}{t}}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 6.2e-137)
   (/ 2.0 (* (/ (* k (* k t)) l) (* k (/ k l))))
   (* 2.0 (* (/ (cos k) (pow (sin k) 2.0)) (/ l (* k (/ k (/ l t))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.2e-137) {
		tmp = 2.0 / (((k * (k * t)) / l) * (k * (k / l)));
	} else {
		tmp = 2.0 * ((cos(k) / pow(sin(k), 2.0)) * (l / (k * (k / (l / 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 <= 6.2d-137) then
        tmp = 2.0d0 / (((k * (k * t)) / l) * (k * (k / l)))
    else
        tmp = 2.0d0 * ((cos(k) / (sin(k) ** 2.0d0)) * (l / (k * (k / (l / 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 <= 6.2e-137) {
		tmp = 2.0 / (((k * (k * t)) / l) * (k * (k / l)));
	} else {
		tmp = 2.0 * ((Math.cos(k) / Math.pow(Math.sin(k), 2.0)) * (l / (k * (k / (l / t)))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 6.2e-137:
		tmp = 2.0 / (((k * (k * t)) / l) * (k * (k / l)))
	else:
		tmp = 2.0 * ((math.cos(k) / math.pow(math.sin(k), 2.0)) * (l / (k * (k / (l / t)))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 6.2e-137)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k * t)) / l) * Float64(k * Float64(k / l))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / (sin(k) ^ 2.0)) * Float64(l / Float64(k * Float64(k / Float64(l / t))))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6.2e-137)
		tmp = 2.0 / (((k * (k * t)) / l) * (k * (k / l)));
	else
		tmp = 2.0 * ((cos(k) / (sin(k) ^ 2.0)) * (l / (k * (k / (l / t)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 6.2e-137], N[(2.0 / N[(N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.19999999999999955e-137

   1. Initial program 30.3%

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

   2. Taylor expanded in t around 0 73.4%

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

      1. associate-*r* 73.3%

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

      2. unpow2 73.3%

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

      3. associate-*l* 73.3%

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

      4. times-frac 87.2%

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

      5. unpow2 87.2%

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

      6. associate-*l* 90.9%

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

   4. Simplified 90.9%

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

      1. div-inv 90.9%

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

   6. Applied egg-rr 90.9%

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

   7. Taylor expanded in k around 0 75.7%

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

      1. unpow2 75.7%

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

      2. associate-*l/ 78.2%

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

      3. *-commutative 78.2%

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

   9. Simplified 78.2%

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

   if 6.19999999999999955e-137 < k

   1. Initial program 33.9%

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

   2. Taylor expanded in t around 0 74.7%

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

      1. associate-*r* 74.7%

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

      2. unpow2 74.7%

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

      3. associate-*l* 74.7%

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

      4. times-frac 78.8%

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

      5. unpow2 78.8%

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

      6. associate-*l* 85.1%

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

   4. Simplified 85.1%

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

      1. div-inv 85.1%

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

   6. Applied egg-rr 85.1%

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

   7. Taylor expanded in k around inf 74.7%

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

      1. associate-*r* 74.7%

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

      2. unpow2 74.7%

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

      3. associate-*r* 80.1%

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

      4. times-frac 79.2%

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

      5. unpow2 79.2%

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

      6. associate-/l* 84.3%

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

      7. associate-*l/ 92.3%

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

      8. *-commutative 92.3%

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

      9. associate-*r* 96.2%

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

      10. associate-/r/ 90.7%

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

   9. Simplified 90.7%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-137}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{\ell}{k \cdot \frac{k}{\frac{\ell}{t}}}\right)\\ \end{array} \]

Alternative 6: 90.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{\cos k}{{\sin k}^{2}}\\ \mathbf{if}\;t \leq 1.65 \cdot 10^{-200}:\\ \;\;\;\;\left(\frac{2}{k} \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{t}\right)\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\ell}{k \cdot \frac{k}{\frac{\ell}{t}}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (cos k) (pow (sin k) 2.0))))
   (if (<= t 1.65e-200)
     (* (* (/ 2.0 k) (* l (/ (/ l k) t))) t_1)
     (* 2.0 (* t_1 (/ l (* k (/ k (/ l t)))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = cos(k) / pow(sin(k), 2.0);
	double tmp;
	if (t <= 1.65e-200) {
		tmp = ((2.0 / k) * (l * ((l / k) / t))) * t_1;
	} else {
		tmp = 2.0 * (t_1 * (l / (k * (k / (l / 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 = cos(k) / (sin(k) ** 2.0d0)
    if (t <= 1.65d-200) then
        tmp = ((2.0d0 / k) * (l * ((l / k) / t))) * t_1
    else
        tmp = 2.0d0 * (t_1 * (l / (k * (k / (l / 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 = Math.cos(k) / Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (t <= 1.65e-200) {
		tmp = ((2.0 / k) * (l * ((l / k) / t))) * t_1;
	} else {
		tmp = 2.0 * (t_1 * (l / (k * (k / (l / t)))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = math.cos(k) / math.pow(math.sin(k), 2.0)
	tmp = 0
	if t <= 1.65e-200:
		tmp = ((2.0 / k) * (l * ((l / k) / t))) * t_1
	else:
		tmp = 2.0 * (t_1 * (l / (k * (k / (l / t)))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(cos(k) / (sin(k) ^ 2.0))
	tmp = 0.0
	if (t <= 1.65e-200)
		tmp = Float64(Float64(Float64(2.0 / k) * Float64(l * Float64(Float64(l / k) / t))) * t_1);
	else
		tmp = Float64(2.0 * Float64(t_1 * Float64(l / Float64(k * Float64(k / Float64(l / t))))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = cos(k) / (sin(k) ^ 2.0);
	tmp = 0.0;
	if (t <= 1.65e-200)
		tmp = ((2.0 / k) * (l * ((l / k) / t))) * t_1;
	else
		tmp = 2.0 * (t_1 * (l / (k * (k / (l / 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[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 1.65e-200], N[(N[(N[(2.0 / k), $MachinePrecision] * N[(l * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(2.0 * N[(t$95$1 * N[(l / N[(k * N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.6499999999999999e-200

   1. Initial program 32.9%

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

   2. Taylor expanded in t around 0 74.8%

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

      1. associate-*r* 74.8%

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

      2. unpow2 74.8%

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

      3. associate-*l* 74.8%

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

      4. times-frac 83.5%

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

      5. unpow2 83.5%

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

      6. associate-*l* 90.1%

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

   4. Simplified 90.1%

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

      1. div-inv 90.1%

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

   6. Applied egg-rr 90.1%

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

   7. Taylor expanded in k around inf 74.8%

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

      1. associate-*r* 74.8%

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

      2. unpow2 74.8%

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

      3. associate-*r* 80.8%

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

      4. times-frac 81.2%

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

      5. associate-/l* 82.0%

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

      6. unpow2 82.0%

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

      7. associate-/l* 90.6%

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

      8. associate-/r/ 90.6%

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

      9. *-commutative 90.6%

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

      10. associate-/r* 91.6%

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

   9. Simplified 91.6%

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

   10. Taylor expanded in k around inf 75.0%

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

       1. associate-*r/ 75.0%

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

       2. associate-/l* 74.8%

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

       3. associate-*r* 74.8%

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

       4. unpow2 74.8%

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

       5. associate-*r* 80.8%

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

       6. times-frac 81.2%

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

       7. associate-/l* 82.0%

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

       8. unpow2 82.0%

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

       9. associate-*r/ 90.6%

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

       10. associate-*r/ 90.6%

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

       11. associate-/l* 90.6%

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

       12. associate-/r/ 90.6%

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

   12. Simplified 94.6%

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

   if 1.6499999999999999e-200 < t

   1. Initial program 29.6%

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

   2. Taylor expanded in t around 0 72.1%

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

      1. associate-*r* 72.1%

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

      2. unpow2 72.1%

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

      3. associate-*l* 72.1%

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

      4. times-frac 84.3%

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

      5. unpow2 84.3%

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

      6. associate-*l* 85.4%

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

   4. Simplified 85.4%

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

      1. div-inv 85.4%

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

   6. Applied egg-rr 85.4%

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

   7. Taylor expanded in k around inf 72.1%

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

      1. associate-*r* 72.1%

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

      2. unpow2 72.1%

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

      3. associate-*r* 73.2%

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

      4. times-frac 73.3%

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

      5. unpow2 73.3%

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

      6. associate-/l* 85.4%

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

      7. associate-*l/ 94.3%

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

      8. *-commutative 94.3%

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

      9. associate-*r* 94.4%

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

      10. associate-/r/ 93.2%

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

   9. Simplified 93.2%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{-200}:\\ \;\;\;\;\left(\frac{2}{k} \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{t}\right)\right) \cdot \frac{\cos k}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{\ell}{k \cdot \frac{k}{\frac{\ell}{t}}}\right)\\ \end{array} \]

Alternative 7: 72.9% accurate, 24.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{+67}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 2.45e+67)
   (/ 2.0 (* (/ (* k (* k t)) l) (* k (/ k l))))
   (* -0.3333333333333333 (* (/ l k) (/ (/ l k) t)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.45e+67) {
		tmp = 2.0 / (((k * (k * t)) / l) * (k * (k / l)));
	} 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 <= 2.45d+67) then
        tmp = 2.0d0 / (((k * (k * t)) / l) * (k * (k / l)))
    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 <= 2.45e+67) {
		tmp = 2.0 / (((k * (k * t)) / l) * (k * (k / l)));
	} else {
		tmp = -0.3333333333333333 * ((l / k) * ((l / k) / t));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 2.45e+67:
		tmp = 2.0 / (((k * (k * t)) / l) * (k * (k / l)))
	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 <= 2.45e+67)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k * t)) / l) * Float64(k * Float64(k / l))));
	else
		tmp = Float64(-0.3333333333333333 * Float64(Float64(l / k) * Float64(Float64(l / k) / t)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.45e+67)
		tmp = 2.0 / (((k * (k * t)) / l) * (k * (k / l)));
	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, 2.45e+67], N[(2.0 / N[(N[(N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.44999999999999995e67

   1. Initial program 30.9%

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

   2. Taylor expanded in t around 0 75.1%

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

      1. associate-*r* 75.0%

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

      2. unpow2 75.0%

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

      3. associate-*l* 75.0%

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

      4. times-frac 87.2%

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

      5. unpow2 87.2%

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

      6. associate-*l* 90.1%

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

   4. Simplified 90.1%

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

      1. div-inv 90.1%

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

   6. Applied egg-rr 90.1%

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

   7. Taylor expanded in k around 0 75.1%

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

      1. unpow2 75.1%

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

      2. associate-*l/ 77.1%

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

      3. *-commutative 77.1%

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

   9. Simplified 77.1%

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

   if 2.44999999999999995e67 < k

   1. Initial program 34.9%

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

      1. associate-/r* 34.9%

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

      2. *-commutative 34.9%

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

      3. associate-/r* 34.9%

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

      4. associate-*r/ 34.9%

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

      5. associate-/l* 34.9%

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

      6. +-commutative 34.9%

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

      7. unpow2 34.9%

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

      8. sqr-neg 34.9%

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

      9. distribute-frac-neg 34.9%

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

      10. distribute-frac-neg 34.9%

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

      11. unpow2 34.9%

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

      12. associate--l+ 50.0%

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

      13. metadata-eval 50.0%

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

      14. +-rgt-identity 50.0%

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

      15. unpow2 50.0%

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

      16. distribute-frac-neg 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in k around 0 59.2%

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

      1. +-commutative 59.2%

         \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} + -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]

      2. fma-def 59.2%

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

      3. unpow2 59.2%

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

      4. *-commutative 59.2%

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

      5. times-frac 58.8%

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

      6. associate-*r/ 58.8%

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

      7. *-commutative 58.8%

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

      8. times-frac 58.8%

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

      9. unpow2 58.8%

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

      10. unpow2 58.8%

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

      11. times-frac 62.1%

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

   6. Simplified 62.1%

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

   7. Taylor expanded in k around inf 62.7%

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

      1. unpow2 62.7%

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

      2. associate-*r* 63.5%

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

      3. associate-*r/ 63.5%

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

      4. unpow2 63.5%

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

   9. Simplified 63.5%

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

   10. Taylor expanded in l around 0 62.7%

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

       1. unpow2 62.7%

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

       2. unpow2 62.7%

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

       3. associate-*r* 63.5%

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

       4. associate-/l* 65.5%

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

       5. associate-*l/ 66.2%

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

       6. *-commutative 66.2%

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

       7. associate-*r* 66.6%

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

       8. associate-/r/ 65.8%

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

   12. Simplified 65.8%

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

   13. Taylor expanded in l around 0 62.7%

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

       1. unpow2 62.7%

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

       2. unpow2 62.7%

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

       3. associate-*r* 63.5%

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

       4. *-commutative 63.5%

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

       5. times-frac 66.3%

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

       6. associate-/r* 66.9%

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

   15. Simplified 66.9%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{+67}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\ell} \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right)\\ \end{array} \]

Alternative 8: 34.6% accurate, 38.3× speedup

Math

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

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(Float64(l / 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[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.8%

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

   1. associate-/r* 31.9%

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

   2. *-commutative 31.9%

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

   3. associate-/r* 36.2%

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

   4. associate-*r/ 37.8%

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

   5. associate-/l* 36.2%

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

   6. +-commutative 36.2%

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

   7. unpow2 36.2%

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

   8. sqr-neg 36.2%

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

   9. distribute-frac-neg 36.2%

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

   10. distribute-frac-neg 36.2%

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

   11. unpow2 36.2%

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

   12. associate--l+ 46.2%

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

   13. metadata-eval 46.2%

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

   14. +-rgt-identity 46.2%

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

   15. unpow2 46.2%

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

   16. distribute-frac-neg 46.2%

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

3. Simplified 46.2%

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

4. Taylor expanded in k around 0 34.1%

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

   1. +-commutative 34.1%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} + -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]

   2. fma-def 34.1%

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

   3. unpow2 34.1%

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

   4. *-commutative 34.1%

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

   5. times-frac 34.0%

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

   6. associate-*r/ 34.0%

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

   7. *-commutative 34.0%

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

   8. times-frac 34.9%

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

   9. unpow2 34.9%

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

   10. unpow2 34.9%

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

   11. times-frac 41.7%

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

6. Simplified 41.7%

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

7. Taylor expanded in k around inf 29.2%

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

   1. unpow2 29.2%

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

   2. associate-*r* 29.9%

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

   3. associate-*r/ 29.9%

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

   4. unpow2 29.9%

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

9. Simplified 29.9%

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

10. Taylor expanded in l around 0 29.2%

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

    1. unpow2 29.2%

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

    2. unpow2 29.2%

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

    3. associate-*r* 29.9%

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

    4. associate-/l* 30.5%

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

    5. associate-*l/ 30.8%

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

    6. *-commutative 30.8%

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

    7. associate-*r* 30.9%

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

    8. associate-/r/ 30.7%

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

12. Simplified 30.7%

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

13. Taylor expanded in l around 0 29.2%

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

    1. unpow2 29.2%

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

    2. unpow2 29.2%

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

    3. associate-*r* 29.9%

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

    4. *-commutative 29.9%

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

    5. times-frac 30.8%

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

    6. associate-/r* 31.0%

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

15. Simplified 31.0%

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

16. Final simplification 31.0%

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

Reproduce

herbie shell --seed 2023283 
(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))))