Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.1% → 98.4%

Time: 28.0s

Alternatives: 9

Speedup: 3.8×

• 33.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{\cos k}{k}\\ \mathbf{if}\;k \leq 5.5 \cdot 10^{-128}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot \left(k \cdot t\right)} \cdot \left(t_1 \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(t_1 \cdot \frac{\ell}{{\sin k}^{2}}\right) \cdot \frac{\frac{\ell}{k}}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (cos k) k)))
   (if (<= k 5.5e-128)
     (* 2.0 (* (/ l (* k (* k t))) (* t_1 (/ l k))))
     (* 2.0 (* (* t_1 (/ l (pow (sin k) 2.0))) (/ (/ l k) t))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = cos(k) / k;
	double tmp;
	if (k <= 5.5e-128) {
		tmp = 2.0 * ((l / (k * (k * t))) * (t_1 * (l / k)));
	} else {
		tmp = 2.0 * ((t_1 * (l / pow(sin(k), 2.0))) * ((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) :: t_1
    real(8) :: tmp
    t_1 = cos(k) / k
    if (k <= 5.5d-128) then
        tmp = 2.0d0 * ((l / (k * (k * t))) * (t_1 * (l / k)))
    else
        tmp = 2.0d0 * ((t_1 * (l / (sin(k) ** 2.0d0))) * ((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 t_1 = Math.cos(k) / k;
	double tmp;
	if (k <= 5.5e-128) {
		tmp = 2.0 * ((l / (k * (k * t))) * (t_1 * (l / k)));
	} else {
		tmp = 2.0 * ((t_1 * (l / Math.pow(Math.sin(k), 2.0))) * ((l / k) / t));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = math.cos(k) / k
	tmp = 0
	if k <= 5.5e-128:
		tmp = 2.0 * ((l / (k * (k * t))) * (t_1 * (l / k)))
	else:
		tmp = 2.0 * ((t_1 * (l / math.pow(math.sin(k), 2.0))) * ((l / k) / t))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(cos(k) / k)
	tmp = 0.0
	if (k <= 5.5e-128)
		tmp = Float64(2.0 * Float64(Float64(l / Float64(k * Float64(k * t))) * Float64(t_1 * Float64(l / k))));
	else
		tmp = Float64(2.0 * Float64(Float64(t_1 * Float64(l / (sin(k) ^ 2.0))) * Float64(Float64(l / k) / t)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = cos(k) / k;
	tmp = 0.0;
	if (k <= 5.5e-128)
		tmp = 2.0 * ((l / (k * (k * t))) * (t_1 * (l / k)));
	else
		tmp = 2.0 * ((t_1 * (l / (sin(k) ^ 2.0))) * ((l / k) / t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]}, If[LessEqual[k, 5.5e-128], N[(2.0 * N[(N[(l / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(t$95$1 * N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.5000000000000004e-128

   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* 31.8%

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

      2. *-commutative 31.8%

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

      3. associate-/r* 37.3%

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

      4. associate-*r/ 38.7%

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

      5. associate-/l* 37.3%

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

      6. +-commutative 37.3%

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

      7. unpow2 37.3%

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

      8. sqr-neg 37.3%

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

      9. distribute-frac-neg 37.3%

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

      10. distribute-frac-neg 37.3%

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

      11. unpow2 37.3%

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

      12. associate--l+ 43.2%

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

      13. metadata-eval 43.2%

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

      14. +-rgt-identity 43.2%

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

      15. unpow2 43.2%

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

      16. distribute-frac-neg 43.2%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in k around inf 60.7%

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

      1. unpow2 60.7%

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

      2. *-commutative 60.7%

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

      3. associate-*l* 60.7%

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

      4. *-commutative 60.7%

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

      5. times-frac 77.4%

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

      6. *-commutative 77.4%

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

      7. unpow2 77.4%

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

   6. Simplified 77.4%

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

      1. times-frac 87.2%

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

   8. Applied egg-rr 87.2%

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

   9. Taylor expanded in k around 0 67.0%

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

       1. unpow2 67.0%

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

       2. associate-*l* 71.3%

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

   11. Simplified 71.3%

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

   if 5.5000000000000004e-128 < k

   1. Initial program 29.4%

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

      1. associate-/r* 29.4%

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

      2. *-commutative 29.4%

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

      3. associate-/r* 32.1%

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

      4. associate-*r/ 32.1%

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

      5. associate-/l* 32.1%

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

      6. +-commutative 32.1%

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

      7. unpow2 32.1%

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

      8. sqr-neg 32.1%

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

      9. distribute-frac-neg 32.1%

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

      10. distribute-frac-neg 32.1%

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

      11. unpow2 32.1%

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

      12. associate--l+ 47.7%

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

      13. metadata-eval 47.7%

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

      14. +-rgt-identity 47.7%

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

      15. unpow2 47.7%

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

      16. distribute-frac-neg 47.7%

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

   3. Simplified 47.7%

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

   4. Taylor expanded in k around inf 71.6%

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

      1. unpow2 71.6%

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

      2. *-commutative 71.6%

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

      3. associate-*l* 71.7%

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

      4. *-commutative 71.7%

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

      5. times-frac 87.1%

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

      6. *-commutative 87.1%

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

      7. unpow2 87.1%

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

   6. Simplified 87.1%

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

      1. times-frac 93.2%

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

   8. Applied egg-rr 93.2%

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

   9. Taylor expanded in l around 0 71.6%

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

       1. unpow2 71.6%

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

       2. times-frac 71.7%

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

       3. unpow2 71.7%

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

       4. associate-*l/ 80.8%

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

       5. associate-/l/ 80.8%

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

       6. associate-*l/ 80.8%

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

       7. *-commutative 80.8%

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

       8. associate-*r* 80.8%

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

       9. associate-*r/ 84.6%

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

       10. *-commutative 84.6%

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

       11. times-frac 90.7%

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

       12. associate-/l/ 93.2%

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

       13. associate-*l/ 90.9%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in k around inf 97.1%

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

       1. times-frac 99.6%

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

   14. Simplified 99.6%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-128}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\cos k}{k} \cdot \frac{\ell}{{\sin k}^{2}}\right) \cdot \frac{\frac{\ell}{k}}{t}\right)\\ \end{array} \]

Alternative 2: 77.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{\cos k}{k} \cdot \frac{\ell}{k}\\ \mathbf{if}\;k \leq 2 \cdot 10^{-124}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot \left(k \cdot t\right)} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \left(\frac{\ell}{t \cdot \left(k \cdot k\right)} + 0.3333333333333333 \cdot \frac{\ell}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ (cos k) k) (/ l k))))
   (if (<= k 2e-124)
     (* 2.0 (* (/ l (* k (* k t))) t_1))
     (* 2.0 (* t_1 (+ (/ l (* t (* k k))) (* 0.3333333333333333 (/ l t))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = (cos(k) / k) * (l / k);
	double tmp;
	if (k <= 2e-124) {
		tmp = 2.0 * ((l / (k * (k * t))) * t_1);
	} else {
		tmp = 2.0 * (t_1 * ((l / (t * (k * k))) + (0.3333333333333333 * (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) / k) * (l / k)
    if (k <= 2d-124) then
        tmp = 2.0d0 * ((l / (k * (k * t))) * t_1)
    else
        tmp = 2.0d0 * (t_1 * ((l / (t * (k * k))) + (0.3333333333333333d0 * (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) / k) * (l / k);
	double tmp;
	if (k <= 2e-124) {
		tmp = 2.0 * ((l / (k * (k * t))) * t_1);
	} else {
		tmp = 2.0 * (t_1 * ((l / (t * (k * k))) + (0.3333333333333333 * (l / t))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = (math.cos(k) / k) * (l / k)
	tmp = 0
	if k <= 2e-124:
		tmp = 2.0 * ((l / (k * (k * t))) * t_1)
	else:
		tmp = 2.0 * (t_1 * ((l / (t * (k * k))) + (0.3333333333333333 * (l / t))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(Float64(cos(k) / k) * Float64(l / k))
	tmp = 0.0
	if (k <= 2e-124)
		tmp = Float64(2.0 * Float64(Float64(l / Float64(k * Float64(k * t))) * t_1));
	else
		tmp = Float64(2.0 * Float64(t_1 * Float64(Float64(l / Float64(t * Float64(k * k))) + Float64(0.3333333333333333 * Float64(l / t)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = (cos(k) / k) * (l / k);
	tmp = 0.0;
	if (k <= 2e-124)
		tmp = 2.0 * ((l / (k * (k * t))) * t_1);
	else
		tmp = 2.0 * (t_1 * ((l / (t * (k * k))) + (0.3333333333333333 * (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[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2e-124], N[(2.0 * N[(N[(l / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$1 * N[(N[(l / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.99999999999999987e-124

   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. Step-by-step derivation

      1. associate-/r* 31.6%

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

      2. *-commutative 31.6%

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

      3. associate-/r* 37.0%

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

      4. associate-*r/ 38.4%

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

      5. associate-/l* 37.0%

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

      6. +-commutative 37.0%

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

      7. unpow2 37.0%

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

      8. sqr-neg 37.0%

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

      9. distribute-frac-neg 37.0%

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

      10. distribute-frac-neg 37.0%

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

      11. unpow2 37.0%

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

      12. associate--l+ 42.9%

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

      13. metadata-eval 42.9%

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

      14. +-rgt-identity 42.9%

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

      15. unpow2 42.9%

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

      16. distribute-frac-neg 42.9%

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

   3. Simplified 42.9%

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

   4. Taylor expanded in k around inf 60.3%

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

      1. unpow2 60.3%

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

      2. *-commutative 60.3%

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

      3. associate-*l* 60.3%

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

      4. *-commutative 60.3%

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

      5. times-frac 77.6%

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

      6. *-commutative 77.6%

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

      7. unpow2 77.6%

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

   6. Simplified 77.6%

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

      1. times-frac 87.2%

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

   8. Applied egg-rr 87.2%

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

   9. Taylor expanded in k around 0 67.2%

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

       1. unpow2 67.2%

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

       2. associate-*l* 71.5%

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

   11. Simplified 71.5%

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

   if 1.99999999999999987e-124 < k

   1. Initial program 29.7%

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

      1. associate-/r* 29.7%

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

      2. *-commutative 29.7%

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

      3. associate-/r* 32.4%

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

      4. associate-*r/ 32.4%

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

      5. associate-/l* 32.4%

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

      6. +-commutative 32.4%

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

      7. unpow2 32.4%

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

      8. sqr-neg 32.4%

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

      9. distribute-frac-neg 32.4%

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

      10. distribute-frac-neg 32.4%

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

      11. unpow2 32.4%

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

      12. associate--l+ 48.1%

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

      13. metadata-eval 48.1%

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

      14. +-rgt-identity 48.1%

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

      15. unpow2 48.1%

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

      16. distribute-frac-neg 48.1%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in k around inf 72.2%

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

      1. unpow2 72.2%

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

      2. *-commutative 72.2%

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

      3. associate-*l* 72.3%

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

      4. *-commutative 72.3%

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

      5. times-frac 87.0%

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

      6. *-commutative 87.0%

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

      7. unpow2 87.0%

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

   6. Simplified 87.0%

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

      1. times-frac 93.1%

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

   8. Applied egg-rr 93.1%

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

   9. Taylor expanded in k around 0 77.2%

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

       1. unpow2 77.2%

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

   11. Simplified 77.2%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-124}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\cos k}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\ell}{t \cdot \left(k \cdot k\right)} + 0.3333333333333333 \cdot \frac{\ell}{t}\right)\right)\\ \end{array} \]

Alternative 3: 73.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := k \cdot \left(k \cdot t\right)\\ \mathbf{if}\;k \leq 7.6 \cdot 10^{+97}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t_1} \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{-0.3333333333333333}{t_1}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (* k t))))
   (if (<= k 7.6e+97)
     (* 2.0 (* (/ l t_1) (* (/ (cos k) k) (/ l k))))
     (* l (* l (/ -0.3333333333333333 t_1))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = k * (k * t);
	double tmp;
	if (k <= 7.6e+97) {
		tmp = 2.0 * ((l / t_1) * ((cos(k) / k) * (l / k)));
	} else {
		tmp = l * (l * (-0.3333333333333333 / t_1));
	}
	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 = k * (k * t)
    if (k <= 7.6d+97) then
        tmp = 2.0d0 * ((l / t_1) * ((cos(k) / k) * (l / k)))
    else
        tmp = l * (l * ((-0.3333333333333333d0) / t_1))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = k * (k * t);
	double tmp;
	if (k <= 7.6e+97) {
		tmp = 2.0 * ((l / t_1) * ((Math.cos(k) / k) * (l / k)));
	} else {
		tmp = l * (l * (-0.3333333333333333 / t_1));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = k * (k * t)
	tmp = 0
	if k <= 7.6e+97:
		tmp = 2.0 * ((l / t_1) * ((math.cos(k) / k) * (l / k)))
	else:
		tmp = l * (l * (-0.3333333333333333 / t_1))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(k * Float64(k * t))
	tmp = 0.0
	if (k <= 7.6e+97)
		tmp = Float64(2.0 * Float64(Float64(l / t_1) * Float64(Float64(cos(k) / k) * Float64(l / k))));
	else
		tmp = Float64(l * Float64(l * Float64(-0.3333333333333333 / t_1)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = k * (k * t);
	tmp = 0.0;
	if (k <= 7.6e+97)
		tmp = 2.0 * ((l / t_1) * ((cos(k) / k) * (l / k)));
	else
		tmp = l * (l * (-0.3333333333333333 / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 7.6e+97], N[(2.0 * N[(N[(l / t$95$1), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(-0.3333333333333333 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 7.60000000000000071e97

   1. Initial program 30.0%

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

      1. associate-/r* 30.0%

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

      2. *-commutative 30.0%

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

      3. associate-/r* 34.3%

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

      4. associate-*r/ 35.3%

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

      5. associate-/l* 34.3%

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

      6. +-commutative 34.3%

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

      7. unpow2 34.3%

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

      8. sqr-neg 34.3%

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

      9. distribute-frac-neg 34.3%

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

      10. distribute-frac-neg 34.3%

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

      11. unpow2 34.3%

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

      12. associate--l+ 42.2%

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

      13. metadata-eval 42.2%

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

      14. +-rgt-identity 42.2%

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

      15. unpow2 42.2%

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

      16. distribute-frac-neg 42.2%

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

   3. Simplified 42.2%

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

   4. Taylor expanded in k around inf 67.0%

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

      1. unpow2 67.0%

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

      2. *-commutative 67.0%

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

      3. associate-*l* 67.1%

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

      4. *-commutative 67.1%

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

      5. times-frac 84.3%

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

      6. *-commutative 84.3%

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

      7. unpow2 84.3%

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

   6. Simplified 84.3%

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

      1. times-frac 91.0%

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

   8. Applied egg-rr 91.0%

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

   9. Taylor expanded in k around 0 70.8%

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

       1. unpow2 70.8%

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

       2. associate-*l* 73.8%

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

   11. Simplified 73.8%

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

   if 7.60000000000000071e97 < k

   1. Initial program 33.8%

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

      1. associate-/r* 33.8%

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

      2. *-commutative 33.8%

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

      3. associate-/r* 37.7%

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

      4. associate-*r/ 37.7%

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

      5. associate-/l* 37.7%

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

      6. +-commutative 37.7%

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

      7. unpow2 37.7%

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

      8. sqr-neg 37.7%

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

      9. distribute-frac-neg 37.7%

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

      10. distribute-frac-neg 37.7%

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

      11. unpow2 37.7%

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

      12. associate--l+ 57.5%

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

      13. metadata-eval 57.5%

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

      14. +-rgt-identity 57.5%

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

      15. unpow2 57.5%

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

      16. distribute-frac-neg 57.5%

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

   3. Simplified 57.5%

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

   4. Taylor expanded in k around 0 40.3%

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

      1. fma-def 40.3%

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

      2. unpow2 40.3%

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

      3. times-frac 48.1%

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

      4. unpow2 48.1%

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

      5. distribute-rgt-out 48.1%

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

      6. metadata-eval 48.1%

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

      7. unpow2 48.1%

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

      8. unpow2 48.1%

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

      9. *-commutative 48.1%

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

   6. Simplified 48.1%

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

   7. Taylor expanded in l around 0 56.2%

      \[\leadsto \color{blue}{{\ell}^{2} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)} \]
   8. Step-by-step derivation

      1. unpow2 56.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

      2. sub-neg 56.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(2 \cdot \frac{1}{{k}^{4} \cdot t} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)} \]

      3. associate-*r/ 56.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot t}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right) \]

      4. metadata-eval 56.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\color{blue}{2}}{{k}^{4} \cdot t} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right) \]

      5. *-commutative 56.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{\color{blue}{t \cdot {k}^{4}}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right) \]

      6. associate-/r* 56.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right) \]

      7. associate-*r/ 56.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\frac{2}{t}}{{k}^{4}} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{k}^{2} \cdot t}}\right)\right) \]

      8. metadata-eval 56.2%

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

      9. distribute-neg-frac 56.2%

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

      10. metadata-eval 56.2%

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

      11. unpow2 56.2%

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

      12. associate-*l* 58.9%

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

   9. Simplified 58.9%

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

       1. pow1 58.9%

          \[\leadsto \color{blue}{{\left(\left(\ell \cdot \ell\right) \cdot \left(\frac{\frac{2}{t}}{{k}^{4}} + \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\right)\right)}^{1}} \]

       2. associate-*l* 67.6%

          \[\leadsto {\color{blue}{\left(\ell \cdot \left(\ell \cdot \left(\frac{\frac{2}{t}}{{k}^{4}} + \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\right)\right)\right)}}^{1} \]

       3. associate-/l/ 67.6%

          \[\leadsto {\left(\ell \cdot \left(\ell \cdot \left(\color{blue}{\frac{2}{{k}^{4} \cdot t}} + \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\right)\right)\right)}^{1} \]

   11. Applied egg-rr 67.6%

       \[\leadsto \color{blue}{{\left(\ell \cdot \left(\ell \cdot \left(\frac{2}{{k}^{4} \cdot t} + \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\right)\right)\right)}^{1}} \]

   12. Taylor expanded in k around inf 64.6%

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

       1. unpow2 64.6%

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

       2. associate-*r* 67.6%

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

       3. associate-*r/ 67.6%

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

       4. associate-*l/ 67.6%

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

       5. *-commutative 67.6%

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

   14. Simplified 67.6%

       \[\leadsto {\left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\right)}\right)}^{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.6 \cdot 10^{+97}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\right)\\ \end{array} \]

Alternative 4: 74.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 8.8 \cdot 10^{-114}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{k \cdot k}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 8.8e-114)
   (* 2.0 (* (/ l (* k (* k t))) (* (/ (cos k) k) (/ l k))))
   (* 2.0 (* (/ (/ l t) (* k k)) (/ (* l (cos k)) (* k k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 8.8e-114) {
		tmp = 2.0 * ((l / (k * (k * t))) * ((cos(k) / k) * (l / k)));
	} else {
		tmp = 2.0 * (((l / t) / (k * k)) * ((l * cos(k)) / (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 <= 8.8d-114) then
        tmp = 2.0d0 * ((l / (k * (k * t))) * ((cos(k) / k) * (l / k)))
    else
        tmp = 2.0d0 * (((l / t) / (k * k)) * ((l * cos(k)) / (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 <= 8.8e-114) {
		tmp = 2.0 * ((l / (k * (k * t))) * ((Math.cos(k) / k) * (l / k)));
	} else {
		tmp = 2.0 * (((l / t) / (k * k)) * ((l * Math.cos(k)) / (k * k)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 8.8e-114:
		tmp = 2.0 * ((l / (k * (k * t))) * ((math.cos(k) / k) * (l / k)))
	else:
		tmp = 2.0 * (((l / t) / (k * k)) * ((l * math.cos(k)) / (k * k)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 8.8e-114)
		tmp = Float64(2.0 * Float64(Float64(l / Float64(k * Float64(k * t))) * Float64(Float64(cos(k) / k) * Float64(l / k))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / t) / Float64(k * k)) * Float64(Float64(l * cos(k)) / Float64(k * k))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 8.8e-114)
		tmp = 2.0 * ((l / (k * (k * t))) * ((cos(k) / k) * (l / k)));
	else
		tmp = 2.0 * (((l / t) / (k * k)) * ((l * cos(k)) / (k * k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 8.8e-114], N[(2.0 * N[(N[(l / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / t), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.80000000000000045e-114

   1. Initial program 31.7%

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

      1. associate-/r* 31.7%

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

      2. *-commutative 31.7%

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

      3. associate-/r* 36.9%

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

      4. associate-*r/ 38.3%

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

      5. associate-/l* 36.9%

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

      6. +-commutative 36.9%

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

      7. unpow2 36.9%

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

      8. sqr-neg 36.9%

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

      9. distribute-frac-neg 36.9%

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

      10. distribute-frac-neg 36.9%

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

      11. unpow2 36.9%

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

      12. associate--l+ 42.7%

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

      13. metadata-eval 42.7%

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

      14. +-rgt-identity 42.7%

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

      15. unpow2 42.7%

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

      16. distribute-frac-neg 42.7%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in k around inf 59.8%

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

      1. unpow2 59.8%

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

      2. *-commutative 59.8%

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

      3. associate-*l* 59.9%

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

      4. *-commutative 59.9%

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

      5. times-frac 78.0%

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

      6. *-commutative 78.0%

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

      7. unpow2 78.0%

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

   6. Simplified 78.0%

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

      1. times-frac 87.5%

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

   8. Applied egg-rr 87.5%

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

   9. Taylor expanded in k around 0 67.9%

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

       1. unpow2 67.9%

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

       2. associate-*l* 72.1%

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

   11. Simplified 72.1%

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

   if 8.80000000000000045e-114 < k

   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. Step-by-step derivation

      1. associate-/r* 29.6%

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

      2. *-commutative 29.6%

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

      3. associate-/r* 32.4%

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

      4. associate-*r/ 32.4%

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

      5. associate-/l* 32.4%

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

      6. +-commutative 32.4%

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

      7. unpow2 32.4%

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

      8. sqr-neg 32.4%

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

      9. distribute-frac-neg 32.4%

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

      10. distribute-frac-neg 32.4%

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

      11. unpow2 32.4%

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

      12. associate--l+ 48.6%

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

      13. metadata-eval 48.6%

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

      14. +-rgt-identity 48.6%

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

      15. unpow2 48.6%

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

      16. distribute-frac-neg 48.6%

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

   3. Simplified 48.6%

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

   4. Taylor expanded in k around inf 73.2%

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

      1. unpow2 73.2%

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

      2. *-commutative 73.2%

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

      3. associate-*l* 73.2%

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

      4. *-commutative 73.2%

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

      5. times-frac 86.6%

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

      6. *-commutative 86.6%

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

      7. unpow2 86.6%

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

   6. Simplified 86.6%

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

      1. div-inv 86.6%

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

   8. Applied egg-rr 86.6%

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

      1. associate-*r/ 86.6%

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

      2. *-rgt-identity 86.6%

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

      3. *-commutative 86.6%

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

      4. associate-/r* 86.6%

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

   10. Simplified 86.6%

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

   11. Taylor expanded in k around 0 72.5%

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

       1. unpow2 72.5%

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

   13. Simplified 72.5%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.8 \cdot 10^{-114}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell \cdot \cos k}{k \cdot k}\right)\\ \end{array} \]

Alternative 5: 67.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.58 \cdot 10^{-112}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{t \cdot {k}^{4}}\right)\\ \mathbf{elif}\;k \leq 3.55 \cdot 10^{+44}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{-0.3333333333333333}{k \cdot k}}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.58e-112)
   (* l (* l (/ 2.0 (* t (pow k 4.0)))))
   (if (<= k 3.55e+44)
     (* 2.0 (* (/ l t) (/ l (pow k 4.0))))
     (* l (* l (/ (/ -0.3333333333333333 (* k k)) t))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.58e-112) {
		tmp = l * (l * (2.0 / (t * pow(k, 4.0))));
	} else if (k <= 3.55e+44) {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	} else {
		tmp = l * (l * ((-0.3333333333333333 / (k * k)) / t));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.58d-112) then
        tmp = l * (l * (2.0d0 / (t * (k ** 4.0d0))))
    else if (k <= 3.55d+44) then
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    else
        tmp = l * (l * (((-0.3333333333333333d0) / (k * k)) / t))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.58e-112) {
		tmp = l * (l * (2.0 / (t * Math.pow(k, 4.0))));
	} else if (k <= 3.55e+44) {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	} else {
		tmp = l * (l * ((-0.3333333333333333 / (k * k)) / t));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.58e-112:
		tmp = l * (l * (2.0 / (t * math.pow(k, 4.0))))
	elif k <= 3.55e+44:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	else:
		tmp = l * (l * ((-0.3333333333333333 / (k * k)) / t))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.58e-112)
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(t * (k ^ 4.0)))));
	elseif (k <= 3.55e+44)
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	else
		tmp = Float64(l * Float64(l * Float64(Float64(-0.3333333333333333 / Float64(k * k)) / t)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.58e-112)
		tmp = l * (l * (2.0 / (t * (k ^ 4.0))));
	elseif (k <= 3.55e+44)
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	else
		tmp = l * (l * ((-0.3333333333333333 / (k * k)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.58e-112], N[(l * N[(l * N[(2.0 / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.55e+44], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(N[(-0.3333333333333333 / N[(k * k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.5800000000000001e-112

   1. Initial program 31.7%

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

      1. associate-/r* 31.7%

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

      2. *-commutative 31.7%

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

      3. associate-/r* 36.9%

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

      4. associate-*r/ 38.3%

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

      5. associate-/l* 36.9%

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

      6. +-commutative 36.9%

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

      7. unpow2 36.9%

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

      8. sqr-neg 36.9%

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

      9. distribute-frac-neg 36.9%

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

      10. distribute-frac-neg 36.9%

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

      11. unpow2 36.9%

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

      12. associate--l+ 42.7%

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

      13. metadata-eval 42.7%

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

      14. +-rgt-identity 42.7%

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

      15. unpow2 42.7%

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

      16. distribute-frac-neg 42.7%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in k around 0 18.3%

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

      1. fma-def 18.3%

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

      2. unpow2 18.3%

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

      3. times-frac 19.9%

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

      4. unpow2 19.9%

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

      5. distribute-rgt-out 19.9%

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

      6. metadata-eval 19.9%

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

      7. unpow2 19.9%

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

      8. unpow2 19.9%

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

      9. *-commutative 19.9%

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

   6. Simplified 19.9%

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

   7. Taylor expanded in l around 0 24.5%

      \[\leadsto \color{blue}{{\ell}^{2} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)} \]
   8. Step-by-step derivation

      1. unpow2 24.5%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

      2. sub-neg 24.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(2 \cdot \frac{1}{{k}^{4} \cdot t} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)} \]

      3. associate-*r/ 24.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot t}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right) \]

      4. metadata-eval 24.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\color{blue}{2}}{{k}^{4} \cdot t} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right) \]

      5. *-commutative 24.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{\color{blue}{t \cdot {k}^{4}}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right) \]

      6. associate-/r* 24.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right) \]

      7. associate-*r/ 24.5%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\frac{2}{t}}{{k}^{4}} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{k}^{2} \cdot t}}\right)\right) \]

      8. metadata-eval 24.5%

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

      9. distribute-neg-frac 24.5%

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

      10. metadata-eval 24.5%

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

      11. unpow2 24.5%

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

      12. associate-*l* 30.9%

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

   9. Simplified 30.9%

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

   10. Taylor expanded in k around 0 52.5%

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

       1. associate-*r/ 52.5%

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

       2. associate-*l/ 52.5%

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

       3. *-commutative 52.5%

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

       4. unpow2 52.5%

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

       5. associate-*l* 60.4%

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

       6. *-commutative 60.4%

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

   12. Simplified 60.4%

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

   if 1.5800000000000001e-112 < k < 3.55e44

   1. Initial program 27.8%

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

      1. associate-/r* 27.8%

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

      2. *-commutative 27.8%

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

      3. associate-/r* 30.0%

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

      4. associate-*r/ 30.0%

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

      5. associate-/l* 29.9%

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

      6. +-commutative 29.9%

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

      7. unpow2 29.9%

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

      8. sqr-neg 29.9%

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

      9. distribute-frac-neg 29.9%

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

      10. distribute-frac-neg 29.9%

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

      11. unpow2 29.9%

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

      12. associate--l+ 43.8%

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

      13. metadata-eval 43.8%

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

      14. +-rgt-identity 43.8%

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

      15. unpow2 43.8%

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

      16. distribute-frac-neg 43.8%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in k around 0 70.2%

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

      1. unpow2 70.2%

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

      2. *-commutative 70.2%

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

   6. Simplified 70.2%

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

      1. times-frac 80.6%

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

   8. Applied egg-rr 80.6%

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

   if 3.55e44 < k

   1. Initial program 31.0%

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

      1. associate-/r* 31.0%

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

      2. *-commutative 31.0%

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

      3. associate-/r* 34.2%

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

      4. associate-*r/ 34.2%

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

      5. associate-/l* 34.2%

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

      6. +-commutative 34.2%

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

      7. unpow2 34.2%

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

      8. sqr-neg 34.2%

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

      9. distribute-frac-neg 34.2%

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

      10. distribute-frac-neg 34.2%

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

      11. unpow2 34.2%

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

      12. associate--l+ 52.2%

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

      13. metadata-eval 52.2%

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

      14. +-rgt-identity 52.2%

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

      15. unpow2 52.2%

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

      16. distribute-frac-neg 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in k around 0 36.5%

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

      1. fma-def 36.5%

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

      2. unpow2 36.5%

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

      3. times-frac 43.2%

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

      4. unpow2 43.2%

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

      5. distribute-rgt-out 43.2%

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

      6. metadata-eval 43.2%

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

      7. unpow2 43.2%

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

      8. unpow2 43.2%

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

      9. *-commutative 43.2%

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

   6. Simplified 43.2%

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

   7. Taylor expanded in k around inf 53.1%

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

      1. associate-*r/ 53.1%

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

      2. times-frac 53.1%

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

      3. unpow2 53.1%

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

      4. unpow2 53.1%

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

      5. associate-*l/ 59.7%

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

   9. Simplified 59.7%

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

   10. Taylor expanded in k around 0 53.1%

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

       1. associate-*r/ 53.1%

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

       2. unpow2 53.1%

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

       3. associate-*r* 55.4%

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

       4. associate-*l/ 55.4%

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

       5. *-commutative 55.4%

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

       6. unpow2 55.4%

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

   12. Simplified 55.4%

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

   13. Taylor expanded in l around 0 53.1%

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

       1. unpow2 53.1%

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

       2. associate-*r* 55.4%

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

       3. associate-*r/ 55.4%

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

       4. *-commutative 55.4%

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

       5. associate-*r/ 55.4%

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

       6. unpow2 55.4%

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

       7. associate-*l* 62.5%

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

       8. associate-/r* 62.5%

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

       9. associate-/r* 60.1%

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

       10. associate-/r* 60.1%

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

   15. Simplified 60.1%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.58 \cdot 10^{-112}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{t \cdot {k}^{4}}\right)\\ \mathbf{elif}\;k \leq 3.55 \cdot 10^{+44}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{-0.3333333333333333}{k \cdot k}}{t}\right)\\ \end{array} \]

Alternative 6: 69.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 7.6 \cdot 10^{+97}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\ell}{{k}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 7.6e+97)
   (* 2.0 (* (/ (/ l k) t) (/ l (pow k 3.0))))
   (* l (* l (/ -0.3333333333333333 (* k (* k t)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 7.6e+97) {
		tmp = 2.0 * (((l / k) / t) * (l / pow(k, 3.0)));
	} else {
		tmp = l * (l * (-0.3333333333333333 / (k * (k * t))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7.6d+97) then
        tmp = 2.0d0 * (((l / k) / t) * (l / (k ** 3.0d0)))
    else
        tmp = l * (l * ((-0.3333333333333333d0) / (k * (k * t))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 7.6e+97) {
		tmp = 2.0 * (((l / k) / t) * (l / Math.pow(k, 3.0)));
	} else {
		tmp = l * (l * (-0.3333333333333333 / (k * (k * t))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 7.6e+97:
		tmp = 2.0 * (((l / k) / t) * (l / math.pow(k, 3.0)))
	else:
		tmp = l * (l * (-0.3333333333333333 / (k * (k * t))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 7.6e+97)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) / t) * Float64(l / (k ^ 3.0))));
	else
		tmp = Float64(l * Float64(l * Float64(-0.3333333333333333 / Float64(k * Float64(k * t)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 7.6e+97)
		tmp = 2.0 * (((l / k) / t) * (l / (k ^ 3.0)));
	else
		tmp = l * (l * (-0.3333333333333333 / (k * (k * t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 7.6e+97], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(-0.3333333333333333 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.60000000000000071e97

   1. Initial program 30.0%

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

      1. associate-/r* 30.0%

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

      2. *-commutative 30.0%

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

      3. associate-/r* 34.3%

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

      4. associate-*r/ 35.3%

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

      5. associate-/l* 34.3%

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

      6. +-commutative 34.3%

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

      7. unpow2 34.3%

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

      8. sqr-neg 34.3%

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

      9. distribute-frac-neg 34.3%

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

      10. distribute-frac-neg 34.3%

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

      11. unpow2 34.3%

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

      12. associate--l+ 42.2%

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

      13. metadata-eval 42.2%

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

      14. +-rgt-identity 42.2%

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

      15. unpow2 42.2%

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

      16. distribute-frac-neg 42.2%

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

   3. Simplified 42.2%

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

   4. Taylor expanded in k around inf 67.0%

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

      1. unpow2 67.0%

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

      2. *-commutative 67.0%

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

      3. associate-*l* 67.1%

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

      4. *-commutative 67.1%

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

      5. times-frac 84.3%

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

      6. *-commutative 84.3%

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

      7. unpow2 84.3%

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

   6. Simplified 84.3%

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

      1. times-frac 91.0%

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

   8. Applied egg-rr 91.0%

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

   9. Taylor expanded in l around 0 67.0%

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

       1. unpow2 67.0%

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

       2. times-frac 68.5%

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

       3. unpow2 68.5%

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

       4. associate-*l/ 78.8%

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

       5. associate-/l/ 78.1%

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

       6. associate-*l/ 78.2%

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

       7. *-commutative 78.2%

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

       8. associate-*r* 78.2%

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

       9. associate-*r/ 82.1%

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

       10. *-commutative 82.1%

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

       11. times-frac 88.8%

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

       12. associate-/l/ 91.0%

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

       13. associate-*l/ 86.4%

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

   11. Simplified 95.1%

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

   12. Taylor expanded in k around 0 67.2%

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

   if 7.60000000000000071e97 < k

   1. Initial program 33.8%

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

      1. associate-/r* 33.8%

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

      2. *-commutative 33.8%

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

      3. associate-/r* 37.7%

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

      4. associate-*r/ 37.7%

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

      5. associate-/l* 37.7%

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

      6. +-commutative 37.7%

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

      7. unpow2 37.7%

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

      8. sqr-neg 37.7%

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

      9. distribute-frac-neg 37.7%

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

      10. distribute-frac-neg 37.7%

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

      11. unpow2 37.7%

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

      12. associate--l+ 57.5%

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

      13. metadata-eval 57.5%

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

      14. +-rgt-identity 57.5%

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

      15. unpow2 57.5%

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

      16. distribute-frac-neg 57.5%

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

   3. Simplified 57.5%

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

   4. Taylor expanded in k around 0 40.3%

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

      1. fma-def 40.3%

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

      2. unpow2 40.3%

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

      3. times-frac 48.1%

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

      4. unpow2 48.1%

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

      5. distribute-rgt-out 48.1%

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

      6. metadata-eval 48.1%

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

      7. unpow2 48.1%

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

      8. unpow2 48.1%

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

      9. *-commutative 48.1%

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

   6. Simplified 48.1%

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

   7. Taylor expanded in l around 0 56.2%

      \[\leadsto \color{blue}{{\ell}^{2} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)} \]
   8. Step-by-step derivation

      1. unpow2 56.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 \cdot \frac{1}{{k}^{4} \cdot t} - 0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right) \]

      2. sub-neg 56.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(2 \cdot \frac{1}{{k}^{4} \cdot t} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right)} \]

      3. associate-*r/ 56.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{2 \cdot 1}{{k}^{4} \cdot t}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right) \]

      4. metadata-eval 56.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\color{blue}{2}}{{k}^{4} \cdot t} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right) \]

      5. *-commutative 56.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{2}{\color{blue}{t \cdot {k}^{4}}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right) \]

      6. associate-/r* 56.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\color{blue}{\frac{\frac{2}{t}}{{k}^{4}}} + \left(-0.3333333333333333 \cdot \frac{1}{{k}^{2} \cdot t}\right)\right) \]

      7. associate-*r/ 56.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \left(\frac{\frac{2}{t}}{{k}^{4}} + \left(-\color{blue}{\frac{0.3333333333333333 \cdot 1}{{k}^{2} \cdot t}}\right)\right) \]

      8. metadata-eval 56.2%

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

      9. distribute-neg-frac 56.2%

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

      10. metadata-eval 56.2%

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

      11. unpow2 56.2%

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

      12. associate-*l* 58.9%

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

   9. Simplified 58.9%

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

       1. pow1 58.9%

          \[\leadsto \color{blue}{{\left(\left(\ell \cdot \ell\right) \cdot \left(\frac{\frac{2}{t}}{{k}^{4}} + \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\right)\right)}^{1}} \]

       2. associate-*l* 67.6%

          \[\leadsto {\color{blue}{\left(\ell \cdot \left(\ell \cdot \left(\frac{\frac{2}{t}}{{k}^{4}} + \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\right)\right)\right)}}^{1} \]

       3. associate-/l/ 67.6%

          \[\leadsto {\left(\ell \cdot \left(\ell \cdot \left(\color{blue}{\frac{2}{{k}^{4} \cdot t}} + \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\right)\right)\right)}^{1} \]

   11. Applied egg-rr 67.6%

       \[\leadsto \color{blue}{{\left(\ell \cdot \left(\ell \cdot \left(\frac{2}{{k}^{4} \cdot t} + \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\right)\right)\right)}^{1}} \]

   12. Taylor expanded in k around inf 64.6%

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

       1. unpow2 64.6%

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

       2. associate-*r* 67.6%

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

       3. associate-*r/ 67.6%

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

       4. associate-*l/ 67.6%

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

       5. *-commutative 67.6%

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

   14. Simplified 67.6%

       \[\leadsto {\left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\right)}\right)}^{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.6 \cdot 10^{+97}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\ell}{{k}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{-0.3333333333333333}{k \cdot \left(k \cdot t\right)}\right)\\ \end{array} \]

Alternative 7: 66.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 3.55 \cdot 10^{+44}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{-0.3333333333333333}{k \cdot k}}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 3.55e+44)
   (* 2.0 (* (/ l t) (/ l (pow k 4.0))))
   (* l (* l (/ (/ -0.3333333333333333 (* k k)) t)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.55e+44) {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	} else {
		tmp = l * (l * ((-0.3333333333333333 / (k * k)) / t));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.55d+44) then
        tmp = 2.0d0 * ((l / t) * (l / (k ** 4.0d0)))
    else
        tmp = l * (l * (((-0.3333333333333333d0) / (k * k)) / t))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.55e+44) {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	} else {
		tmp = l * (l * ((-0.3333333333333333 / (k * k)) / t));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 3.55e+44:
		tmp = 2.0 * ((l / t) * (l / math.pow(k, 4.0)))
	else:
		tmp = l * (l * ((-0.3333333333333333 / (k * k)) / t))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 3.55e+44)
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	else
		tmp = Float64(l * Float64(l * Float64(Float64(-0.3333333333333333 / Float64(k * k)) / t)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.55e+44)
		tmp = 2.0 * ((l / t) * (l / (k ^ 4.0)));
	else
		tmp = l * (l * ((-0.3333333333333333 / (k * k)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 3.55e+44], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(N[(-0.3333333333333333 / N[(k * k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.55e44

   1. Initial program 30.7%

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

      1. associate-/r* 30.7%

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

      2. *-commutative 30.7%

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

      3. associate-/r* 35.2%

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

      4. associate-*r/ 36.3%

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

      5. associate-/l* 35.2%

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

      6. +-commutative 35.2%

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

      7. unpow2 35.2%

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

      8. sqr-neg 35.2%

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

      9. distribute-frac-neg 35.2%

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

      10. distribute-frac-neg 35.2%

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

      11. unpow2 35.2%

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

      12. associate--l+ 43.0%

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

      13. metadata-eval 43.0%

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

      14. +-rgt-identity 43.0%

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

      15. unpow2 43.0%

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

      16. distribute-frac-neg 43.0%

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

   3. Simplified 43.0%

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

   4. Taylor expanded in k around 0 56.9%

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

      1. unpow2 56.9%

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

      2. *-commutative 56.9%

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

   6. Simplified 56.9%

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

      1. times-frac 64.5%

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

   8. Applied egg-rr 64.5%

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

   if 3.55e44 < k

   1. Initial program 31.0%

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

      1. associate-/r* 31.0%

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

      2. *-commutative 31.0%

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

      3. associate-/r* 34.2%

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

      4. associate-*r/ 34.2%

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

      5. associate-/l* 34.2%

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

      6. +-commutative 34.2%

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

      7. unpow2 34.2%

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

      8. sqr-neg 34.2%

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

      9. distribute-frac-neg 34.2%

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

      10. distribute-frac-neg 34.2%

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

      11. unpow2 34.2%

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

      12. associate--l+ 52.2%

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

      13. metadata-eval 52.2%

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

      14. +-rgt-identity 52.2%

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

      15. unpow2 52.2%

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

      16. distribute-frac-neg 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in k around 0 36.5%

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

      1. fma-def 36.5%

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

      2. unpow2 36.5%

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

      3. times-frac 43.2%

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

      4. unpow2 43.2%

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

      5. distribute-rgt-out 43.2%

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

      6. metadata-eval 43.2%

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

      7. unpow2 43.2%

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

      8. unpow2 43.2%

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

      9. *-commutative 43.2%

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

   6. Simplified 43.2%

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

   7. Taylor expanded in k around inf 53.1%

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

      1. associate-*r/ 53.1%

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

      2. times-frac 53.1%

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

      3. unpow2 53.1%

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

      4. unpow2 53.1%

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

      5. associate-*l/ 59.7%

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

   9. Simplified 59.7%

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

   10. Taylor expanded in k around 0 53.1%

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

       1. associate-*r/ 53.1%

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

       2. unpow2 53.1%

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

       3. associate-*r* 55.4%

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

       4. associate-*l/ 55.4%

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

       5. *-commutative 55.4%

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

       6. unpow2 55.4%

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

   12. Simplified 55.4%

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

   13. Taylor expanded in l around 0 53.1%

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

       1. unpow2 53.1%

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

       2. associate-*r* 55.4%

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

       3. associate-*r/ 55.4%

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

       4. *-commutative 55.4%

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

       5. associate-*r/ 55.4%

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

       6. unpow2 55.4%

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

       7. associate-*l* 62.5%

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

       8. associate-/r* 62.5%

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

       9. associate-/r* 60.1%

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

       10. associate-/r* 60.1%

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

   15. Simplified 60.1%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.55 \cdot 10^{+44}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{-0.3333333333333333}{k \cdot k}}{t}\right)\\ \end{array} \]

Alternative 8: 68.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\ell}{{k}^{3}}\right) \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* 2.0 (* (/ (/ l k) t) (/ l (pow k 3.0)))))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * (((l / k) / t) * (l / pow(k, 3.0)));
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * (((l / k) / t) * (l / (k ** 3.0d0)))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * (((l / k) / t) * (l / Math.pow(k, 3.0)));
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 * (((l / k) / t) * (l / math.pow(k, 3.0)))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64(Float64(l / k) / t) * Float64(l / (k ^ 3.0))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * (((l / k) / t) * (l / (k ^ 3.0)));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision] * N[(l / N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.8%

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

   1. associate-/r* 30.8%

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

   2. *-commutative 30.8%

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

   3. associate-/r* 35.0%

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

   4. associate-*r/ 35.8%

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

   5. associate-/l* 35.0%

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

   6. +-commutative 35.0%

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

   7. unpow2 35.0%

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

   8. sqr-neg 35.0%

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

   9. distribute-frac-neg 35.0%

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

   10. distribute-frac-neg 35.0%

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

   11. unpow2 35.0%

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

   12. associate--l+ 45.2%

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

   13. metadata-eval 45.2%

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

   14. +-rgt-identity 45.2%

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

   15. unpow2 45.2%

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

   16. distribute-frac-neg 45.2%

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

3. Simplified 45.2%

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

4. Taylor expanded in k around inf 65.6%

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

   1. unpow2 65.6%

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

   2. *-commutative 65.6%

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

   3. associate-*l* 65.6%

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

   4. *-commutative 65.6%

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

   5. times-frac 81.7%

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

   6. *-commutative 81.7%

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

   7. unpow2 81.7%

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

6. Simplified 81.7%

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

   1. times-frac 89.8%

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

8. Applied egg-rr 89.8%

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

9. Taylor expanded in l around 0 65.6%

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

    1. unpow2 65.6%

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

    2. times-frac 66.7%

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

    3. unpow2 66.7%

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

    4. associate-*l/ 76.6%

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

    5. associate-/l/ 76.0%

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

    6. associate-*l/ 76.1%

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

    7. *-commutative 76.1%

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

    8. associate-*r* 76.1%

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

    9. associate-*r/ 80.0%

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

    10. *-commutative 80.0%

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

    11. times-frac 88.1%

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

    12. associate-/l/ 89.8%

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

    13. associate-*l/ 87.1%

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

11. Simplified 96.0%

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

12. Taylor expanded in k around 0 66.2%

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

13. Final simplification 66.2%

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

Alternative 9: 34.0% accurate, 38.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.8%

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

   1. associate-/r* 30.8%

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

   2. *-commutative 30.8%

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

   3. associate-/r* 35.0%

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

   4. associate-*r/ 35.8%

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

   5. associate-/l* 35.0%

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

   6. +-commutative 35.0%

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

   7. unpow2 35.0%

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

   8. sqr-neg 35.0%

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

   9. distribute-frac-neg 35.0%

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

   10. distribute-frac-neg 35.0%

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

   11. unpow2 35.0%

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

   12. associate--l+ 45.2%

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

   13. metadata-eval 45.2%

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

   14. +-rgt-identity 45.2%

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

   15. unpow2 45.2%

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

   16. distribute-frac-neg 45.2%

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

3. Simplified 45.2%

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

4. Taylor expanded in k around 0 24.2%

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

   1. fma-def 24.2%

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

   2. unpow2 24.2%

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

   3. times-frac 27.5%

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

   4. unpow2 27.5%

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

   5. distribute-rgt-out 27.5%

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

   6. metadata-eval 27.5%

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

   7. unpow2 27.5%

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

   8. unpow2 27.5%

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

   9. *-commutative 27.5%

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

6. Simplified 27.5%

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

7. Taylor expanded in k around inf 28.9%

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

   1. associate-*r/ 28.9%

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

   2. times-frac 28.9%

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

   3. unpow2 28.9%

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

   4. unpow2 28.9%

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

   5. associate-*l/ 31.2%

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

9. Simplified 31.2%

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

10. Taylor expanded in k around 0 28.9%

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

    1. associate-*r/ 28.9%

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

    2. unpow2 28.9%

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

    3. associate-*r* 29.6%

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

    4. associate-*l/ 29.6%

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

    5. *-commutative 29.6%

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

    6. unpow2 29.6%

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

12. Simplified 29.6%

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

13. Taylor expanded in l around 0 28.9%

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

    1. unpow2 28.9%

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

    2. associate-*r* 29.6%

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

    3. associate-*r/ 29.6%

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

    4. *-commutative 29.6%

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

    5. associate-*r/ 29.6%

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

    6. unpow2 29.6%

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

    7. associate-*l* 32.3%

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

    8. associate-/r* 32.3%

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

    9. associate-/r* 31.5%

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

    10. associate-/r* 31.5%

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

15. Simplified 31.5%

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

16. Final simplification 31.5%

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

Reproduce

herbie shell --seed 2023275 
(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))))