Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.5% → 88.6%

Time: 32.4s

Alternatives: 19

Speedup: 3.8×

• 34.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 34.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{k_m}{\frac{\ell \cdot \sqrt{\cos k_m}}{\sin k_m \cdot \sqrt{t_m}}}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 950:\\ \;\;\;\;\frac{2}{t_2 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{t_m} \cdot \left(k_m \cdot \sin k_m\right)}}{\sqrt{t_m} \cdot \left(\left(k_m \cdot {\ell}^{-2}\right) \cdot \tan k_m\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ k_m (/ (* l (sqrt (cos k_m))) (* (sin k_m) (sqrt t_m))))))
   (*
    t_s
    (if (<= k_m 950.0)
      (/ 2.0 (* t_2 t_2))
      (/
       (/ 2.0 (* (sqrt t_m) (* k_m (sin k_m))))
       (* (sqrt t_m) (* (* k_m (pow l -2.0)) (tan k_m))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = k_m / ((l * sqrt(cos(k_m))) / (sin(k_m) * sqrt(t_m)));
	double tmp;
	if (k_m <= 950.0) {
		tmp = 2.0 / (t_2 * t_2);
	} else {
		tmp = (2.0 / (sqrt(t_m) * (k_m * sin(k_m)))) / (sqrt(t_m) * ((k_m * pow(l, -2.0)) * tan(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = k_m / ((l * sqrt(cos(k_m))) / (sin(k_m) * sqrt(t_m)))
    if (k_m <= 950.0d0) then
        tmp = 2.0d0 / (t_2 * t_2)
    else
        tmp = (2.0d0 / (sqrt(t_m) * (k_m * sin(k_m)))) / (sqrt(t_m) * ((k_m * (l ** (-2.0d0))) * tan(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = k_m / ((l * Math.sqrt(Math.cos(k_m))) / (Math.sin(k_m) * Math.sqrt(t_m)));
	double tmp;
	if (k_m <= 950.0) {
		tmp = 2.0 / (t_2 * t_2);
	} else {
		tmp = (2.0 / (Math.sqrt(t_m) * (k_m * Math.sin(k_m)))) / (Math.sqrt(t_m) * ((k_m * Math.pow(l, -2.0)) * Math.tan(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = k_m / ((l * math.sqrt(math.cos(k_m))) / (math.sin(k_m) * math.sqrt(t_m)))
	tmp = 0
	if k_m <= 950.0:
		tmp = 2.0 / (t_2 * t_2)
	else:
		tmp = (2.0 / (math.sqrt(t_m) * (k_m * math.sin(k_m)))) / (math.sqrt(t_m) * ((k_m * math.pow(l, -2.0)) * math.tan(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(k_m / Float64(Float64(l * sqrt(cos(k_m))) / Float64(sin(k_m) * sqrt(t_m))))
	tmp = 0.0
	if (k_m <= 950.0)
		tmp = Float64(2.0 / Float64(t_2 * t_2));
	else
		tmp = Float64(Float64(2.0 / Float64(sqrt(t_m) * Float64(k_m * sin(k_m)))) / Float64(sqrt(t_m) * Float64(Float64(k_m * (l ^ -2.0)) * tan(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = k_m / ((l * sqrt(cos(k_m))) / (sin(k_m) * sqrt(t_m)));
	tmp = 0.0;
	if (k_m <= 950.0)
		tmp = 2.0 / (t_2 * t_2);
	else
		tmp = (2.0 / (sqrt(t_m) * (k_m * sin(k_m)))) / (sqrt(t_m) * ((k_m * (l ^ -2.0)) * tan(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(k$95$m / N[(N[(l * N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k$95$m], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 950.0], N[(2.0 / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[(k$95$m * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 950

   1. Initial program 36.9%

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

      1. associate-*l* 36.9%

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

      2. sqr-neg 36.9%

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

      3. associate-*l* 36.9%

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

      4. associate-*l* 36.9%

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

      5. associate-*l/ 37.9%

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

      6. sqr-neg 37.9%

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

      7. associate-*l/ 36.9%

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

      8. associate--l+ 36.9%

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

   3. Simplified 36.9%

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

   4. Taylor expanded in t around 0 76.9%

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

      1. add-sqr-sqrt 32.2%

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

   6. Applied egg-rr 31.5%

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

      1. associate-/l* 31.6%

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

      2. associate-/l* 33.0%

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

   8. Simplified 33.0%

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

   if 950 < k

   1. Initial program 41.8%

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

      1. associate-*l* 41.8%

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

      2. sqr-neg 41.8%

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

      3. associate-*l* 41.8%

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

      4. associate-*l* 41.8%

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

      5. associate-*l/ 41.8%

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

      6. sqr-neg 41.8%

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

      7. associate-*l/ 41.8%

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

      8. associate--l+ 41.8%

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

   3. Simplified 41.8%

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

   4. Taylor expanded in t around 0 74.3%

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

      1. add-sqr-sqrt 37.3%

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

      2. *-un-lft-identity 37.3%

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

      3. times-frac 37.3%

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

      4. sqrt-prod 37.3%

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

      5. unpow2 37.3%

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

      6. sqrt-prod 37.3%

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

      7. add-sqr-sqrt 37.3%

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

      8. *-commutative 37.3%

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

      9. sqrt-prod 37.3%

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

      10. unpow2 37.3%

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

      11. sqrt-prod 18.0%

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

      12. add-sqr-sqrt 28.5%

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

   6. Applied egg-rr 41.7%

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

      1. times-frac 41.6%

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

      2. *-commutative 41.6%

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

      3. *-commutative 41.6%

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

      4. *-un-lft-identity 41.6%

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

      5. times-frac 41.6%

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

      6. tan-quot 41.7%

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

      7. div-inv 41.7%

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

      8. metadata-eval 41.7%

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

      9. unpow2 41.7%

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

      10. frac-times 41.7%

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

      11. inv-pow 41.7%

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

      12. metadata-eval 41.7%

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

      13. inv-pow 41.7%

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

      14. metadata-eval 41.7%

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

      15. pow-sqr 41.6%

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

      16. metadata-eval 41.6%

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

      17. metadata-eval 41.6%

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

   8. Applied egg-rr 41.6%

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

      1. associate-/r* 41.7%

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

      2. div-inv 41.7%

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

      3. /-rgt-identity 41.7%

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

      4. associate-*r* 41.7%

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

      5. *-commutative 41.7%

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

      6. *-commutative 41.7%

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

      7. /-rgt-identity 41.7%

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

   10. Applied egg-rr 41.7%

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

       1. associate-*r/ 41.7%

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

       2. *-rgt-identity 41.7%

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

       3. /-rgt-identity 41.7%

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

       4. associate-/l* 41.7%

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

       5. associate-/r* 41.7%

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

       6. associate-/l* 41.7%

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

       7. associate-/r/ 41.7%

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

       8. /-rgt-identity 41.7%

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

   12. Simplified 41.7%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 950:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell \cdot \sqrt{\cos k}}{\sin k \cdot \sqrt{t}}} \cdot \frac{k}{\frac{\ell \cdot \sqrt{\cos k}}{\sin k \cdot \sqrt{t}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{t} \cdot \left(k \cdot \sin k\right)}}{\sqrt{t} \cdot \left(\left(k \cdot {\ell}^{-2}\right) \cdot \tan k\right)}\\ \end{array} \]

Alternative 2: 74.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt{t_m}}{\frac{\ell}{{k_m}^{2}}}\\ t_3 := {\left(\frac{k_m}{t_m}\right)}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;\left(\tan k_m \cdot \left(\sin k_m \cdot \frac{{t_m}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(1 + t_3\right) + -1\right) \leq 5 \cdot 10^{+187}:\\ \;\;\;\;\frac{2}{\frac{t_3 \cdot \frac{\sin k_m \cdot {t_m}^{3}}{\frac{\ell}{\tan k_m}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot t_2}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (sqrt t_m) (/ l (pow k_m 2.0)))) (t_3 (pow (/ k_m t_m) 2.0)))
   (*
    t_s
    (if (<=
         (*
          (* (tan k_m) (* (sin k_m) (/ (pow t_m 3.0) (* l l))))
          (+ (+ 1.0 t_3) -1.0))
         5e+187)
      (/ 2.0 (/ (* t_3 (/ (* (sin k_m) (pow t_m 3.0)) (/ l (tan k_m)))) l))
      (/ 2.0 (* t_2 t_2))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = sqrt(t_m) / (l / pow(k_m, 2.0));
	double t_3 = pow((k_m / t_m), 2.0);
	double tmp;
	if (((tan(k_m) * (sin(k_m) * (pow(t_m, 3.0) / (l * l)))) * ((1.0 + t_3) + -1.0)) <= 5e+187) {
		tmp = 2.0 / ((t_3 * ((sin(k_m) * pow(t_m, 3.0)) / (l / tan(k_m)))) / l);
	} else {
		tmp = 2.0 / (t_2 * t_2);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = sqrt(t_m) / (l / (k_m ** 2.0d0))
    t_3 = (k_m / t_m) ** 2.0d0
    if (((tan(k_m) * (sin(k_m) * ((t_m ** 3.0d0) / (l * l)))) * ((1.0d0 + t_3) + (-1.0d0))) <= 5d+187) then
        tmp = 2.0d0 / ((t_3 * ((sin(k_m) * (t_m ** 3.0d0)) / (l / tan(k_m)))) / l)
    else
        tmp = 2.0d0 / (t_2 * t_2)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.sqrt(t_m) / (l / Math.pow(k_m, 2.0));
	double t_3 = Math.pow((k_m / t_m), 2.0);
	double tmp;
	if (((Math.tan(k_m) * (Math.sin(k_m) * (Math.pow(t_m, 3.0) / (l * l)))) * ((1.0 + t_3) + -1.0)) <= 5e+187) {
		tmp = 2.0 / ((t_3 * ((Math.sin(k_m) * Math.pow(t_m, 3.0)) / (l / Math.tan(k_m)))) / l);
	} else {
		tmp = 2.0 / (t_2 * t_2);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = math.sqrt(t_m) / (l / math.pow(k_m, 2.0))
	t_3 = math.pow((k_m / t_m), 2.0)
	tmp = 0
	if ((math.tan(k_m) * (math.sin(k_m) * (math.pow(t_m, 3.0) / (l * l)))) * ((1.0 + t_3) + -1.0)) <= 5e+187:
		tmp = 2.0 / ((t_3 * ((math.sin(k_m) * math.pow(t_m, 3.0)) / (l / math.tan(k_m)))) / l)
	else:
		tmp = 2.0 / (t_2 * t_2)
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(sqrt(t_m) / Float64(l / (k_m ^ 2.0)))
	t_3 = Float64(k_m / t_m) ^ 2.0
	tmp = 0.0
	if (Float64(Float64(tan(k_m) * Float64(sin(k_m) * Float64((t_m ^ 3.0) / Float64(l * l)))) * Float64(Float64(1.0 + t_3) + -1.0)) <= 5e+187)
		tmp = Float64(2.0 / Float64(Float64(t_3 * Float64(Float64(sin(k_m) * (t_m ^ 3.0)) / Float64(l / tan(k_m)))) / l));
	else
		tmp = Float64(2.0 / Float64(t_2 * t_2));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = sqrt(t_m) / (l / (k_m ^ 2.0));
	t_3 = (k_m / t_m) ^ 2.0;
	tmp = 0.0;
	if (((tan(k_m) * (sin(k_m) * ((t_m ^ 3.0) / (l * l)))) * ((1.0 + t_3) + -1.0)) <= 5e+187)
		tmp = 2.0 / ((t_3 * ((sin(k_m) * (t_m ^ 3.0)) / (l / tan(k_m)))) / l);
	else
		tmp = 2.0 / (t_2 * t_2);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[Sqrt[t$95$m], $MachinePrecision] / N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + t$95$3), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 5e+187], N[(2.0 / N[(N[(t$95$3 * N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[(l / N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < 5.0000000000000001e187

   1. Initial program 91.1%

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

      1. associate-*l* 91.1%

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

      2. sqr-neg 91.1%

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

      3. associate-*l* 91.1%

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

      4. associate-*l* 91.1%

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

      5. associate-*l/ 91.1%

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

      6. sqr-neg 91.1%

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

      7. associate-*l/ 91.1%

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

      8. associate--l+ 91.1%

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

   3. Simplified 91.1%

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

      1. associate-*r* 91.1%

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

      2. associate-*l/ 91.1%

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

      3. associate-*l/ 91.1%

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

      4. +-commutative 91.1%

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

      5. sub-neg 91.1%

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

      6. associate-+l+ 94.5%

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

      7. metadata-eval 94.5%

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

      8. metadata-eval 94.5%

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

      9. *-commutative 94.5%

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

      10. +-rgt-identity 94.5%

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

      11. associate-/r* 96.4%

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

      12. associate-*r/ 96.9%

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

   5. Applied egg-rr 96.8%

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

   if 5.0000000000000001e187 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))

   1. Initial program 13.2%

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

      1. associate-*l* 13.2%

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

      2. sqr-neg 13.2%

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

      3. associate-*l* 13.2%

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

      4. associate-*l* 13.2%

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

      5. associate-*l/ 14.4%

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

      6. sqr-neg 14.4%

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

      7. associate-*l/ 13.2%

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

      8. associate--l+ 13.2%

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

   3. Simplified 13.2%

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

   4. Taylor expanded in k around 0 56.5%

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

      1. associate-/l* 54.7%

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

      2. associate-/r/ 55.6%

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

   6. Simplified 55.6%

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

      1. associate-*l/ 56.5%

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

      2. unpow2 56.5%

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

      3. associate-/r* 64.5%

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

      4. *-commutative 64.5%

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

   8. Applied egg-rr 64.5%

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

      1. associate-/l/ 56.5%

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

      2. add-sqr-sqrt 27.5%

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

      3. times-frac 31.7%

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

      4. sqrt-prod 31.7%

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

      5. sqrt-pow1 31.7%

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

      6. metadata-eval 31.7%

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

      7. sqrt-prod 32.4%

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

      8. sqrt-pow1 34.0%

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

      9. metadata-eval 34.0%

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

   10. Applied egg-rr 34.0%

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

       1. associate-/l* 34.0%

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

       2. associate-/l* 34.0%

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

   12. Simplified 34.0%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + -1\right) \leq 5 \cdot 10^{+187}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{t}\right)}^{2} \cdot \frac{\sin k \cdot {t}^{3}}{\frac{\ell}{\tan k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}}\\ \end{array} \]

Alternative 3: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{\sqrt{2}}{\sqrt{t_m}}}{\frac{{k_m}^{2}}{\ell}}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 2.7 \cdot 10^{-10}:\\ \;\;\;\;t_2 \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{t_m} \cdot \left(k_m \cdot \sin k_m\right)}}{\sqrt{t_m} \cdot \left(\left(k_m \cdot {\ell}^{-2}\right) \cdot \tan k_m\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (/ (sqrt 2.0) (sqrt t_m)) (/ (pow k_m 2.0) l))))
   (*
    t_s
    (if (<= k_m 2.7e-10)
      (* t_2 t_2)
      (/
       (/ 2.0 (* (sqrt t_m) (* k_m (sin k_m))))
       (* (sqrt t_m) (* (* k_m (pow l -2.0)) (tan k_m))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = (sqrt(2.0) / sqrt(t_m)) / (pow(k_m, 2.0) / l);
	double tmp;
	if (k_m <= 2.7e-10) {
		tmp = t_2 * t_2;
	} else {
		tmp = (2.0 / (sqrt(t_m) * (k_m * sin(k_m)))) / (sqrt(t_m) * ((k_m * pow(l, -2.0)) * tan(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (sqrt(2.0d0) / sqrt(t_m)) / ((k_m ** 2.0d0) / l)
    if (k_m <= 2.7d-10) then
        tmp = t_2 * t_2
    else
        tmp = (2.0d0 / (sqrt(t_m) * (k_m * sin(k_m)))) / (sqrt(t_m) * ((k_m * (l ** (-2.0d0))) * tan(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = (Math.sqrt(2.0) / Math.sqrt(t_m)) / (Math.pow(k_m, 2.0) / l);
	double tmp;
	if (k_m <= 2.7e-10) {
		tmp = t_2 * t_2;
	} else {
		tmp = (2.0 / (Math.sqrt(t_m) * (k_m * Math.sin(k_m)))) / (Math.sqrt(t_m) * ((k_m * Math.pow(l, -2.0)) * Math.tan(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = (math.sqrt(2.0) / math.sqrt(t_m)) / (math.pow(k_m, 2.0) / l)
	tmp = 0
	if k_m <= 2.7e-10:
		tmp = t_2 * t_2
	else:
		tmp = (2.0 / (math.sqrt(t_m) * (k_m * math.sin(k_m)))) / (math.sqrt(t_m) * ((k_m * math.pow(l, -2.0)) * math.tan(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(Float64(sqrt(2.0) / sqrt(t_m)) / Float64((k_m ^ 2.0) / l))
	tmp = 0.0
	if (k_m <= 2.7e-10)
		tmp = Float64(t_2 * t_2);
	else
		tmp = Float64(Float64(2.0 / Float64(sqrt(t_m) * Float64(k_m * sin(k_m)))) / Float64(sqrt(t_m) * Float64(Float64(k_m * (l ^ -2.0)) * tan(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = (sqrt(2.0) / sqrt(t_m)) / ((k_m ^ 2.0) / l);
	tmp = 0.0;
	if (k_m <= 2.7e-10)
		tmp = t_2 * t_2;
	else
		tmp = (2.0 / (sqrt(t_m) * (k_m * sin(k_m)))) / (sqrt(t_m) * ((k_m * (l ^ -2.0)) * tan(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 2.7e-10], N[(t$95$2 * t$95$2), $MachinePrecision], N[(N[(2.0 / N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[(k$95$m * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.7e-10

   1. Initial program 37.7%

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

      1. associate-*l* 37.7%

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

      2. sqr-neg 37.7%

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

      3. associate-*l* 37.7%

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

      4. associate-*l* 37.7%

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

      5. associate-*l/ 38.8%

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

      6. sqr-neg 38.8%

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

      7. associate-*l/ 37.7%

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

      8. associate--l+ 37.7%

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

   3. Simplified 37.7%

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

   4. Taylor expanded in k around 0 68.0%

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

      1. associate-/l* 65.7%

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

      2. associate-/r/ 66.5%

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

   6. Simplified 66.5%

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

      1. add-sqr-sqrt 44.0%

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

      2. sqrt-div 31.2%

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

      3. *-commutative 31.2%

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

      4. sqrt-prod 31.2%

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

      5. sqrt-div 31.2%

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

      6. sqrt-pow1 31.2%

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

      7. metadata-eval 31.2%

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

      8. unpow2 31.2%

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

      9. sqrt-prod 14.2%

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

      10. add-sqr-sqrt 22.5%

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

      11. sqrt-div 22.5%

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

      12. *-commutative 22.5%

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

      13. sqrt-prod 22.5%

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

      14. sqrt-div 22.5%

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

      15. sqrt-pow1 23.1%

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

      16. metadata-eval 23.1%

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

   8. Applied egg-rr 37.2%

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

      1. associate-/r* 37.2%

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

      2. associate-/r* 37.2%

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

   10. Simplified 37.2%

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

   if 2.7e-10 < k

   1. Initial program 39.5%

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

      1. associate-*l* 39.5%

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

      2. sqr-neg 39.5%

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

      3. associate-*l* 39.5%

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

      4. associate-*l* 39.5%

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

      5. associate-*l/ 39.5%

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

      6. sqr-neg 39.5%

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

      7. associate-*l/ 39.5%

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

      8. associate--l+ 39.5%

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

   3. Simplified 39.5%

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

   4. Taylor expanded in t around 0 74.7%

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

      1. add-sqr-sqrt 38.4%

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

      2. *-un-lft-identity 38.4%

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

      3. times-frac 38.4%

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

      4. sqrt-prod 38.4%

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

      5. unpow2 38.4%

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

      6. sqrt-prod 38.4%

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

      7. add-sqr-sqrt 38.4%

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

      8. *-commutative 38.4%

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

      9. sqrt-prod 38.4%

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

      10. unpow2 38.4%

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

      11. sqrt-prod 19.7%

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

      12. add-sqr-sqrt 30.1%

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

   6. Applied egg-rr 42.5%

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

      1. times-frac 42.5%

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

      2. *-commutative 42.5%

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

      3. *-commutative 42.5%

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

      4. *-un-lft-identity 42.5%

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

      5. times-frac 42.5%

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

      6. tan-quot 42.6%

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

      7. div-inv 42.5%

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

      8. metadata-eval 42.5%

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

      9. unpow2 42.5%

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

      10. frac-times 42.5%

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

      11. inv-pow 42.5%

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

      12. metadata-eval 42.5%

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

      13. inv-pow 42.5%

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

      14. metadata-eval 42.5%

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

      15. pow-sqr 42.5%

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

      16. metadata-eval 42.5%

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

      17. metadata-eval 42.5%

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

   8. Applied egg-rr 42.5%

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

      1. associate-/r* 43.5%

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

      2. div-inv 43.5%

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

      3. /-rgt-identity 43.5%

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

      4. associate-*r* 43.5%

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

      5. *-commutative 43.5%

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

      6. *-commutative 43.5%

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

      7. /-rgt-identity 43.5%

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

   10. Applied egg-rr 43.5%

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

       1. associate-*r/ 43.5%

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

       2. *-rgt-identity 43.5%

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

       3. /-rgt-identity 43.5%

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

       4. associate-/l* 43.5%

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

       5. associate-/r* 43.6%

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

       6. associate-/l* 43.5%

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

       7. associate-/r/ 43.5%

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

       8. /-rgt-identity 43.5%

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

   12. Simplified 43.5%

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

4. Final simplification 39.0%

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

Alternative 4: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt{t_m}}{\frac{\ell}{{k_m}^{2}}}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{t_2 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sqrt{t_m} \cdot \left(k_m \cdot \sin k_m\right)\right) \cdot \left(\sqrt{t_m} \cdot \left(\left(k_m \cdot {\ell}^{-2}\right) \cdot \tan k_m\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (sqrt t_m) (/ l (pow k_m 2.0)))))
   (*
    t_s
    (if (<= k_m 1.2e-12)
      (/ 2.0 (* t_2 t_2))
      (/
       2.0
       (*
        (* (sqrt t_m) (* k_m (sin k_m)))
        (* (sqrt t_m) (* (* k_m (pow l -2.0)) (tan k_m)))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = sqrt(t_m) / (l / pow(k_m, 2.0));
	double tmp;
	if (k_m <= 1.2e-12) {
		tmp = 2.0 / (t_2 * t_2);
	} else {
		tmp = 2.0 / ((sqrt(t_m) * (k_m * sin(k_m))) * (sqrt(t_m) * ((k_m * pow(l, -2.0)) * tan(k_m))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sqrt(t_m) / (l / (k_m ** 2.0d0))
    if (k_m <= 1.2d-12) then
        tmp = 2.0d0 / (t_2 * t_2)
    else
        tmp = 2.0d0 / ((sqrt(t_m) * (k_m * sin(k_m))) * (sqrt(t_m) * ((k_m * (l ** (-2.0d0))) * tan(k_m))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.sqrt(t_m) / (l / Math.pow(k_m, 2.0));
	double tmp;
	if (k_m <= 1.2e-12) {
		tmp = 2.0 / (t_2 * t_2);
	} else {
		tmp = 2.0 / ((Math.sqrt(t_m) * (k_m * Math.sin(k_m))) * (Math.sqrt(t_m) * ((k_m * Math.pow(l, -2.0)) * Math.tan(k_m))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = math.sqrt(t_m) / (l / math.pow(k_m, 2.0))
	tmp = 0
	if k_m <= 1.2e-12:
		tmp = 2.0 / (t_2 * t_2)
	else:
		tmp = 2.0 / ((math.sqrt(t_m) * (k_m * math.sin(k_m))) * (math.sqrt(t_m) * ((k_m * math.pow(l, -2.0)) * math.tan(k_m))))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(sqrt(t_m) / Float64(l / (k_m ^ 2.0)))
	tmp = 0.0
	if (k_m <= 1.2e-12)
		tmp = Float64(2.0 / Float64(t_2 * t_2));
	else
		tmp = Float64(2.0 / Float64(Float64(sqrt(t_m) * Float64(k_m * sin(k_m))) * Float64(sqrt(t_m) * Float64(Float64(k_m * (l ^ -2.0)) * tan(k_m)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = sqrt(t_m) / (l / (k_m ^ 2.0));
	tmp = 0.0;
	if (k_m <= 1.2e-12)
		tmp = 2.0 / (t_2 * t_2);
	else
		tmp = 2.0 / ((sqrt(t_m) * (k_m * sin(k_m))) * (sqrt(t_m) * ((k_m * (l ^ -2.0)) * tan(k_m))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[Sqrt[t$95$m], $MachinePrecision] / N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.2e-12], N[(2.0 / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[(k$95$m * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.19999999999999994e-12

   1. Initial program 37.9%

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

      1. associate-*l* 37.9%

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

      2. sqr-neg 37.9%

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

      3. associate-*l* 37.9%

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

      4. associate-*l* 37.9%

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

      5. associate-*l/ 39.0%

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

      6. sqr-neg 39.0%

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

      7. associate-*l/ 37.9%

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

      8. associate--l+ 37.9%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in k around 0 68.2%

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

      1. associate-/l* 65.9%

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

      2. associate-/r/ 66.7%

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

   6. Simplified 66.7%

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

      1. associate-*l/ 68.2%

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

      2. unpow2 68.2%

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

      3. associate-/r* 75.7%

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

      4. *-commutative 75.7%

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

   8. Applied egg-rr 75.7%

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

      1. associate-/l/ 68.2%

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

      2. add-sqr-sqrt 31.3%

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

      3. times-frac 35.3%

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

      4. sqrt-prod 35.3%

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

      5. sqrt-pow1 35.3%

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

      6. metadata-eval 35.3%

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

      7. sqrt-prod 35.8%

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

      8. sqrt-pow1 37.4%

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

      9. metadata-eval 37.4%

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

   10. Applied egg-rr 37.4%

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

       1. associate-/l* 37.4%

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

       2. associate-/l* 37.4%

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

   12. Simplified 37.4%

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

   if 1.19999999999999994e-12 < k

   1. Initial program 38.9%

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

      1. associate-*l* 38.9%

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

      2. sqr-neg 38.9%

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

      3. associate-*l* 38.9%

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

      4. associate-*l* 38.9%

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

      5. associate-*l/ 38.9%

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

      6. sqr-neg 38.9%

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

      7. associate-*l/ 38.9%

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

      8. associate--l+ 38.9%

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

   3. Simplified 38.9%

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

   4. Taylor expanded in t around 0 74.1%

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

      1. add-sqr-sqrt 37.9%

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

      2. *-un-lft-identity 37.9%

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

      3. times-frac 37.9%

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

      4. sqrt-prod 37.9%

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

      5. unpow2 37.9%

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

      6. sqrt-prod 37.9%

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

      7. add-sqr-sqrt 37.9%

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

      8. *-commutative 37.9%

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

      9. sqrt-prod 37.9%

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

      10. unpow2 37.9%

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

      11. sqrt-prod 19.5%

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

      12. add-sqr-sqrt 29.7%

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

   6. Applied egg-rr 42.0%

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

      1. times-frac 41.9%

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

      2. *-commutative 41.9%

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

      3. *-commutative 41.9%

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

      4. *-un-lft-identity 41.9%

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

      5. times-frac 41.9%

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

      6. tan-quot 42.0%

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

      7. div-inv 42.0%

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

      8. metadata-eval 42.0%

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

      9. unpow2 42.0%

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

      10. frac-times 41.9%

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

      11. inv-pow 41.9%

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

      12. metadata-eval 41.9%

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

      13. inv-pow 41.9%

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

      14. metadata-eval 41.9%

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

      15. pow-sqr 41.9%

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

      16. metadata-eval 41.9%

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

      17. metadata-eval 41.9%

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

   8. Applied egg-rr 41.9%

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

      1. /-rgt-identity 41.9%

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

      2. *-commutative 41.9%

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

      3. associate-/l* 41.9%

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

      4. *-commutative 41.9%

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

      5. /-rgt-identity 41.9%

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

      6. /-rgt-identity 41.9%

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

      7. associate-*r* 41.9%

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

      8. *-commutative 41.9%

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

   10. Applied egg-rr 41.9%

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

       1. associate-/r/ 41.9%

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

       2. associate-/l* 41.9%

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

       3. associate-/r* 41.9%

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

       4. associate-/l* 41.9%

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

       5. associate-/r/ 41.9%

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

       6. /-rgt-identity 41.9%

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

   12. Simplified 41.9%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sqrt{t} \cdot \left(k \cdot \sin k\right)\right) \cdot \left(\sqrt{t} \cdot \left(\left(k \cdot {\ell}^{-2}\right) \cdot \tan k\right)\right)}\\ \end{array} \]

Alternative 5: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt{t_m}}{\frac{\ell}{{k_m}^{2}}}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 7 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{t_2 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k_m \cdot {\ell}^{-2}}}{\left(k_m \cdot \left(\sin k_m \cdot \sqrt{t_m}\right)\right) \cdot \left(\sqrt{t_m} \cdot \tan k_m\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (sqrt t_m) (/ l (pow k_m 2.0)))))
   (*
    t_s
    (if (<= k_m 7e-13)
      (/ 2.0 (* t_2 t_2))
      (/
       (/ 2.0 (* k_m (pow l -2.0)))
       (* (* k_m (* (sin k_m) (sqrt t_m))) (* (sqrt t_m) (tan k_m))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = sqrt(t_m) / (l / pow(k_m, 2.0));
	double tmp;
	if (k_m <= 7e-13) {
		tmp = 2.0 / (t_2 * t_2);
	} else {
		tmp = (2.0 / (k_m * pow(l, -2.0))) / ((k_m * (sin(k_m) * sqrt(t_m))) * (sqrt(t_m) * tan(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sqrt(t_m) / (l / (k_m ** 2.0d0))
    if (k_m <= 7d-13) then
        tmp = 2.0d0 / (t_2 * t_2)
    else
        tmp = (2.0d0 / (k_m * (l ** (-2.0d0)))) / ((k_m * (sin(k_m) * sqrt(t_m))) * (sqrt(t_m) * tan(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.sqrt(t_m) / (l / Math.pow(k_m, 2.0));
	double tmp;
	if (k_m <= 7e-13) {
		tmp = 2.0 / (t_2 * t_2);
	} else {
		tmp = (2.0 / (k_m * Math.pow(l, -2.0))) / ((k_m * (Math.sin(k_m) * Math.sqrt(t_m))) * (Math.sqrt(t_m) * Math.tan(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = math.sqrt(t_m) / (l / math.pow(k_m, 2.0))
	tmp = 0
	if k_m <= 7e-13:
		tmp = 2.0 / (t_2 * t_2)
	else:
		tmp = (2.0 / (k_m * math.pow(l, -2.0))) / ((k_m * (math.sin(k_m) * math.sqrt(t_m))) * (math.sqrt(t_m) * math.tan(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(sqrt(t_m) / Float64(l / (k_m ^ 2.0)))
	tmp = 0.0
	if (k_m <= 7e-13)
		tmp = Float64(2.0 / Float64(t_2 * t_2));
	else
		tmp = Float64(Float64(2.0 / Float64(k_m * (l ^ -2.0))) / Float64(Float64(k_m * Float64(sin(k_m) * sqrt(t_m))) * Float64(sqrt(t_m) * tan(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = sqrt(t_m) / (l / (k_m ^ 2.0));
	tmp = 0.0;
	if (k_m <= 7e-13)
		tmp = 2.0 / (t_2 * t_2);
	else
		tmp = (2.0 / (k_m * (l ^ -2.0))) / ((k_m * (sin(k_m) * sqrt(t_m))) * (sqrt(t_m) * tan(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[Sqrt[t$95$m], $MachinePrecision] / N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 7e-13], N[(2.0 / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k$95$m * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.0000000000000005e-13

   1. Initial program 37.9%

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

      1. associate-*l* 37.9%

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

      2. sqr-neg 37.9%

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

      3. associate-*l* 37.9%

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

      4. associate-*l* 37.9%

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

      5. associate-*l/ 39.0%

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

      6. sqr-neg 39.0%

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

      7. associate-*l/ 37.9%

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

      8. associate--l+ 37.9%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in k around 0 68.2%

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

      1. associate-/l* 65.9%

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

      2. associate-/r/ 66.7%

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

   6. Simplified 66.7%

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

      1. associate-*l/ 68.2%

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

      2. unpow2 68.2%

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

      3. associate-/r* 75.7%

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

      4. *-commutative 75.7%

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

   8. Applied egg-rr 75.7%

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

      1. associate-/l/ 68.2%

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

      2. add-sqr-sqrt 31.3%

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

      3. times-frac 35.3%

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

      4. sqrt-prod 35.3%

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

      5. sqrt-pow1 35.3%

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

      6. metadata-eval 35.3%

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

      7. sqrt-prod 35.8%

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

      8. sqrt-pow1 37.4%

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

      9. metadata-eval 37.4%

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

   10. Applied egg-rr 37.4%

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

       1. associate-/l* 37.4%

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

       2. associate-/l* 37.4%

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

   12. Simplified 37.4%

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

   if 7.0000000000000005e-13 < k

   1. Initial program 38.9%

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

      1. associate-*l* 38.9%

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

      2. sqr-neg 38.9%

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

      3. associate-*l* 38.9%

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

      4. associate-*l* 38.9%

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

      5. associate-*l/ 38.9%

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

      6. sqr-neg 38.9%

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

      7. associate-*l/ 38.9%

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

      8. associate--l+ 38.9%

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

   3. Simplified 38.9%

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

   4. Taylor expanded in t around 0 74.1%

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

      1. add-sqr-sqrt 37.9%

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

      2. *-un-lft-identity 37.9%

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

      3. times-frac 37.9%

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

      4. sqrt-prod 37.9%

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

      5. unpow2 37.9%

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

      6. sqrt-prod 37.9%

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

      7. add-sqr-sqrt 37.9%

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

      8. *-commutative 37.9%

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

      9. sqrt-prod 37.9%

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

      10. unpow2 37.9%

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

      11. sqrt-prod 19.5%

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

      12. add-sqr-sqrt 29.7%

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

   6. Applied egg-rr 42.0%

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

      1. times-frac 41.9%

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

      2. *-commutative 41.9%

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

      3. *-commutative 41.9%

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

      4. *-un-lft-identity 41.9%

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

      5. times-frac 41.9%

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

      6. tan-quot 42.0%

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

      7. div-inv 42.0%

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

      8. metadata-eval 42.0%

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

      9. unpow2 42.0%

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

      10. frac-times 41.9%

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

      11. inv-pow 41.9%

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

      12. metadata-eval 41.9%

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

      13. inv-pow 41.9%

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

      14. metadata-eval 41.9%

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

      15. pow-sqr 41.9%

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

      16. metadata-eval 41.9%

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

      17. metadata-eval 41.9%

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

   8. Applied egg-rr 41.9%

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

      1. *-un-lft-identity 41.9%

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

      2. associate-*r* 41.9%

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

      3. *-commutative 41.9%

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

      4. /-rgt-identity 41.9%

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

      5. *-commutative 41.9%

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

      6. /-rgt-identity 41.9%

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

      7. associate-*l* 41.9%

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

   10. Applied egg-rr 41.9%

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

       1. *-lft-identity 41.9%

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

       2. associate-/r* 42.9%

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

       3. associate-*r* 42.9%

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

   12. Simplified 42.9%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k \cdot {\ell}^{-2}}}{\left(k \cdot \left(\sin k \cdot \sqrt{t}\right)\right) \cdot \left(\sqrt{t} \cdot \tan k\right)}\\ \end{array} \]

Alternative 6: 86.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt{t_m}}{\frac{\ell}{{k_m}^{2}}}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{t_2 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{t_m} \cdot \left(k_m \cdot \sin k_m\right)}}{\sqrt{t_m} \cdot \left(\left(k_m \cdot {\ell}^{-2}\right) \cdot \tan k_m\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (sqrt t_m) (/ l (pow k_m 2.0)))))
   (*
    t_s
    (if (<= k_m 1.2e-12)
      (/ 2.0 (* t_2 t_2))
      (/
       (/ 2.0 (* (sqrt t_m) (* k_m (sin k_m))))
       (* (sqrt t_m) (* (* k_m (pow l -2.0)) (tan k_m))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = sqrt(t_m) / (l / pow(k_m, 2.0));
	double tmp;
	if (k_m <= 1.2e-12) {
		tmp = 2.0 / (t_2 * t_2);
	} else {
		tmp = (2.0 / (sqrt(t_m) * (k_m * sin(k_m)))) / (sqrt(t_m) * ((k_m * pow(l, -2.0)) * tan(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sqrt(t_m) / (l / (k_m ** 2.0d0))
    if (k_m <= 1.2d-12) then
        tmp = 2.0d0 / (t_2 * t_2)
    else
        tmp = (2.0d0 / (sqrt(t_m) * (k_m * sin(k_m)))) / (sqrt(t_m) * ((k_m * (l ** (-2.0d0))) * tan(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.sqrt(t_m) / (l / Math.pow(k_m, 2.0));
	double tmp;
	if (k_m <= 1.2e-12) {
		tmp = 2.0 / (t_2 * t_2);
	} else {
		tmp = (2.0 / (Math.sqrt(t_m) * (k_m * Math.sin(k_m)))) / (Math.sqrt(t_m) * ((k_m * Math.pow(l, -2.0)) * Math.tan(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = math.sqrt(t_m) / (l / math.pow(k_m, 2.0))
	tmp = 0
	if k_m <= 1.2e-12:
		tmp = 2.0 / (t_2 * t_2)
	else:
		tmp = (2.0 / (math.sqrt(t_m) * (k_m * math.sin(k_m)))) / (math.sqrt(t_m) * ((k_m * math.pow(l, -2.0)) * math.tan(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(sqrt(t_m) / Float64(l / (k_m ^ 2.0)))
	tmp = 0.0
	if (k_m <= 1.2e-12)
		tmp = Float64(2.0 / Float64(t_2 * t_2));
	else
		tmp = Float64(Float64(2.0 / Float64(sqrt(t_m) * Float64(k_m * sin(k_m)))) / Float64(sqrt(t_m) * Float64(Float64(k_m * (l ^ -2.0)) * tan(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = sqrt(t_m) / (l / (k_m ^ 2.0));
	tmp = 0.0;
	if (k_m <= 1.2e-12)
		tmp = 2.0 / (t_2 * t_2);
	else
		tmp = (2.0 / (sqrt(t_m) * (k_m * sin(k_m)))) / (sqrt(t_m) * ((k_m * (l ^ -2.0)) * tan(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[Sqrt[t$95$m], $MachinePrecision] / N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.2e-12], N[(2.0 / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[(k$95$m * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.19999999999999994e-12

   1. Initial program 37.9%

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

      1. associate-*l* 37.9%

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

      2. sqr-neg 37.9%

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

      3. associate-*l* 37.9%

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

      4. associate-*l* 37.9%

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

      5. associate-*l/ 39.0%

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

      6. sqr-neg 39.0%

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

      7. associate-*l/ 37.9%

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

      8. associate--l+ 37.9%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in k around 0 68.2%

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

      1. associate-/l* 65.9%

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

      2. associate-/r/ 66.7%

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

   6. Simplified 66.7%

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

      1. associate-*l/ 68.2%

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

      2. unpow2 68.2%

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

      3. associate-/r* 75.7%

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

      4. *-commutative 75.7%

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

   8. Applied egg-rr 75.7%

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

      1. associate-/l/ 68.2%

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

      2. add-sqr-sqrt 31.3%

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

      3. times-frac 35.3%

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

      4. sqrt-prod 35.3%

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

      5. sqrt-pow1 35.3%

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

      6. metadata-eval 35.3%

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

      7. sqrt-prod 35.8%

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

      8. sqrt-pow1 37.4%

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

      9. metadata-eval 37.4%

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

   10. Applied egg-rr 37.4%

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

       1. associate-/l* 37.4%

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

       2. associate-/l* 37.4%

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

   12. Simplified 37.4%

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

   if 1.19999999999999994e-12 < k

   1. Initial program 38.9%

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

      1. associate-*l* 38.9%

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

      2. sqr-neg 38.9%

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

      3. associate-*l* 38.9%

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

      4. associate-*l* 38.9%

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

      5. associate-*l/ 38.9%

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

      6. sqr-neg 38.9%

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

      7. associate-*l/ 38.9%

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

      8. associate--l+ 38.9%

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

   3. Simplified 38.9%

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

   4. Taylor expanded in t around 0 74.1%

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

      1. add-sqr-sqrt 37.9%

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

      2. *-un-lft-identity 37.9%

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

      3. times-frac 37.9%

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

      4. sqrt-prod 37.9%

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

      5. unpow2 37.9%

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

      6. sqrt-prod 37.9%

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

      7. add-sqr-sqrt 37.9%

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

      8. *-commutative 37.9%

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

      9. sqrt-prod 37.9%

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

      10. unpow2 37.9%

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

      11. sqrt-prod 19.5%

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

      12. add-sqr-sqrt 29.7%

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

   6. Applied egg-rr 42.0%

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

      1. times-frac 41.9%

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

      2. *-commutative 41.9%

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

      3. *-commutative 41.9%

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

      4. *-un-lft-identity 41.9%

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

      5. times-frac 41.9%

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

      6. tan-quot 42.0%

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

      7. div-inv 42.0%

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

      8. metadata-eval 42.0%

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

      9. unpow2 42.0%

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

      10. frac-times 41.9%

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

      11. inv-pow 41.9%

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

      12. metadata-eval 41.9%

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

      13. inv-pow 41.9%

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

      14. metadata-eval 41.9%

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

      15. pow-sqr 41.9%

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

      16. metadata-eval 41.9%

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

      17. metadata-eval 41.9%

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

   8. Applied egg-rr 41.9%

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

      1. associate-/r* 42.9%

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

      2. div-inv 42.9%

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

      3. /-rgt-identity 42.9%

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

      4. associate-*r* 42.9%

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

      5. *-commutative 42.9%

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

      6. *-commutative 42.9%

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

      7. /-rgt-identity 42.9%

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

   10. Applied egg-rr 42.9%

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

       1. associate-*r/ 42.9%

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

       2. *-rgt-identity 42.9%

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

       3. /-rgt-identity 42.9%

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

       4. associate-/l* 42.9%

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

       5. associate-/r* 43.0%

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

       6. associate-/l* 42.9%

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

       7. associate-/r/ 42.9%

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

       8. /-rgt-identity 42.9%

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

   12. Simplified 42.9%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{t} \cdot \left(k \cdot \sin k\right)}}{\sqrt{t} \cdot \left(\left(k \cdot {\ell}^{-2}\right) \cdot \tan k\right)}\\ \end{array} \]

Alternative 7: 84.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt{t_m}}{\frac{\ell}{{k_m}^{2}}}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{t_2 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sin k_m \cdot \left(k_m \cdot \sqrt{t_m}\right)\right)}^{2}} \cdot \left(\cos k_m \cdot {\ell}^{2}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (sqrt t_m) (/ l (pow k_m 2.0)))))
   (*
    t_s
    (if (<= k_m 1.2e-12)
      (/ 2.0 (* t_2 t_2))
      (*
       (/ 2.0 (pow (* (sin k_m) (* k_m (sqrt t_m))) 2.0))
       (* (cos k_m) (pow l 2.0)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = sqrt(t_m) / (l / pow(k_m, 2.0));
	double tmp;
	if (k_m <= 1.2e-12) {
		tmp = 2.0 / (t_2 * t_2);
	} else {
		tmp = (2.0 / pow((sin(k_m) * (k_m * sqrt(t_m))), 2.0)) * (cos(k_m) * pow(l, 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sqrt(t_m) / (l / (k_m ** 2.0d0))
    if (k_m <= 1.2d-12) then
        tmp = 2.0d0 / (t_2 * t_2)
    else
        tmp = (2.0d0 / ((sin(k_m) * (k_m * sqrt(t_m))) ** 2.0d0)) * (cos(k_m) * (l ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.sqrt(t_m) / (l / Math.pow(k_m, 2.0));
	double tmp;
	if (k_m <= 1.2e-12) {
		tmp = 2.0 / (t_2 * t_2);
	} else {
		tmp = (2.0 / Math.pow((Math.sin(k_m) * (k_m * Math.sqrt(t_m))), 2.0)) * (Math.cos(k_m) * Math.pow(l, 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = math.sqrt(t_m) / (l / math.pow(k_m, 2.0))
	tmp = 0
	if k_m <= 1.2e-12:
		tmp = 2.0 / (t_2 * t_2)
	else:
		tmp = (2.0 / math.pow((math.sin(k_m) * (k_m * math.sqrt(t_m))), 2.0)) * (math.cos(k_m) * math.pow(l, 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(sqrt(t_m) / Float64(l / (k_m ^ 2.0)))
	tmp = 0.0
	if (k_m <= 1.2e-12)
		tmp = Float64(2.0 / Float64(t_2 * t_2));
	else
		tmp = Float64(Float64(2.0 / (Float64(sin(k_m) * Float64(k_m * sqrt(t_m))) ^ 2.0)) * Float64(cos(k_m) * (l ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = sqrt(t_m) / (l / (k_m ^ 2.0));
	tmp = 0.0;
	if (k_m <= 1.2e-12)
		tmp = 2.0 / (t_2 * t_2);
	else
		tmp = (2.0 / ((sin(k_m) * (k_m * sqrt(t_m))) ^ 2.0)) * (cos(k_m) * (l ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[Sqrt[t$95$m], $MachinePrecision] / N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.2e-12], N[(2.0 / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.19999999999999994e-12

   1. Initial program 37.9%

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

      1. associate-*l* 37.9%

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

      2. sqr-neg 37.9%

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

      3. associate-*l* 37.9%

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

      4. associate-*l* 37.9%

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

      5. associate-*l/ 39.0%

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

      6. sqr-neg 39.0%

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

      7. associate-*l/ 37.9%

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

      8. associate--l+ 37.9%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in k around 0 68.2%

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

      1. associate-/l* 65.9%

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

      2. associate-/r/ 66.7%

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

   6. Simplified 66.7%

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

      1. associate-*l/ 68.2%

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

      2. unpow2 68.2%

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

      3. associate-/r* 75.7%

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

      4. *-commutative 75.7%

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

   8. Applied egg-rr 75.7%

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

      1. associate-/l/ 68.2%

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

      2. add-sqr-sqrt 31.3%

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

      3. times-frac 35.3%

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

      4. sqrt-prod 35.3%

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

      5. sqrt-pow1 35.3%

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

      6. metadata-eval 35.3%

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

      7. sqrt-prod 35.8%

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

      8. sqrt-pow1 37.4%

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

      9. metadata-eval 37.4%

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

   10. Applied egg-rr 37.4%

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

       1. associate-/l* 37.4%

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

       2. associate-/l* 37.4%

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

   12. Simplified 37.4%

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

   if 1.19999999999999994e-12 < k

   1. Initial program 38.9%

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

      1. associate-*l* 38.9%

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

      2. sqr-neg 38.9%

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

      3. associate-*l* 38.9%

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

      4. associate-*l* 38.9%

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

      5. associate-*l/ 38.9%

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

      6. sqr-neg 38.9%

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

      7. associate-*l/ 38.9%

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

      8. associate--l+ 38.9%

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

   3. Simplified 38.9%

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

   4. Taylor expanded in t around 0 74.1%

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

      1. add-sqr-sqrt 37.9%

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

      2. *-un-lft-identity 37.9%

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

      3. times-frac 37.9%

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

      4. sqrt-prod 37.9%

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

      5. unpow2 37.9%

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

      6. sqrt-prod 37.9%

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

      7. add-sqr-sqrt 37.9%

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

      8. *-commutative 37.9%

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

      9. sqrt-prod 37.9%

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

      10. unpow2 37.9%

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

      11. sqrt-prod 19.5%

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

      12. add-sqr-sqrt 29.7%

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

   6. Applied egg-rr 42.0%

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

      1. associate-*r/ 39.1%

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

      2. /-rgt-identity 39.1%

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

      3. associate-/r/ 40.1%

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

      4. pow1 40.1%

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

      5. pow1 40.1%

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

      6. pow-sqr 40.1%

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

      7. *-commutative 40.1%

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

      8. associate-*l* 40.1%

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

      9. metadata-eval 40.1%

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

   8. Applied egg-rr 40.1%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sin k \cdot \left(k \cdot \sqrt{t}\right)\right)}^{2}} \cdot \left(\cos k \cdot {\ell}^{2}\right)\\ \end{array} \]

Alternative 8: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 5.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{4}}{\ell} \cdot \frac{t_m}{\ell}}\\ \mathbf{elif}\;t_m \leq 4.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{2}{{\left(\frac{k_m}{t_m}\right)}^{2} \cdot \frac{\tan k_m \cdot \frac{{t_m}^{3}}{\frac{\ell}{\sin k_m}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\sqrt{t_m}}{\frac{\ell}{{k_m}^{2}}} \cdot \left(\sqrt{t_m} \cdot {k_m}^{2}\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.8e-91)
    (/ 2.0 (* (/ (pow k_m 4.0) l) (/ t_m l)))
    (if (<= t_m 4.5e+86)
      (/
       2.0
       (*
        (pow (/ k_m t_m) 2.0)
        (/ (* (tan k_m) (/ (pow t_m 3.0) (/ l (sin k_m)))) l)))
      (/
       2.0
       (/
        (* (/ (sqrt t_m) (/ l (pow k_m 2.0))) (* (sqrt t_m) (pow k_m 2.0)))
        l))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 5.8e-91) {
		tmp = 2.0 / ((pow(k_m, 4.0) / l) * (t_m / l));
	} else if (t_m <= 4.5e+86) {
		tmp = 2.0 / (pow((k_m / t_m), 2.0) * ((tan(k_m) * (pow(t_m, 3.0) / (l / sin(k_m)))) / l));
	} else {
		tmp = 2.0 / (((sqrt(t_m) / (l / pow(k_m, 2.0))) * (sqrt(t_m) * pow(k_m, 2.0))) / l);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 5.8d-91) then
        tmp = 2.0d0 / (((k_m ** 4.0d0) / l) * (t_m / l))
    else if (t_m <= 4.5d+86) then
        tmp = 2.0d0 / (((k_m / t_m) ** 2.0d0) * ((tan(k_m) * ((t_m ** 3.0d0) / (l / sin(k_m)))) / l))
    else
        tmp = 2.0d0 / (((sqrt(t_m) / (l / (k_m ** 2.0d0))) * (sqrt(t_m) * (k_m ** 2.0d0))) / l)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 5.8e-91) {
		tmp = 2.0 / ((Math.pow(k_m, 4.0) / l) * (t_m / l));
	} else if (t_m <= 4.5e+86) {
		tmp = 2.0 / (Math.pow((k_m / t_m), 2.0) * ((Math.tan(k_m) * (Math.pow(t_m, 3.0) / (l / Math.sin(k_m)))) / l));
	} else {
		tmp = 2.0 / (((Math.sqrt(t_m) / (l / Math.pow(k_m, 2.0))) * (Math.sqrt(t_m) * Math.pow(k_m, 2.0))) / l);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 5.8e-91:
		tmp = 2.0 / ((math.pow(k_m, 4.0) / l) * (t_m / l))
	elif t_m <= 4.5e+86:
		tmp = 2.0 / (math.pow((k_m / t_m), 2.0) * ((math.tan(k_m) * (math.pow(t_m, 3.0) / (l / math.sin(k_m)))) / l))
	else:
		tmp = 2.0 / (((math.sqrt(t_m) / (l / math.pow(k_m, 2.0))) * (math.sqrt(t_m) * math.pow(k_m, 2.0))) / l)
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 5.8e-91)
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 4.0) / l) * Float64(t_m / l)));
	elseif (t_m <= 4.5e+86)
		tmp = Float64(2.0 / Float64((Float64(k_m / t_m) ^ 2.0) * Float64(Float64(tan(k_m) * Float64((t_m ^ 3.0) / Float64(l / sin(k_m)))) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sqrt(t_m) / Float64(l / (k_m ^ 2.0))) * Float64(sqrt(t_m) * (k_m ^ 2.0))) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 5.8e-91)
		tmp = 2.0 / (((k_m ^ 4.0) / l) * (t_m / l));
	elseif (t_m <= 4.5e+86)
		tmp = 2.0 / (((k_m / t_m) ^ 2.0) * ((tan(k_m) * ((t_m ^ 3.0) / (l / sin(k_m)))) / l));
	else
		tmp = 2.0 / (((sqrt(t_m) / (l / (k_m ^ 2.0))) * (sqrt(t_m) * (k_m ^ 2.0))) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 5.8e-91], N[(2.0 / N[(N[(N[Power[k$95$m, 4.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.5e+86], N[(2.0 / N[(N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Sqrt[t$95$m], $MachinePrecision] / N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.8000000000000001e-91

   1. Initial program 36.9%

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

      1. associate-*l* 36.9%

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

      2. sqr-neg 36.9%

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

      3. associate-*l* 36.9%

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

      4. associate-*l* 36.9%

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

      5. associate-*l/ 38.1%

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

      6. sqr-neg 38.1%

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

      7. associate-*l/ 36.9%

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

      8. associate--l+ 36.9%

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

   3. Simplified 36.9%

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

   4. Taylor expanded in k around 0 63.0%

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

      1. associate-/l* 62.3%

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

      2. associate-/r/ 61.4%

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

   6. Simplified 61.4%

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

      1. associate-*l/ 63.0%

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

      2. unpow2 63.0%

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

      3. associate-/r* 68.8%

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

      4. *-commutative 68.8%

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

   8. Applied egg-rr 68.8%

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

      1. associate-/l/ 63.0%

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

      2. *-commutative 63.0%

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

      3. times-frac 67.2%

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

   10. Applied egg-rr 67.2%

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

   if 5.8000000000000001e-91 < t < 4.49999999999999993e86

   1. Initial program 71.2%

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

   3. Simplified 77.7%

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

      1. associate-*r* 77.7%

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

      2. associate-/r* 76.2%

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

      3. *-commutative 76.2%

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

      4. associate-*l/ 76.2%

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

      5. associate-*l/ 76.2%

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

      6. times-frac 82.7%

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

      7. associate-*r/ 82.7%

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

      8. associate-/l* 84.9%

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

   5. Applied egg-rr 84.9%

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

   if 4.49999999999999993e86 < t

   1. Initial program 14.6%

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

      1. associate-*l* 14.6%

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

      2. sqr-neg 14.6%

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

      3. associate-*l* 14.6%

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

      4. associate-*l* 14.6%

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

      5. associate-*l/ 14.6%

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

      6. sqr-neg 14.6%

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

      7. associate-*l/ 14.6%

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

      8. associate--l+ 14.6%

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

   3. Simplified 14.6%

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

   4. Taylor expanded in k around 0 69.7%

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

      1. associate-/l* 67.8%

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

      2. associate-/r/ 72.1%

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

   6. Simplified 72.1%

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

      1. associate-*l/ 69.7%

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

      2. unpow2 69.7%

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

      3. associate-/r* 76.1%

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

      4. *-commutative 76.1%

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

   8. Applied egg-rr 76.1%

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

      1. add-sqr-sqrt 76.1%

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

      2. *-un-lft-identity 76.1%

         \[\leadsto \frac{2}{\frac{\frac{\sqrt{t \cdot {k}^{4}} \cdot \sqrt{t \cdot {k}^{4}}}{\color{blue}{1 \cdot \ell}}}{\ell}} \]

      3. times-frac 76.1%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{\sqrt{t \cdot {k}^{4}}}{1} \cdot \frac{\sqrt{t \cdot {k}^{4}}}{\ell}}}{\ell}} \]

      4. sqrt-prod 76.1%

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{\sqrt{t} \cdot \sqrt{{k}^{4}}}}{1} \cdot \frac{\sqrt{t \cdot {k}^{4}}}{\ell}}{\ell}} \]

      5. sqrt-pow1 76.1%

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

      6. metadata-eval 76.1%

         \[\leadsto \frac{2}{\frac{\frac{\sqrt{t} \cdot {k}^{\color{blue}{2}}}{1} \cdot \frac{\sqrt{t \cdot {k}^{4}}}{\ell}}{\ell}} \]

      7. sqrt-prod 76.3%

         \[\leadsto \frac{2}{\frac{\frac{\sqrt{t} \cdot {k}^{2}}{1} \cdot \frac{\color{blue}{\sqrt{t} \cdot \sqrt{{k}^{4}}}}{\ell}}{\ell}} \]

      8. sqrt-pow1 80.4%

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

      9. metadata-eval 80.4%

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

   10. Applied egg-rr 80.4%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{\sqrt{t} \cdot {k}^{2}}{1} \cdot \frac{\sqrt{t} \cdot {k}^{2}}{\ell}}}{\ell}} \]
   11. Step-by-step derivation

       1. /-rgt-identity 80.4%

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

       2. *-commutative 80.4%

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

       3. associate-/l* 80.4%

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

   12. Simplified 80.4%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t}\right)}^{2} \cdot \frac{\tan k \cdot \frac{{t}^{3}}{\frac{\ell}{\sin k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}} \cdot \left(\sqrt{t} \cdot {k}^{2}\right)}{\ell}}\\ \end{array} \]

Alternative 9: 70.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt{t_m}}{\frac{\ell}{{k_m}^{2}}}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\frac{2}{t_2 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_m \cdot {k_m}^{4}}{\cos k_m \cdot {\ell}^{2}}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (sqrt t_m) (/ l (pow k_m 2.0)))))
   (*
    t_s
    (if (<= (* l l) 5e+53)
      (/ 2.0 (* t_2 t_2))
      (/ 2.0 (/ (* t_m (pow k_m 4.0)) (* (cos k_m) (pow l 2.0))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = sqrt(t_m) / (l / pow(k_m, 2.0));
	double tmp;
	if ((l * l) <= 5e+53) {
		tmp = 2.0 / (t_2 * t_2);
	} else {
		tmp = 2.0 / ((t_m * pow(k_m, 4.0)) / (cos(k_m) * pow(l, 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sqrt(t_m) / (l / (k_m ** 2.0d0))
    if ((l * l) <= 5d+53) then
        tmp = 2.0d0 / (t_2 * t_2)
    else
        tmp = 2.0d0 / ((t_m * (k_m ** 4.0d0)) / (cos(k_m) * (l ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.sqrt(t_m) / (l / Math.pow(k_m, 2.0));
	double tmp;
	if ((l * l) <= 5e+53) {
		tmp = 2.0 / (t_2 * t_2);
	} else {
		tmp = 2.0 / ((t_m * Math.pow(k_m, 4.0)) / (Math.cos(k_m) * Math.pow(l, 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = math.sqrt(t_m) / (l / math.pow(k_m, 2.0))
	tmp = 0
	if (l * l) <= 5e+53:
		tmp = 2.0 / (t_2 * t_2)
	else:
		tmp = 2.0 / ((t_m * math.pow(k_m, 4.0)) / (math.cos(k_m) * math.pow(l, 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(sqrt(t_m) / Float64(l / (k_m ^ 2.0)))
	tmp = 0.0
	if (Float64(l * l) <= 5e+53)
		tmp = Float64(2.0 / Float64(t_2 * t_2));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * (k_m ^ 4.0)) / Float64(cos(k_m) * (l ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = sqrt(t_m) / (l / (k_m ^ 2.0));
	tmp = 0.0;
	if ((l * l) <= 5e+53)
		tmp = 2.0 / (t_2 * t_2);
	else
		tmp = 2.0 / ((t_m * (k_m ^ 4.0)) / (cos(k_m) * (l ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[Sqrt[t$95$m], $MachinePrecision] / N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e+53], N[(2.0 / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.0000000000000004e53

   1. Initial program 33.0%

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

      1. associate-*l* 33.0%

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

      2. sqr-neg 33.0%

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

      3. associate-*l* 33.0%

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

      4. associate-*l* 33.0%

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

      5. associate-*l/ 34.4%

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

      6. sqr-neg 34.4%

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

      7. associate-*l/ 33.0%

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

      8. associate--l+ 33.0%

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

   3. Simplified 33.0%

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

   4. Taylor expanded in k around 0 71.2%

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

      1. associate-/l* 68.2%

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

      2. associate-/r/ 71.8%

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

   6. Simplified 71.8%

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

      1. associate-*l/ 71.2%

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

      2. unpow2 71.2%

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

      3. associate-/r* 79.7%

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

      4. *-commutative 79.7%

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

   8. Applied egg-rr 79.7%

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

      1. associate-/l/ 71.2%

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

      2. add-sqr-sqrt 33.7%

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

      3. times-frac 38.0%

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

      4. sqrt-prod 38.0%

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

      5. sqrt-pow1 38.0%

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

      6. metadata-eval 38.0%

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

      7. sqrt-prod 38.7%

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

      8. sqrt-pow1 40.7%

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

      9. metadata-eval 40.7%

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

   10. Applied egg-rr 40.7%

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

       1. associate-/l* 40.7%

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

       2. associate-/l* 40.7%

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

   12. Simplified 40.7%

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

   if 5.0000000000000004e53 < (*.f64 l l)

   1. Initial program 44.8%

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

      1. associate-*l* 44.8%

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

      2. sqr-neg 44.8%

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

      3. associate-*l* 44.8%

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

      4. associate-*l* 44.8%

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

      5. associate-*l/ 44.8%

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

      6. sqr-neg 44.8%

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

      7. associate-*l/ 44.8%

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

      8. associate--l+ 44.8%

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

   3. Simplified 44.8%

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

   4. Taylor expanded in t around 0 72.6%

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

   5. Taylor expanded in k around 0 64.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{4} \cdot t}}{{\ell}^{2} \cdot \cos k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]

Alternative 10: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{k_m}^{2}}{\ell}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-58}:\\ \;\;\;\;\frac{2}{t_m \cdot \left(t_2 \cdot t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{2} \cdot \left(t_m \cdot {k_m}^{2}\right)}{\cos k_m \cdot {\ell}^{2}}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (pow k_m 2.0) l)))
   (*
    t_s
    (if (<= (* l l) 1e-58)
      (/ 2.0 (* t_m (* t_2 t_2)))
      (/
       2.0
       (/
        (* (pow k_m 2.0) (* t_m (pow k_m 2.0)))
        (* (cos k_m) (pow l 2.0))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = pow(k_m, 2.0) / l;
	double tmp;
	if ((l * l) <= 1e-58) {
		tmp = 2.0 / (t_m * (t_2 * t_2));
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t_m * pow(k_m, 2.0))) / (cos(k_m) * pow(l, 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k_m ** 2.0d0) / l
    if ((l * l) <= 1d-58) then
        tmp = 2.0d0 / (t_m * (t_2 * t_2))
    else
        tmp = 2.0d0 / (((k_m ** 2.0d0) * (t_m * (k_m ** 2.0d0))) / (cos(k_m) * (l ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.pow(k_m, 2.0) / l;
	double tmp;
	if ((l * l) <= 1e-58) {
		tmp = 2.0 / (t_m * (t_2 * t_2));
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t_m * Math.pow(k_m, 2.0))) / (Math.cos(k_m) * Math.pow(l, 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = math.pow(k_m, 2.0) / l
	tmp = 0
	if (l * l) <= 1e-58:
		tmp = 2.0 / (t_m * (t_2 * t_2))
	else:
		tmp = 2.0 / ((math.pow(k_m, 2.0) * (t_m * math.pow(k_m, 2.0))) / (math.cos(k_m) * math.pow(l, 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64((k_m ^ 2.0) / l)
	tmp = 0.0
	if (Float64(l * l) <= 1e-58)
		tmp = Float64(2.0 / Float64(t_m * Float64(t_2 * t_2)));
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t_m * (k_m ^ 2.0))) / Float64(cos(k_m) * (l ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = (k_m ^ 2.0) / l;
	tmp = 0.0;
	if ((l * l) <= 1e-58)
		tmp = 2.0 / (t_m * (t_2 * t_2));
	else
		tmp = 2.0 / (((k_m ^ 2.0) * (t_m * (k_m ^ 2.0))) / (cos(k_m) * (l ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-58], N[(2.0 / N[(t$95$m * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1e-58

   1. Initial program 33.2%

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

      1. associate-*l* 33.3%

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

      2. sqr-neg 33.3%

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

      3. associate-*l* 33.2%

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

      4. associate-*l* 33.3%

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

      5. associate-*l/ 34.9%

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

      6. sqr-neg 34.9%

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

      7. associate-*l/ 33.3%

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

      8. associate--l+ 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in k around 0 70.7%

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

      1. associate-/l* 67.3%

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

      2. associate-/r/ 71.5%

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

   6. Simplified 71.5%

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

      1. metadata-eval 71.5%

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

      2. pow-sqr 71.4%

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

      3. unpow2 71.4%

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

      4. times-frac 86.1%

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

   8. Applied egg-rr 86.1%

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

   if 1e-58 < (*.f64 l l)

   1. Initial program 42.7%

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

      1. associate-*l* 42.7%

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

      2. sqr-neg 42.7%

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

      3. associate-*l* 42.7%

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

      4. associate-*l* 42.7%

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

      5. associate-*l/ 42.7%

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

      6. sqr-neg 42.7%

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

      7. associate-*l/ 42.7%

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

      8. associate--l+ 42.7%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in t around 0 75.6%

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

   5. Taylor expanded in k around 0 66.3%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-58}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{{k}^{2}}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]

Alternative 11: 71.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3.9 \cdot 10^{-91}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{4}}{\ell} \cdot \frac{t_m}{\ell}}\\ \mathbf{elif}\;t_m \leq 2.75 \cdot 10^{+84}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t_m}^{3}}{\frac{\ell}{\sin k_m}}}{\ell} \cdot \left(\tan k_m \cdot \frac{k_m}{t_m \cdot \frac{t_m}{k_m}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{2} \cdot \frac{t_m \cdot {k_m}^{2}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.9e-91)
    (/ 2.0 (* (/ (pow k_m 4.0) l) (/ t_m l)))
    (if (<= t_m 2.75e+84)
      (/
       2.0
       (*
        (/ (/ (pow t_m 3.0) (/ l (sin k_m))) l)
        (* (tan k_m) (/ k_m (* t_m (/ t_m k_m))))))
      (/ 2.0 (/ (* (pow k_m 2.0) (/ (* t_m (pow k_m 2.0)) l)) l))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 3.9e-91) {
		tmp = 2.0 / ((pow(k_m, 4.0) / l) * (t_m / l));
	} else if (t_m <= 2.75e+84) {
		tmp = 2.0 / (((pow(t_m, 3.0) / (l / sin(k_m))) / l) * (tan(k_m) * (k_m / (t_m * (t_m / k_m)))));
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) * ((t_m * pow(k_m, 2.0)) / l)) / l);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 3.9d-91) then
        tmp = 2.0d0 / (((k_m ** 4.0d0) / l) * (t_m / l))
    else if (t_m <= 2.75d+84) then
        tmp = 2.0d0 / ((((t_m ** 3.0d0) / (l / sin(k_m))) / l) * (tan(k_m) * (k_m / (t_m * (t_m / k_m)))))
    else
        tmp = 2.0d0 / (((k_m ** 2.0d0) * ((t_m * (k_m ** 2.0d0)) / l)) / l)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 3.9e-91) {
		tmp = 2.0 / ((Math.pow(k_m, 4.0) / l) * (t_m / l));
	} else if (t_m <= 2.75e+84) {
		tmp = 2.0 / (((Math.pow(t_m, 3.0) / (l / Math.sin(k_m))) / l) * (Math.tan(k_m) * (k_m / (t_m * (t_m / k_m)))));
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * ((t_m * Math.pow(k_m, 2.0)) / l)) / l);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 3.9e-91:
		tmp = 2.0 / ((math.pow(k_m, 4.0) / l) * (t_m / l))
	elif t_m <= 2.75e+84:
		tmp = 2.0 / (((math.pow(t_m, 3.0) / (l / math.sin(k_m))) / l) * (math.tan(k_m) * (k_m / (t_m * (t_m / k_m)))))
	else:
		tmp = 2.0 / ((math.pow(k_m, 2.0) * ((t_m * math.pow(k_m, 2.0)) / l)) / l)
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 3.9e-91)
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 4.0) / l) * Float64(t_m / l)));
	elseif (t_m <= 2.75e+84)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) / Float64(l / sin(k_m))) / l) * Float64(tan(k_m) * Float64(k_m / Float64(t_m * Float64(t_m / k_m))))));
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(Float64(t_m * (k_m ^ 2.0)) / l)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 3.9e-91)
		tmp = 2.0 / (((k_m ^ 4.0) / l) * (t_m / l));
	elseif (t_m <= 2.75e+84)
		tmp = 2.0 / ((((t_m ^ 3.0) / (l / sin(k_m))) / l) * (tan(k_m) * (k_m / (t_m * (t_m / k_m)))));
	else
		tmp = 2.0 / (((k_m ^ 2.0) * ((t_m * (k_m ^ 2.0)) / l)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.9e-91], N[(2.0 / N[(N[(N[Power[k$95$m, 4.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.75e+84], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m / N[(t$95$m * N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.89999999999999994e-91

   1. Initial program 36.9%

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

      1. associate-*l* 36.9%

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

      2. sqr-neg 36.9%

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

      3. associate-*l* 36.9%

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

      4. associate-*l* 36.9%

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

      5. associate-*l/ 38.1%

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

      6. sqr-neg 38.1%

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

      7. associate-*l/ 36.9%

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

      8. associate--l+ 36.9%

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

   3. Simplified 36.9%

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

   4. Taylor expanded in k around 0 63.0%

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

      1. associate-/l* 62.3%

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

      2. associate-/r/ 61.4%

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

   6. Simplified 61.4%

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

      1. associate-*l/ 63.0%

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

      2. unpow2 63.0%

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

      3. associate-/r* 68.8%

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

      4. *-commutative 68.8%

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

   8. Applied egg-rr 68.8%

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

      1. associate-/l/ 63.0%

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

      2. *-commutative 63.0%

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

      3. times-frac 67.2%

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

   10. Applied egg-rr 67.2%

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

   if 3.89999999999999994e-91 < t < 2.7500000000000002e84

   1. Initial program 71.2%

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

      1. associate-*l* 71.1%

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

      2. sqr-neg 71.1%

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

      3. associate-*l* 71.2%

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

      4. associate-*l* 71.1%

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

      5. associate-*l/ 71.1%

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

      6. sqr-neg 71.1%

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

      7. associate-*l/ 71.1%

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

      8. associate--l+ 71.1%

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

   3. Simplified 71.1%

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

      1. +-commutative 71.1%

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

      2. associate-+l- 76.1%

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

      3. metadata-eval 76.1%

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

      4. --rgt-identity 76.1%

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

      5. unpow2 76.1%

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

      6. clear-num 76.1%

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

      7. frac-times 76.2%

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

      8. *-un-lft-identity 76.2%

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

   5. Applied egg-rr 76.2%

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

      1. associate-*l/ 76.2%

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

      2. associate-/r* 77.8%

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

      3. associate-/l* 77.8%

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

   7. Applied egg-rr 77.8%

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

   if 2.7500000000000002e84 < t

   1. Initial program 14.6%

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

      1. associate-*l* 14.6%

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

      2. sqr-neg 14.6%

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

      3. associate-*l* 14.6%

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

      4. associate-*l* 14.6%

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

      5. associate-*l/ 14.6%

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

      6. sqr-neg 14.6%

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

      7. associate-*l/ 14.6%

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

      8. associate--l+ 14.6%

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

   3. Simplified 14.6%

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

   4. Taylor expanded in k around 0 69.7%

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

      1. associate-/l* 67.8%

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

      2. associate-/r/ 72.1%

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

   6. Simplified 72.1%

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

      1. associate-*l/ 69.7%

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

      2. unpow2 69.7%

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

      3. associate-/r* 76.1%

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

      4. *-commutative 76.1%

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

   8. Applied egg-rr 76.1%

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

      1. expm1-log1p-u 52.3%

         \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot {k}^{4}}{\ell}\right)\right)}}{\ell}} \]

      2. associate-/l* 52.5%

         \[\leadsto \frac{2}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{t}{\frac{\ell}{{k}^{4}}}}\right)\right)}{\ell}} \]

      3. associate-/r/ 48.6%

         \[\leadsto \frac{2}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{t}{\ell} \cdot {k}^{4}}\right)\right)}{\ell}} \]

   10. Applied egg-rr 48.6%

       \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\ell} \cdot {k}^{4}\right)\right)}}{\ell}} \]
   11. Step-by-step derivation

       1. expm1-log1p 72.3%

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

       2. add-sqr-sqrt 72.3%

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

       3. associate-*r* 72.3%

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

       4. add-sqr-sqrt 72.3%

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

       5. *-un-lft-identity 72.3%

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

       6. times-frac 72.3%

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

       7. /-rgt-identity 72.3%

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

       8. sqrt-pow1 72.3%

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

       9. metadata-eval 72.3%

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

       10. associate-*r* 74.3%

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

       11. associate-/r/ 76.3%

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

       12. sqrt-pow1 80.4%

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

       13. metadata-eval 80.4%

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

       14. *-commutative 80.4%

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

       15. associate-*l* 80.4%

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

       16. associate-*r/ 80.3%

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

       17. associate-/r/ 80.3%

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

   12. Applied egg-rr 80.3%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.9 \cdot 10^{-91}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell} \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{+84}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\frac{\ell}{\sin k}}}{\ell} \cdot \left(\tan k \cdot \frac{k}{t \cdot \frac{t}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \frac{t \cdot {k}^{2}}{\ell}}{\ell}}\\ \end{array} \]

Alternative 12: 69.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{2} \cdot \frac{t_m \cdot {k_m}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_m \cdot {k_m}^{4}}{\cos k_m \cdot {\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 5e+53)
    (/ 2.0 (/ (* (pow k_m 2.0) (/ (* t_m (pow k_m 2.0)) l)) l))
    (/ 2.0 (/ (* t_m (pow k_m 4.0)) (* (cos k_m) (pow l 2.0)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 5e+53) {
		tmp = 2.0 / ((pow(k_m, 2.0) * ((t_m * pow(k_m, 2.0)) / l)) / l);
	} else {
		tmp = 2.0 / ((t_m * pow(k_m, 4.0)) / (cos(k_m) * pow(l, 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 5d+53) then
        tmp = 2.0d0 / (((k_m ** 2.0d0) * ((t_m * (k_m ** 2.0d0)) / l)) / l)
    else
        tmp = 2.0d0 / ((t_m * (k_m ** 4.0d0)) / (cos(k_m) * (l ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 5e+53) {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * ((t_m * Math.pow(k_m, 2.0)) / l)) / l);
	} else {
		tmp = 2.0 / ((t_m * Math.pow(k_m, 4.0)) / (Math.cos(k_m) * Math.pow(l, 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 5e+53:
		tmp = 2.0 / ((math.pow(k_m, 2.0) * ((t_m * math.pow(k_m, 2.0)) / l)) / l)
	else:
		tmp = 2.0 / ((t_m * math.pow(k_m, 4.0)) / (math.cos(k_m) * math.pow(l, 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 5e+53)
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(Float64(t_m * (k_m ^ 2.0)) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * (k_m ^ 4.0)) / Float64(cos(k_m) * (l ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 5e+53)
		tmp = 2.0 / (((k_m ^ 2.0) * ((t_m * (k_m ^ 2.0)) / l)) / l);
	else
		tmp = 2.0 / ((t_m * (k_m ^ 4.0)) / (cos(k_m) * (l ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e+53], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.0000000000000004e53

   1. Initial program 33.0%

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

      1. associate-*l* 33.0%

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

      2. sqr-neg 33.0%

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

      3. associate-*l* 33.0%

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

      4. associate-*l* 33.0%

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

      5. associate-*l/ 34.4%

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

      6. sqr-neg 34.4%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      7. associate-*l/ 33.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      8. associate--l+ 33.0%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)}\right)} \]

   3. Simplified 33.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 71.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 68.2%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}}} \]

      2. associate-/r/ 71.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]

   6. Simplified 71.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]
   7. Step-by-step derivation

      1. associate-*l/ 71.2%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

      2. unpow2 71.2%

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      3. associate-/r* 79.7%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{4} \cdot t}{\ell}}{\ell}}} \]

      4. *-commutative 79.7%

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{4}}}{\ell}}{\ell}} \]

   8. Applied egg-rr 79.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t \cdot {k}^{4}}{\ell}}{\ell}}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 54.9%

         \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot {k}^{4}}{\ell}\right)\right)}}{\ell}} \]

      2. associate-/l* 55.6%

         \[\leadsto \frac{2}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{t}{\frac{\ell}{{k}^{4}}}}\right)\right)}{\ell}} \]

      3. associate-/r/ 52.1%

         \[\leadsto \frac{2}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{t}{\ell} \cdot {k}^{4}}\right)\right)}{\ell}} \]

   10. Applied egg-rr 52.1%

       \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\ell} \cdot {k}^{4}\right)\right)}}{\ell}} \]
   11. Step-by-step derivation

       1. expm1-log1p 76.9%

          \[\leadsto \frac{2}{\frac{\color{blue}{\frac{t}{\ell} \cdot {k}^{4}}}{\ell}} \]

       2. add-sqr-sqrt 76.9%

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell} \cdot \color{blue}{\left(\sqrt{{k}^{4}} \cdot \sqrt{{k}^{4}}\right)}}{\ell}} \]

       3. associate-*r* 76.9%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(\frac{t}{\ell} \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}}{\ell}} \]

       4. add-sqr-sqrt 37.4%

          \[\leadsto \frac{2}{\frac{\left(\frac{\color{blue}{\sqrt{t} \cdot \sqrt{t}}}{\ell} \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       5. *-un-lft-identity 37.4%

          \[\leadsto \frac{2}{\frac{\left(\frac{\sqrt{t} \cdot \sqrt{t}}{\color{blue}{1 \cdot \ell}} \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       6. times-frac 37.3%

          \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(\frac{\sqrt{t}}{1} \cdot \frac{\sqrt{t}}{\ell}\right)} \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       7. /-rgt-identity 37.3%

          \[\leadsto \frac{2}{\frac{\left(\left(\color{blue}{\sqrt{t}} \cdot \frac{\sqrt{t}}{\ell}\right) \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       8. sqrt-pow1 37.4%

          \[\leadsto \frac{2}{\frac{\left(\left(\sqrt{t} \cdot \frac{\sqrt{t}}{\ell}\right) \cdot \color{blue}{{k}^{\left(\frac{4}{2}\right)}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       9. metadata-eval 37.4%

          \[\leadsto \frac{2}{\frac{\left(\left(\sqrt{t} \cdot \frac{\sqrt{t}}{\ell}\right) \cdot {k}^{\color{blue}{2}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       10. associate-*r* 38.0%

           \[\leadsto \frac{2}{\frac{\color{blue}{\left(\sqrt{t} \cdot \left(\frac{\sqrt{t}}{\ell} \cdot {k}^{2}\right)\right)} \cdot \sqrt{{k}^{4}}}{\ell}} \]

       11. associate-/r/ 38.7%

           \[\leadsto \frac{2}{\frac{\left(\sqrt{t} \cdot \color{blue}{\frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       12. sqrt-pow1 40.7%

           \[\leadsto \frac{2}{\frac{\left(\sqrt{t} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}\right) \cdot \color{blue}{{k}^{\left(\frac{4}{2}\right)}}}{\ell}} \]

       13. metadata-eval 40.7%

           \[\leadsto \frac{2}{\frac{\left(\sqrt{t} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}\right) \cdot {k}^{\color{blue}{2}}}{\ell}} \]

       14. *-commutative 40.7%

           \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{2} \cdot \left(\sqrt{t} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}\right)}}{\ell}} \]

       15. associate-*l* 40.7%

           \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot \sqrt{t}\right) \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}}}{\ell}} \]

       16. associate-*r/ 40.7%

           \[\leadsto \frac{2}{\frac{\color{blue}{\frac{\left({k}^{2} \cdot \sqrt{t}\right) \cdot \sqrt{t}}{\frac{\ell}{{k}^{2}}}}}{\ell}} \]

       17. associate-/r/ 40.7%

           \[\leadsto \frac{2}{\frac{\color{blue}{\frac{\left({k}^{2} \cdot \sqrt{t}\right) \cdot \sqrt{t}}{\ell} \cdot {k}^{2}}}{\ell}} \]

   12. Applied egg-rr 84.4%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot {k}^{2}}}{\ell}} \]

   if 5.0000000000000004e53 < (*.f64 l l)

   1. Initial program 44.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 44.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. sqr-neg 44.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      3. associate-*l* 44.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-*l* 44.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      5. associate-*l/ 44.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      6. sqr-neg 44.8%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      7. associate-*l/ 44.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      8. associate--l+ 44.8%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)}\right)} \]

   3. Simplified 44.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)}} \]

   4. Taylor expanded in t around 0 72.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]

   5. Taylor expanded in k around 0 64.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{4} \cdot t}}{{\ell}^{2} \cdot \cos k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \frac{t \cdot {k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]

Alternative 13: 69.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 5.8 \cdot 10^{-152}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \ell}{t_m \cdot {k_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{2}}{\ell} \cdot \left({k_m}^{2} \cdot \frac{t_m}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 5.8e-152)
    (* l (/ (* 2.0 l) (* t_m (pow k_m 4.0))))
    (/ 2.0 (* (/ (pow k_m 2.0) l) (* (pow k_m 2.0) (/ t_m l)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5.8e-152) {
		tmp = l * ((2.0 * l) / (t_m * pow(k_m, 4.0)));
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) / l) * (pow(k_m, 2.0) * (t_m / l)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5.8d-152) then
        tmp = l * ((2.0d0 * l) / (t_m * (k_m ** 4.0d0)))
    else
        tmp = 2.0d0 / (((k_m ** 2.0d0) / l) * ((k_m ** 2.0d0) * (t_m / l)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5.8e-152) {
		tmp = l * ((2.0 * l) / (t_m * Math.pow(k_m, 4.0)));
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) / l) * (Math.pow(k_m, 2.0) * (t_m / l)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 5.8e-152:
		tmp = l * ((2.0 * l) / (t_m * math.pow(k_m, 4.0)))
	else:
		tmp = 2.0 / ((math.pow(k_m, 2.0) / l) * (math.pow(k_m, 2.0) * (t_m / l)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 5.8e-152)
		tmp = Float64(l * Float64(Float64(2.0 * l) / Float64(t_m * (k_m ^ 4.0))));
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) / l) * Float64((k_m ^ 2.0) * Float64(t_m / l))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 5.8e-152)
		tmp = l * ((2.0 * l) / (t_m * (k_m ^ 4.0)));
	else
		tmp = 2.0 / (((k_m ^ 2.0) / l) * ((k_m ^ 2.0) * (t_m / l)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 5.8e-152], N[(l * N[(N[(2.0 * l), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.8000000000000003e-152

   1. Initial program 36.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 36.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. sqr-neg 36.3%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      3. associate-*l* 36.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-*l* 36.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      5. associate-*l/ 37.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      6. sqr-neg 37.6%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      7. associate-*l/ 36.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      8. associate--l+ 36.3%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)}\right)} \]

   3. Simplified 36.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 66.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 62.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}}} \]

      2. associate-/r/ 64.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]

   6. Simplified 64.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]
   7. Step-by-step derivation

      1. associate-*l/ 66.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

      2. unpow2 66.1%

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      3. associate-/r* 73.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{4} \cdot t}{\ell}}{\ell}}} \]

      4. *-commutative 73.3%

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{4}}}{\ell}}{\ell}} \]

   8. Applied egg-rr 73.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t \cdot {k}^{4}}{\ell}}{\ell}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 73.3%

         \[\leadsto \color{blue}{\frac{2}{\frac{t \cdot {k}^{4}}{\ell}} \cdot \ell} \]

      2. clear-num 73.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{1}{\frac{\ell}{t \cdot {k}^{4}}}}} \cdot \ell \]

      3. associate-/r/ 73.3%

         \[\leadsto \color{blue}{\left(\frac{2}{1} \cdot \frac{\ell}{t \cdot {k}^{4}}\right)} \cdot \ell \]

      4. metadata-eval 73.3%

         \[\leadsto \left(\color{blue}{2} \cdot \frac{\ell}{t \cdot {k}^{4}}\right) \cdot \ell \]

      5. associate-/r* 71.2%

         \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\ell}{t}}{{k}^{4}}}\right) \cdot \ell \]

   10. Applied egg-rr 71.2%

       \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right) \cdot \ell} \]
   11. Step-by-step derivation

       1. *-commutative 71.2%

          \[\leadsto \color{blue}{\left(\frac{\frac{\ell}{t}}{{k}^{4}} \cdot 2\right)} \cdot \ell \]

       2. associate-/l/ 73.3%

          \[\leadsto \left(\color{blue}{\frac{\ell}{{k}^{4} \cdot t}} \cdot 2\right) \cdot \ell \]

       3. associate-*l/ 73.3%

          \[\leadsto \color{blue}{\frac{\ell \cdot 2}{{k}^{4} \cdot t}} \cdot \ell \]

       4. *-commutative 73.3%

          \[\leadsto \frac{\ell \cdot 2}{\color{blue}{t \cdot {k}^{4}}} \cdot \ell \]

   12. Applied egg-rr 73.3%

       \[\leadsto \color{blue}{\frac{\ell \cdot 2}{t \cdot {k}^{4}}} \cdot \ell \]

   if 5.8000000000000003e-152 < k

   1. Initial program 41.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 41.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. sqr-neg 41.0%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      3. associate-*l* 41.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-*l* 41.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      5. associate-*l/ 41.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      6. sqr-neg 41.0%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      7. associate-*l/ 41.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      8. associate--l+ 41.0%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)}\right)} \]

   3. Simplified 41.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 61.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 63.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}}} \]

      2. associate-/r/ 62.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]

   6. Simplified 62.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]
   7. Step-by-step derivation

      1. associate-*l/ 61.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

      2. unpow2 61.0%

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      3. associate-/r* 64.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{4} \cdot t}{\ell}}{\ell}}} \]

      4. *-commutative 64.0%

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{4}}}{\ell}}{\ell}} \]

   8. Applied egg-rr 64.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t \cdot {k}^{4}}{\ell}}{\ell}}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 47.8%

         \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot {k}^{4}}{\ell}\right)\right)}}{\ell}} \]

      2. associate-/l* 47.0%

         \[\leadsto \frac{2}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{t}{\frac{\ell}{{k}^{4}}}}\right)\right)}{\ell}} \]

      3. associate-/r/ 47.9%

         \[\leadsto \frac{2}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{t}{\ell} \cdot {k}^{4}}\right)\right)}{\ell}} \]

   10. Applied egg-rr 47.9%

       \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\ell} \cdot {k}^{4}\right)\right)}}{\ell}} \]
   11. Step-by-step derivation

       1. expm1-log1p 64.1%

          \[\leadsto \frac{2}{\frac{\color{blue}{\frac{t}{\ell} \cdot {k}^{4}}}{\ell}} \]

       2. add-sqr-sqrt 64.1%

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell} \cdot \color{blue}{\left(\sqrt{{k}^{4}} \cdot \sqrt{{k}^{4}}\right)}}{\ell}} \]

       3. associate-*r* 64.1%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(\frac{t}{\ell} \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}}{\ell}} \]

       4. add-sqr-sqrt 33.6%

          \[\leadsto \frac{2}{\frac{\left(\frac{\color{blue}{\sqrt{t} \cdot \sqrt{t}}}{\ell} \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       5. *-un-lft-identity 33.6%

          \[\leadsto \frac{2}{\frac{\left(\frac{\sqrt{t} \cdot \sqrt{t}}{\color{blue}{1 \cdot \ell}} \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       6. times-frac 33.5%

          \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(\frac{\sqrt{t}}{1} \cdot \frac{\sqrt{t}}{\ell}\right)} \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       7. /-rgt-identity 33.5%

          \[\leadsto \frac{2}{\frac{\left(\left(\color{blue}{\sqrt{t}} \cdot \frac{\sqrt{t}}{\ell}\right) \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       8. sqrt-pow1 33.6%

          \[\leadsto \frac{2}{\frac{\left(\left(\sqrt{t} \cdot \frac{\sqrt{t}}{\ell}\right) \cdot \color{blue}{{k}^{\left(\frac{4}{2}\right)}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       9. metadata-eval 33.6%

          \[\leadsto \frac{2}{\frac{\left(\left(\sqrt{t} \cdot \frac{\sqrt{t}}{\ell}\right) \cdot {k}^{\color{blue}{2}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       10. associate-*r* 34.4%

           \[\leadsto \frac{2}{\frac{\color{blue}{\left(\sqrt{t} \cdot \left(\frac{\sqrt{t}}{\ell} \cdot {k}^{2}\right)\right)} \cdot \sqrt{{k}^{4}}}{\ell}} \]

       11. associate-/r/ 34.5%

           \[\leadsto \frac{2}{\frac{\left(\sqrt{t} \cdot \color{blue}{\frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       12. sqrt-pow1 36.3%

           \[\leadsto \frac{2}{\frac{\left(\sqrt{t} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}\right) \cdot \color{blue}{{k}^{\left(\frac{4}{2}\right)}}}{\ell}} \]

       13. metadata-eval 36.3%

           \[\leadsto \frac{2}{\frac{\left(\sqrt{t} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}\right) \cdot {k}^{\color{blue}{2}}}{\ell}} \]

       14. *-commutative 36.3%

           \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{2} \cdot \left(\sqrt{t} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}\right)}}{\ell}} \]

       15. associate-*l* 36.3%

           \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot \sqrt{t}\right) \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}}}{\ell}} \]

       16. associate-/l* 35.2%

           \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \sqrt{t}}{\frac{\ell}{\frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}}}}} \]

   12. Applied egg-rr 67.7%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot {k}^{2}\right) \cdot \frac{{k}^{2}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-152}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \ell}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\ell} \cdot \left({k}^{2} \cdot \frac{t}{\ell}\right)}\\ \end{array} \]

Alternative 14: 69.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{k_m}^{2}}{\ell}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 4.1 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{t_m \cdot \left(t_2 \cdot t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left({k_m}^{2} \cdot \frac{t_m}{\ell}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (pow k_m 2.0) l)))
   (*
    t_s
    (if (<= k_m 4.1e-138)
      (/ 2.0 (* t_m (* t_2 t_2)))
      (/ 2.0 (* t_2 (* (pow k_m 2.0) (/ t_m l))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = pow(k_m, 2.0) / l;
	double tmp;
	if (k_m <= 4.1e-138) {
		tmp = 2.0 / (t_m * (t_2 * t_2));
	} else {
		tmp = 2.0 / (t_2 * (pow(k_m, 2.0) * (t_m / l)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k_m ** 2.0d0) / l
    if (k_m <= 4.1d-138) then
        tmp = 2.0d0 / (t_m * (t_2 * t_2))
    else
        tmp = 2.0d0 / (t_2 * ((k_m ** 2.0d0) * (t_m / l)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.pow(k_m, 2.0) / l;
	double tmp;
	if (k_m <= 4.1e-138) {
		tmp = 2.0 / (t_m * (t_2 * t_2));
	} else {
		tmp = 2.0 / (t_2 * (Math.pow(k_m, 2.0) * (t_m / l)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = math.pow(k_m, 2.0) / l
	tmp = 0
	if k_m <= 4.1e-138:
		tmp = 2.0 / (t_m * (t_2 * t_2))
	else:
		tmp = 2.0 / (t_2 * (math.pow(k_m, 2.0) * (t_m / l)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64((k_m ^ 2.0) / l)
	tmp = 0.0
	if (k_m <= 4.1e-138)
		tmp = Float64(2.0 / Float64(t_m * Float64(t_2 * t_2)));
	else
		tmp = Float64(2.0 / Float64(t_2 * Float64((k_m ^ 2.0) * Float64(t_m / l))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = (k_m ^ 2.0) / l;
	tmp = 0.0;
	if (k_m <= 4.1e-138)
		tmp = 2.0 / (t_m * (t_2 * t_2));
	else
		tmp = 2.0 / (t_2 * ((k_m ^ 2.0) * (t_m / l)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 4.1e-138], N[(2.0 / N[(t$95$m * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.09999999999999999e-138

   1. Initial program 36.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 36.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. sqr-neg 36.2%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      3. associate-*l* 36.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-*l* 36.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      5. associate-*l/ 37.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      6. sqr-neg 37.5%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      7. associate-*l/ 36.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      8. associate--l+ 36.2%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)}\right)} \]

   3. Simplified 36.2%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 66.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 62.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}}} \]

      2. associate-/r/ 64.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]

   6. Simplified 64.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]
   7. Step-by-step derivation

      1. metadata-eval 64.4%

         \[\leadsto \frac{2}{\frac{{k}^{\color{blue}{\left(2 \cdot 2\right)}}}{{\ell}^{2}} \cdot t} \]

      2. pow-sqr 64.4%

         \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{2} \cdot {k}^{2}}}{{\ell}^{2}} \cdot t} \]

      3. unpow2 64.4%

         \[\leadsto \frac{2}{\frac{{k}^{2} \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot t} \]

      4. times-frac 74.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot t} \]

   8. Applied egg-rr 74.9%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot t} \]

   if 4.09999999999999999e-138 < k

   1. Initial program 41.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 41.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. sqr-neg 41.2%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      3. associate-*l* 41.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-*l* 41.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      5. associate-*l/ 41.2%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      6. sqr-neg 41.2%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      7. associate-*l/ 41.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      8. associate--l+ 41.2%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)}\right)} \]

   3. Simplified 41.2%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 60.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 63.7%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}}} \]

      2. associate-/r/ 62.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]

   6. Simplified 62.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]
   7. Step-by-step derivation

      1. associate-*l/ 60.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

      2. unpow2 60.8%

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      3. associate-/r* 63.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{4} \cdot t}{\ell}}{\ell}}} \]

      4. *-commutative 63.9%

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{4}}}{\ell}}{\ell}} \]

   8. Applied egg-rr 63.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t \cdot {k}^{4}}{\ell}}{\ell}}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 47.2%

         \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot {k}^{4}}{\ell}\right)\right)}}{\ell}} \]

      2. associate-/l* 46.4%

         \[\leadsto \frac{2}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{t}{\frac{\ell}{{k}^{4}}}}\right)\right)}{\ell}} \]

      3. associate-/r/ 47.3%

         \[\leadsto \frac{2}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{t}{\ell} \cdot {k}^{4}}\right)\right)}{\ell}} \]

   10. Applied egg-rr 47.3%

       \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\ell} \cdot {k}^{4}\right)\right)}}{\ell}} \]
   11. Step-by-step derivation

       1. expm1-log1p 64.0%

          \[\leadsto \frac{2}{\frac{\color{blue}{\frac{t}{\ell} \cdot {k}^{4}}}{\ell}} \]

       2. add-sqr-sqrt 64.0%

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell} \cdot \color{blue}{\left(\sqrt{{k}^{4}} \cdot \sqrt{{k}^{4}}\right)}}{\ell}} \]

       3. associate-*r* 64.0%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(\frac{t}{\ell} \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}}{\ell}} \]

       4. add-sqr-sqrt 33.5%

          \[\leadsto \frac{2}{\frac{\left(\frac{\color{blue}{\sqrt{t} \cdot \sqrt{t}}}{\ell} \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       5. *-un-lft-identity 33.5%

          \[\leadsto \frac{2}{\frac{\left(\frac{\sqrt{t} \cdot \sqrt{t}}{\color{blue}{1 \cdot \ell}} \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       6. times-frac 33.5%

          \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(\frac{\sqrt{t}}{1} \cdot \frac{\sqrt{t}}{\ell}\right)} \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       7. /-rgt-identity 33.5%

          \[\leadsto \frac{2}{\frac{\left(\left(\color{blue}{\sqrt{t}} \cdot \frac{\sqrt{t}}{\ell}\right) \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       8. sqrt-pow1 33.5%

          \[\leadsto \frac{2}{\frac{\left(\left(\sqrt{t} \cdot \frac{\sqrt{t}}{\ell}\right) \cdot \color{blue}{{k}^{\left(\frac{4}{2}\right)}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       9. metadata-eval 33.5%

          \[\leadsto \frac{2}{\frac{\left(\left(\sqrt{t} \cdot \frac{\sqrt{t}}{\ell}\right) \cdot {k}^{\color{blue}{2}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       10. associate-*r* 34.4%

           \[\leadsto \frac{2}{\frac{\color{blue}{\left(\sqrt{t} \cdot \left(\frac{\sqrt{t}}{\ell} \cdot {k}^{2}\right)\right)} \cdot \sqrt{{k}^{4}}}{\ell}} \]

       11. associate-/r/ 34.5%

           \[\leadsto \frac{2}{\frac{\left(\sqrt{t} \cdot \color{blue}{\frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       12. sqrt-pow1 35.4%

           \[\leadsto \frac{2}{\frac{\left(\sqrt{t} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}\right) \cdot \color{blue}{{k}^{\left(\frac{4}{2}\right)}}}{\ell}} \]

       13. metadata-eval 35.4%

           \[\leadsto \frac{2}{\frac{\left(\sqrt{t} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}\right) \cdot {k}^{\color{blue}{2}}}{\ell}} \]

       14. *-commutative 35.4%

           \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{2} \cdot \left(\sqrt{t} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}\right)}}{\ell}} \]

       15. associate-*l* 35.4%

           \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot \sqrt{t}\right) \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}}}{\ell}} \]

       16. associate-/l* 34.3%

           \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \sqrt{t}}{\frac{\ell}{\frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}}}}} \]

   12. Applied egg-rr 66.8%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot {k}^{2}\right) \cdot \frac{{k}^{2}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.1 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{{k}^{2}}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2}}{\ell} \cdot \left({k}^{2} \cdot \frac{t}{\ell}\right)}\\ \end{array} \]

Alternative 15: 68.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \frac{2}{\frac{{k_m}^{2} \cdot \frac{t_m \cdot {k_m}^{2}}{\ell}}{\ell}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ 2.0 (/ (* (pow k_m 2.0) (/ (* t_m (pow k_m 2.0)) l)) l))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / ((pow(k_m, 2.0) * ((t_m * pow(k_m, 2.0)) / l)) / l));
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 / (((k_m ** 2.0d0) * ((t_m * (k_m ** 2.0d0)) / l)) / l))
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 / ((Math.pow(k_m, 2.0) * ((t_m * Math.pow(k_m, 2.0)) / l)) / l));
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 / ((math.pow(k_m, 2.0) * ((t_m * math.pow(k_m, 2.0)) / l)) / l))

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(Float64(t_m * (k_m ^ 2.0)) / l)) / l)))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 / (((k_m ^ 2.0) * ((t_m * (k_m ^ 2.0)) / l)) / l));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 38.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   2. sqr-neg 38.2%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

   3. associate-*l* 38.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   4. associate-*l* 38.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   5. associate-*l/ 39.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

   6. sqr-neg 39.0%

      \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

   7. associate-*l/ 38.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

   8. associate--l+ 38.2%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)}\right)} \]

3. Simplified 38.2%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)}} \]

4. Taylor expanded in k around 0 64.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
5. Step-by-step derivation

   1. associate-/l* 63.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}}} \]

   2. associate-/r/ 63.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]

6. Simplified 63.4%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]
7. Step-by-step derivation

   1. associate-*l/ 64.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

   2. unpow2 64.0%

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   3. associate-/r* 69.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{4} \cdot t}{\ell}}{\ell}}} \]

   4. *-commutative 69.5%

      \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{4}}}{\ell}}{\ell}} \]

8. Applied egg-rr 69.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t \cdot {k}^{4}}{\ell}}{\ell}}} \]
9. Step-by-step derivation

   1. expm1-log1p-u 53.7%

      \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot {k}^{4}}{\ell}\right)\right)}}{\ell}} \]

   2. associate-/l* 53.0%

      \[\leadsto \frac{2}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{t}{\frac{\ell}{{k}^{4}}}}\right)\right)}{\ell}} \]

   3. associate-/r/ 52.5%

      \[\leadsto \frac{2}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{t}{\ell} \cdot {k}^{4}}\right)\right)}{\ell}} \]

10. Applied egg-rr 52.5%

    \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\ell} \cdot {k}^{4}\right)\right)}}{\ell}} \]
11. Step-by-step derivation

    1. expm1-log1p 68.3%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{t}{\ell} \cdot {k}^{4}}}{\ell}} \]

    2. add-sqr-sqrt 68.3%

       \[\leadsto \frac{2}{\frac{\frac{t}{\ell} \cdot \color{blue}{\left(\sqrt{{k}^{4}} \cdot \sqrt{{k}^{4}}\right)}}{\ell}} \]

    3. associate-*r* 68.3%

       \[\leadsto \frac{2}{\frac{\color{blue}{\left(\frac{t}{\ell} \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}}{\ell}} \]

    4. add-sqr-sqrt 32.2%

       \[\leadsto \frac{2}{\frac{\left(\frac{\color{blue}{\sqrt{t} \cdot \sqrt{t}}}{\ell} \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

    5. *-un-lft-identity 32.2%

       \[\leadsto \frac{2}{\frac{\left(\frac{\sqrt{t} \cdot \sqrt{t}}{\color{blue}{1 \cdot \ell}} \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

    6. times-frac 32.2%

       \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(\frac{\sqrt{t}}{1} \cdot \frac{\sqrt{t}}{\ell}\right)} \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

    7. /-rgt-identity 32.2%

       \[\leadsto \frac{2}{\frac{\left(\left(\color{blue}{\sqrt{t}} \cdot \frac{\sqrt{t}}{\ell}\right) \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

    8. sqrt-pow1 32.2%

       \[\leadsto \frac{2}{\frac{\left(\left(\sqrt{t} \cdot \frac{\sqrt{t}}{\ell}\right) \cdot \color{blue}{{k}^{\left(\frac{4}{2}\right)}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

    9. metadata-eval 32.2%

       \[\leadsto \frac{2}{\frac{\left(\left(\sqrt{t} \cdot \frac{\sqrt{t}}{\ell}\right) \cdot {k}^{\color{blue}{2}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

    10. associate-*r* 32.6%

        \[\leadsto \frac{2}{\frac{\color{blue}{\left(\sqrt{t} \cdot \left(\frac{\sqrt{t}}{\ell} \cdot {k}^{2}\right)\right)} \cdot \sqrt{{k}^{4}}}{\ell}} \]

    11. associate-/r/ 33.0%

        \[\leadsto \frac{2}{\frac{\left(\sqrt{t} \cdot \color{blue}{\frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

    12. sqrt-pow1 34.1%

        \[\leadsto \frac{2}{\frac{\left(\sqrt{t} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}\right) \cdot \color{blue}{{k}^{\left(\frac{4}{2}\right)}}}{\ell}} \]

    13. metadata-eval 34.1%

        \[\leadsto \frac{2}{\frac{\left(\sqrt{t} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}\right) \cdot {k}^{\color{blue}{2}}}{\ell}} \]

    14. *-commutative 34.1%

        \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{2} \cdot \left(\sqrt{t} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}\right)}}{\ell}} \]

    15. associate-*l* 34.1%

        \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot \sqrt{t}\right) \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}}}{\ell}} \]

    16. associate-*r/ 34.1%

        \[\leadsto \frac{2}{\frac{\color{blue}{\frac{\left({k}^{2} \cdot \sqrt{t}\right) \cdot \sqrt{t}}{\frac{\ell}{{k}^{2}}}}}{\ell}} \]

    17. associate-/r/ 34.1%

        \[\leadsto \frac{2}{\frac{\color{blue}{\frac{\left({k}^{2} \cdot \sqrt{t}\right) \cdot \sqrt{t}}{\ell} \cdot {k}^{2}}}{\ell}} \]

12. Applied egg-rr 72.2%

    \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot {k}^{2}}}{\ell}} \]

13. Final simplification 72.2%

    \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \frac{t \cdot {k}^{2}}{\ell}}{\ell}} \]

Alternative 16: 68.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 5 \cdot 10^{-150}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \ell}{t_m \cdot {k_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k_m \cdot \left(k_m \cdot \left({k_m}^{2} \cdot \frac{t_m}{\ell}\right)\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 5e-150)
    (* l (/ (* 2.0 l) (* t_m (pow k_m 4.0))))
    (/ 2.0 (/ (* k_m (* k_m (* (pow k_m 2.0) (/ t_m l)))) l)))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5e-150) {
		tmp = l * ((2.0 * l) / (t_m * pow(k_m, 4.0)));
	} else {
		tmp = 2.0 / ((k_m * (k_m * (pow(k_m, 2.0) * (t_m / l)))) / l);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5d-150) then
        tmp = l * ((2.0d0 * l) / (t_m * (k_m ** 4.0d0)))
    else
        tmp = 2.0d0 / ((k_m * (k_m * ((k_m ** 2.0d0) * (t_m / l)))) / l)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5e-150) {
		tmp = l * ((2.0 * l) / (t_m * Math.pow(k_m, 4.0)));
	} else {
		tmp = 2.0 / ((k_m * (k_m * (Math.pow(k_m, 2.0) * (t_m / l)))) / l);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 5e-150:
		tmp = l * ((2.0 * l) / (t_m * math.pow(k_m, 4.0)))
	else:
		tmp = 2.0 / ((k_m * (k_m * (math.pow(k_m, 2.0) * (t_m / l)))) / l)
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 5e-150)
		tmp = Float64(l * Float64(Float64(2.0 * l) / Float64(t_m * (k_m ^ 4.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * Float64(k_m * Float64((k_m ^ 2.0) * Float64(t_m / l)))) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 5e-150)
		tmp = l * ((2.0 * l) / (t_m * (k_m ^ 4.0)));
	else
		tmp = 2.0 / ((k_m * (k_m * ((k_m ^ 2.0) * (t_m / l)))) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 5e-150], N[(l * N[(N[(2.0 * l), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * N[(k$95$m * N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.9999999999999999e-150

   1. Initial program 36.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 36.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. sqr-neg 36.3%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      3. associate-*l* 36.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-*l* 36.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      5. associate-*l/ 37.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      6. sqr-neg 37.6%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      7. associate-*l/ 36.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      8. associate--l+ 36.3%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)}\right)} \]

   3. Simplified 36.3%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 66.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 62.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}}} \]

      2. associate-/r/ 64.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]

   6. Simplified 64.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]
   7. Step-by-step derivation

      1. associate-*l/ 66.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

      2. unpow2 66.1%

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      3. associate-/r* 73.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{4} \cdot t}{\ell}}{\ell}}} \]

      4. *-commutative 73.3%

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{4}}}{\ell}}{\ell}} \]

   8. Applied egg-rr 73.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t \cdot {k}^{4}}{\ell}}{\ell}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 73.3%

         \[\leadsto \color{blue}{\frac{2}{\frac{t \cdot {k}^{4}}{\ell}} \cdot \ell} \]

      2. clear-num 73.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{1}{\frac{\ell}{t \cdot {k}^{4}}}}} \cdot \ell \]

      3. associate-/r/ 73.3%

         \[\leadsto \color{blue}{\left(\frac{2}{1} \cdot \frac{\ell}{t \cdot {k}^{4}}\right)} \cdot \ell \]

      4. metadata-eval 73.3%

         \[\leadsto \left(\color{blue}{2} \cdot \frac{\ell}{t \cdot {k}^{4}}\right) \cdot \ell \]

      5. associate-/r* 71.2%

         \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\ell}{t}}{{k}^{4}}}\right) \cdot \ell \]

   10. Applied egg-rr 71.2%

       \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right) \cdot \ell} \]
   11. Step-by-step derivation

       1. *-commutative 71.2%

          \[\leadsto \color{blue}{\left(\frac{\frac{\ell}{t}}{{k}^{4}} \cdot 2\right)} \cdot \ell \]

       2. associate-/l/ 73.3%

          \[\leadsto \left(\color{blue}{\frac{\ell}{{k}^{4} \cdot t}} \cdot 2\right) \cdot \ell \]

       3. associate-*l/ 73.3%

          \[\leadsto \color{blue}{\frac{\ell \cdot 2}{{k}^{4} \cdot t}} \cdot \ell \]

       4. *-commutative 73.3%

          \[\leadsto \frac{\ell \cdot 2}{\color{blue}{t \cdot {k}^{4}}} \cdot \ell \]

   12. Applied egg-rr 73.3%

       \[\leadsto \color{blue}{\frac{\ell \cdot 2}{t \cdot {k}^{4}}} \cdot \ell \]

   if 4.9999999999999999e-150 < k

   1. Initial program 41.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 41.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. sqr-neg 41.0%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      3. associate-*l* 41.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      4. associate-*l* 41.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      5. associate-*l/ 41.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      6. sqr-neg 41.0%

         \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      7. associate-*l/ 41.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

      8. associate--l+ 41.0%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)}\right)} \]

   3. Simplified 41.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)}} \]

   4. Taylor expanded in k around 0 61.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   5. Step-by-step derivation

      1. associate-/l* 63.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}}} \]

      2. associate-/r/ 62.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]

   6. Simplified 62.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]
   7. Step-by-step derivation

      1. associate-*l/ 61.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

      2. unpow2 61.0%

         \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

      3. associate-/r* 64.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{4} \cdot t}{\ell}}{\ell}}} \]

      4. *-commutative 64.0%

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{4}}}{\ell}}{\ell}} \]

   8. Applied egg-rr 64.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t \cdot {k}^{4}}{\ell}}{\ell}}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 47.8%

         \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot {k}^{4}}{\ell}\right)\right)}}{\ell}} \]

      2. associate-/l* 47.0%

         \[\leadsto \frac{2}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{t}{\frac{\ell}{{k}^{4}}}}\right)\right)}{\ell}} \]

      3. associate-/r/ 47.9%

         \[\leadsto \frac{2}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{t}{\ell} \cdot {k}^{4}}\right)\right)}{\ell}} \]

   10. Applied egg-rr 47.9%

       \[\leadsto \frac{2}{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t}{\ell} \cdot {k}^{4}\right)\right)}}{\ell}} \]
   11. Step-by-step derivation

       1. expm1-log1p 64.1%

          \[\leadsto \frac{2}{\frac{\color{blue}{\frac{t}{\ell} \cdot {k}^{4}}}{\ell}} \]

       2. add-sqr-sqrt 64.1%

          \[\leadsto \frac{2}{\frac{\frac{t}{\ell} \cdot \color{blue}{\left(\sqrt{{k}^{4}} \cdot \sqrt{{k}^{4}}\right)}}{\ell}} \]

       3. associate-*r* 64.1%

          \[\leadsto \frac{2}{\frac{\color{blue}{\left(\frac{t}{\ell} \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}}{\ell}} \]

       4. add-sqr-sqrt 33.6%

          \[\leadsto \frac{2}{\frac{\left(\frac{\color{blue}{\sqrt{t} \cdot \sqrt{t}}}{\ell} \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       5. *-un-lft-identity 33.6%

          \[\leadsto \frac{2}{\frac{\left(\frac{\sqrt{t} \cdot \sqrt{t}}{\color{blue}{1 \cdot \ell}} \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       6. times-frac 33.5%

          \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(\frac{\sqrt{t}}{1} \cdot \frac{\sqrt{t}}{\ell}\right)} \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       7. /-rgt-identity 33.5%

          \[\leadsto \frac{2}{\frac{\left(\left(\color{blue}{\sqrt{t}} \cdot \frac{\sqrt{t}}{\ell}\right) \cdot \sqrt{{k}^{4}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       8. sqrt-pow1 33.6%

          \[\leadsto \frac{2}{\frac{\left(\left(\sqrt{t} \cdot \frac{\sqrt{t}}{\ell}\right) \cdot \color{blue}{{k}^{\left(\frac{4}{2}\right)}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       9. metadata-eval 33.6%

          \[\leadsto \frac{2}{\frac{\left(\left(\sqrt{t} \cdot \frac{\sqrt{t}}{\ell}\right) \cdot {k}^{\color{blue}{2}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       10. associate-*r* 34.4%

           \[\leadsto \frac{2}{\frac{\color{blue}{\left(\sqrt{t} \cdot \left(\frac{\sqrt{t}}{\ell} \cdot {k}^{2}\right)\right)} \cdot \sqrt{{k}^{4}}}{\ell}} \]

       11. associate-/r/ 34.5%

           \[\leadsto \frac{2}{\frac{\left(\sqrt{t} \cdot \color{blue}{\frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}}\right) \cdot \sqrt{{k}^{4}}}{\ell}} \]

       12. sqrt-pow1 36.3%

           \[\leadsto \frac{2}{\frac{\left(\sqrt{t} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}\right) \cdot \color{blue}{{k}^{\left(\frac{4}{2}\right)}}}{\ell}} \]

       13. metadata-eval 36.3%

           \[\leadsto \frac{2}{\frac{\left(\sqrt{t} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}\right) \cdot {k}^{\color{blue}{2}}}{\ell}} \]

       14. *-commutative 36.3%

           \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{2} \cdot \left(\sqrt{t} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}\right)}}{\ell}} \]

       15. unpow2 36.3%

           \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \left(\sqrt{t} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}\right)}{\ell}} \]

       16. associate-*l* 36.3%

           \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot \left(k \cdot \left(\sqrt{t} \cdot \frac{\sqrt{t}}{\frac{\ell}{{k}^{2}}}\right)\right)}}{\ell}} \]

       17. associate-/r/ 36.2%

           \[\leadsto \frac{2}{\frac{k \cdot \left(k \cdot \left(\sqrt{t} \cdot \color{blue}{\left(\frac{\sqrt{t}}{\ell} \cdot {k}^{2}\right)}\right)\right)}{\ell}} \]

   12. Applied egg-rr 66.9%

       \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot \left(k \cdot \left(\frac{t}{\ell} \cdot {k}^{2}\right)\right)}}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-150}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot \ell}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot \left({k}^{2} \cdot \frac{t}{\ell}\right)\right)}{\ell}}\\ \end{array} \]

Alternative 17: 65.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(\ell \cdot \left(2 \cdot \left(\ell \cdot \frac{\frac{1}{t_m}}{{k_m}^{4}}\right)\right)\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* l (* 2.0 (* l (/ (/ 1.0 t_m) (pow k_m 4.0)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (l * (2.0 * (l * ((1.0 / t_m) / pow(k_m, 4.0)))));
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (l * (2.0d0 * (l * ((1.0d0 / t_m) / (k_m ** 4.0d0)))))
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (l * (2.0 * (l * ((1.0 / t_m) / Math.pow(k_m, 4.0)))));
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (l * (2.0 * (l * ((1.0 / t_m) / math.pow(k_m, 4.0)))))

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(l * Float64(2.0 * Float64(l * Float64(Float64(1.0 / t_m) / (k_m ^ 4.0))))))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (l * (2.0 * (l * ((1.0 / t_m) / (k_m ^ 4.0)))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(l * N[(2.0 * N[(l * N[(N[(1.0 / t$95$m), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 38.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   2. sqr-neg 38.2%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

   3. associate-*l* 38.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   4. associate-*l* 38.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   5. associate-*l/ 39.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

   6. sqr-neg 39.0%

      \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

   7. associate-*l/ 38.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

   8. associate--l+ 38.2%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)}\right)} \]

3. Simplified 38.2%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)}} \]

4. Taylor expanded in k around 0 64.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
5. Step-by-step derivation

   1. associate-/l* 63.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}}} \]

   2. associate-/r/ 63.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]

6. Simplified 63.4%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]
7. Step-by-step derivation

   1. associate-*l/ 64.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

   2. unpow2 64.0%

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   3. associate-/r* 69.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{4} \cdot t}{\ell}}{\ell}}} \]

   4. *-commutative 69.5%

      \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{4}}}{\ell}}{\ell}} \]

8. Applied egg-rr 69.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t \cdot {k}^{4}}{\ell}}{\ell}}} \]
9. Step-by-step derivation

   1. associate-/r/ 69.8%

      \[\leadsto \color{blue}{\frac{2}{\frac{t \cdot {k}^{4}}{\ell}} \cdot \ell} \]

   2. clear-num 69.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{1}{\frac{\ell}{t \cdot {k}^{4}}}}} \cdot \ell \]

   3. associate-/r/ 69.8%

      \[\leadsto \color{blue}{\left(\frac{2}{1} \cdot \frac{\ell}{t \cdot {k}^{4}}\right)} \cdot \ell \]

   4. metadata-eval 69.8%

      \[\leadsto \left(\color{blue}{2} \cdot \frac{\ell}{t \cdot {k}^{4}}\right) \cdot \ell \]

   5. associate-/r* 68.6%

      \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\ell}{t}}{{k}^{4}}}\right) \cdot \ell \]

10. Applied egg-rr 68.6%

    \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right) \cdot \ell} \]
11. Step-by-step derivation

    1. div-inv 68.6%

       \[\leadsto \left(2 \cdot \frac{\color{blue}{\ell \cdot \frac{1}{t}}}{{k}^{4}}\right) \cdot \ell \]

    2. *-un-lft-identity 68.6%

       \[\leadsto \left(2 \cdot \frac{\ell \cdot \frac{1}{t}}{\color{blue}{1 \cdot {k}^{4}}}\right) \cdot \ell \]

    3. times-frac 69.8%

       \[\leadsto \left(2 \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\frac{1}{t}}{{k}^{4}}\right)}\right) \cdot \ell \]

12. Applied egg-rr 69.8%

    \[\leadsto \left(2 \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\frac{1}{t}}{{k}^{4}}\right)}\right) \cdot \ell \]

13. Final simplification 69.8%

    \[\leadsto \ell \cdot \left(2 \cdot \left(\ell \cdot \frac{\frac{1}{t}}{{k}^{4}}\right)\right) \]

Alternative 18: 64.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(\ell \cdot \left(2 \cdot \frac{\frac{\ell}{t_m}}{{k_m}^{4}}\right)\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* l (* 2.0 (/ (/ l t_m) (pow k_m 4.0))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (l * (2.0 * ((l / t_m) / pow(k_m, 4.0))));
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (l * (2.0d0 * ((l / t_m) / (k_m ** 4.0d0))))
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (l * (2.0 * ((l / t_m) / Math.pow(k_m, 4.0))));
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (l * (2.0 * ((l / t_m) / math.pow(k_m, 4.0))))

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(l * Float64(2.0 * Float64(Float64(l / t_m) / (k_m ^ 4.0)))))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (l * (2.0 * ((l / t_m) / (k_m ^ 4.0))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(l * N[(2.0 * N[(N[(l / t$95$m), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 38.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   2. sqr-neg 38.2%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

   3. associate-*l* 38.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   4. associate-*l* 38.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   5. associate-*l/ 39.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

   6. sqr-neg 39.0%

      \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

   7. associate-*l/ 38.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

   8. associate--l+ 38.2%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)}\right)} \]

3. Simplified 38.2%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)}} \]

4. Taylor expanded in k around 0 64.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
5. Step-by-step derivation

   1. associate-/l* 63.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}}} \]

   2. associate-/r/ 63.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]

6. Simplified 63.4%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]
7. Step-by-step derivation

   1. associate-*l/ 64.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

   2. unpow2 64.0%

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   3. associate-/r* 69.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{4} \cdot t}{\ell}}{\ell}}} \]

   4. *-commutative 69.5%

      \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{4}}}{\ell}}{\ell}} \]

8. Applied egg-rr 69.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t \cdot {k}^{4}}{\ell}}{\ell}}} \]
9. Step-by-step derivation

   1. associate-/r/ 69.8%

      \[\leadsto \color{blue}{\frac{2}{\frac{t \cdot {k}^{4}}{\ell}} \cdot \ell} \]

   2. clear-num 69.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{1}{\frac{\ell}{t \cdot {k}^{4}}}}} \cdot \ell \]

   3. associate-/r/ 69.8%

      \[\leadsto \color{blue}{\left(\frac{2}{1} \cdot \frac{\ell}{t \cdot {k}^{4}}\right)} \cdot \ell \]

   4. metadata-eval 69.8%

      \[\leadsto \left(\color{blue}{2} \cdot \frac{\ell}{t \cdot {k}^{4}}\right) \cdot \ell \]

   5. associate-/r* 68.6%

      \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\ell}{t}}{{k}^{4}}}\right) \cdot \ell \]

10. Applied egg-rr 68.6%

    \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right) \cdot \ell} \]

11. Final simplification 68.6%

    \[\leadsto \ell \cdot \left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right) \]

Alternative 19: 65.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(\ell \cdot \frac{2 \cdot \ell}{t_m \cdot {k_m}^{4}}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* l (/ (* 2.0 l) (* t_m (pow k_m 4.0))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (l * ((2.0 * l) / (t_m * pow(k_m, 4.0))));
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (l * ((2.0d0 * l) / (t_m * (k_m ** 4.0d0))))
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (l * ((2.0 * l) / (t_m * Math.pow(k_m, 4.0))));
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (l * ((2.0 * l) / (t_m * math.pow(k_m, 4.0))))

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(l * Float64(Float64(2.0 * l) / Float64(t_m * (k_m ^ 4.0)))))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (l * ((2.0 * l) / (t_m * (k_m ^ 4.0))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(l * N[(N[(2.0 * l), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 38.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   2. sqr-neg 38.2%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

   3. associate-*l* 38.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   4. associate-*l* 38.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   5. associate-*l/ 39.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

   6. sqr-neg 39.0%

      \[\leadsto \frac{2}{\frac{{t}^{3} \cdot \sin k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

   7. associate-*l/ 38.2%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]

   8. associate--l+ 38.2%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)}\right)} \]

3. Simplified 38.2%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)\right)\right)}} \]

4. Taylor expanded in k around 0 64.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
5. Step-by-step derivation

   1. associate-/l* 63.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}}} \]

   2. associate-/r/ 63.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]

6. Simplified 63.4%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4}}{{\ell}^{2}} \cdot t}} \]
7. Step-by-step derivation

   1. associate-*l/ 64.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

   2. unpow2 64.0%

      \[\leadsto \frac{2}{\frac{{k}^{4} \cdot t}{\color{blue}{\ell \cdot \ell}}} \]

   3. associate-/r* 69.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{k}^{4} \cdot t}{\ell}}{\ell}}} \]

   4. *-commutative 69.5%

      \[\leadsto \frac{2}{\frac{\frac{\color{blue}{t \cdot {k}^{4}}}{\ell}}{\ell}} \]

8. Applied egg-rr 69.5%

   \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t \cdot {k}^{4}}{\ell}}{\ell}}} \]
9. Step-by-step derivation

   1. associate-/r/ 69.8%

      \[\leadsto \color{blue}{\frac{2}{\frac{t \cdot {k}^{4}}{\ell}} \cdot \ell} \]

   2. clear-num 69.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{1}{\frac{\ell}{t \cdot {k}^{4}}}}} \cdot \ell \]

   3. associate-/r/ 69.8%

      \[\leadsto \color{blue}{\left(\frac{2}{1} \cdot \frac{\ell}{t \cdot {k}^{4}}\right)} \cdot \ell \]

   4. metadata-eval 69.8%

      \[\leadsto \left(\color{blue}{2} \cdot \frac{\ell}{t \cdot {k}^{4}}\right) \cdot \ell \]

   5. associate-/r* 68.6%

      \[\leadsto \left(2 \cdot \color{blue}{\frac{\frac{\ell}{t}}{{k}^{4}}}\right) \cdot \ell \]

10. Applied egg-rr 68.6%

    \[\leadsto \color{blue}{\left(2 \cdot \frac{\frac{\ell}{t}}{{k}^{4}}\right) \cdot \ell} \]
11. Step-by-step derivation

    1. *-commutative 68.6%

       \[\leadsto \color{blue}{\left(\frac{\frac{\ell}{t}}{{k}^{4}} \cdot 2\right)} \cdot \ell \]

    2. associate-/l/ 69.8%

       \[\leadsto \left(\color{blue}{\frac{\ell}{{k}^{4} \cdot t}} \cdot 2\right) \cdot \ell \]

    3. associate-*l/ 69.8%

       \[\leadsto \color{blue}{\frac{\ell \cdot 2}{{k}^{4} \cdot t}} \cdot \ell \]

    4. *-commutative 69.8%

       \[\leadsto \frac{\ell \cdot 2}{\color{blue}{t \cdot {k}^{4}}} \cdot \ell \]

12. Applied egg-rr 69.8%

    \[\leadsto \color{blue}{\frac{\ell \cdot 2}{t \cdot {k}^{4}}} \cdot \ell \]

13. Final simplification 69.8%

    \[\leadsto \ell \cdot \frac{2 \cdot \ell}{t \cdot {k}^{4}} \]

Reproduce

herbie shell --seed 2023336 
(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))))