Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.8% → 98.5%

Time: 27.5s

Alternatives: 15

Speedup: 2.0×

• 39.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.85 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k\_m} \cdot \frac{\frac{\sqrt{\cos k\_m}}{\sin k\_m}}{\sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\frac{\ell}{\sqrt{t\_m}}}{k\_m}\right)}^{2} \cdot \frac{\cos k\_m}{{\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \]

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 1.85e-24)
    (*
     2.0
     (pow (* (/ l k_m) (/ (/ (sqrt (cos k_m)) (sin k_m)) (sqrt t_m))) 2.0))
    (*
     2.0
     (*
      (pow (/ (/ l (sqrt t_m)) k_m) 2.0)
      (/ (cos k_m) (pow (sin k_m) 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 (k_m <= 1.85e-24) {
		tmp = 2.0 * pow(((l / k_m) * ((sqrt(cos(k_m)) / sin(k_m)) / sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * (pow(((l / sqrt(t_m)) / k_m), 2.0) * (cos(k_m) / pow(sin(k_m), 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 (k_m <= 1.85d-24) then
        tmp = 2.0d0 * (((l / k_m) * ((sqrt(cos(k_m)) / sin(k_m)) / sqrt(t_m))) ** 2.0d0)
    else
        tmp = 2.0d0 * ((((l / sqrt(t_m)) / k_m) ** 2.0d0) * (cos(k_m) / (sin(k_m) ** 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 (k_m <= 1.85e-24) {
		tmp = 2.0 * Math.pow(((l / k_m) * ((Math.sqrt(Math.cos(k_m)) / Math.sin(k_m)) / Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * (Math.pow(((l / Math.sqrt(t_m)) / k_m), 2.0) * (Math.cos(k_m) / Math.pow(Math.sin(k_m), 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 k_m <= 1.85e-24:
		tmp = 2.0 * math.pow(((l / k_m) * ((math.sqrt(math.cos(k_m)) / math.sin(k_m)) / math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 * (math.pow(((l / math.sqrt(t_m)) / k_m), 2.0) * (math.cos(k_m) / math.pow(math.sin(k_m), 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 (k_m <= 1.85e-24)
		tmp = Float64(2.0 * (Float64(Float64(l / k_m) * Float64(Float64(sqrt(cos(k_m)) / sin(k_m)) / sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64((Float64(Float64(l / sqrt(t_m)) / k_m) ^ 2.0) * Float64(cos(k_m) / (sin(k_m) ^ 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 (k_m <= 1.85e-24)
		tmp = 2.0 * (((l / k_m) * ((sqrt(cos(k_m)) / sin(k_m)) / sqrt(t_m))) ^ 2.0);
	else
		tmp = 2.0 * ((((l / sqrt(t_m)) / k_m) ^ 2.0) * (cos(k_m) / (sin(k_m) ^ 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[k$95$m, 1.85e-24], N[(2.0 * N[Power[N[(N[(l / k$95$m), $MachinePrecision] * N[(N[(N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(N[(l / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.8499999999999999e-24

   1. Initial program 30.7%

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

      1. associate-*l* 31.1%

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

      2. associate-/r* 30.7%

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

      3. sub-neg 30.7%

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

      4. distribute-rgt-in 23.4%

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

      5. unpow2 23.4%

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

      6. times-frac 17.7%

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

      7. sqr-neg 17.7%

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

      8. times-frac 23.4%

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

      9. unpow2 23.4%

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

      10. distribute-rgt-in 30.7%

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

      11. +-commutative 30.7%

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

      12. associate-+l+ 36.9%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in t around 0 72.5%

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

      1. associate-*r* 72.5%

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

      2. times-frac 72.6%

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

      3. *-commutative 72.6%

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

      4. associate-/r* 68.7%

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

   7. Simplified 68.7%

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

      1. add-sqr-sqrt 40.8%

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

      2. pow2 40.8%

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

      3. sqrt-div 37.6%

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

      4. sqrt-div 33.6%

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

      5. pow2 33.6%

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

      6. sqrt-prod 22.4%

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

      7. add-sqr-sqrt 39.4%

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

      8. unpow2 39.4%

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

      9. sqrt-prod 13.9%

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

      10. add-sqr-sqrt 41.7%

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

   9. Applied egg-rr 41.7%

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

       1. expm1-log1p-u 39.8%

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

       2. expm1-udef 37.1%

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

   11. Applied egg-rr 35.2%

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

       1. expm1-def 36.2%

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

       2. expm1-log1p 36.6%

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

       3. associate-*l/ 36.7%

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

       4. times-frac 37.0%

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

   13. Simplified 37.0%

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

   if 1.8499999999999999e-24 < k

   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* 33.2%

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

      2. associate-/r* 33.2%

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

      3. sub-neg 33.2%

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

      4. distribute-rgt-in 33.2%

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

      5. unpow2 33.2%

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

      6. times-frac 31.6%

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

      7. sqr-neg 31.6%

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

      8. times-frac 33.2%

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

      9. unpow2 33.2%

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

      10. distribute-rgt-in 33.2%

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

      11. +-commutative 33.2%

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

      12. associate-+l+ 44.7%

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

   3. Simplified 44.7%

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

   5. Taylor expanded in t around 0 80.6%

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

      1. associate-*r* 80.5%

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

      2. times-frac 80.5%

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

      3. *-commutative 80.5%

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

      4. associate-/r* 83.7%

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

   7. Simplified 83.7%

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

      1. add-sqr-sqrt 63.9%

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

      2. pow2 63.9%

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

      3. sqrt-div 55.8%

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

      4. sqrt-div 41.4%

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

      5. pow2 41.4%

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

      6. sqrt-prod 21.0%

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

      7. add-sqr-sqrt 43.1%

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

      8. unpow2 43.1%

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

      9. sqrt-prod 45.0%

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

      10. add-sqr-sqrt 45.0%

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

   9. Applied egg-rr 45.0%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k} \cdot \frac{\frac{\sqrt{\cos k}}{\sin k}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\frac{\ell}{\sqrt{t}}}{k}\right)}^{2} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-29}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k\_m} \cdot \frac{\frac{\sqrt{\cos k\_m}}{\sin k\_m}}{\sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\frac{\ell}{\sqrt{t\_m}}}{k\_m}\right)}^{2} \cdot \left(\cos k\_m \cdot {\sin k\_m}^{-2}\right)\right)\\ \end{array} \end{array} \]

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 2.4e-29)
    (*
     2.0
     (pow (* (/ l k_m) (/ (/ (sqrt (cos k_m)) (sin k_m)) (sqrt t_m))) 2.0))
    (*
     2.0
     (*
      (pow (/ (/ l (sqrt t_m)) k_m) 2.0)
      (* (cos k_m) (pow (sin k_m) -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 (k_m <= 2.4e-29) {
		tmp = 2.0 * pow(((l / k_m) * ((sqrt(cos(k_m)) / sin(k_m)) / sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * (pow(((l / sqrt(t_m)) / k_m), 2.0) * (cos(k_m) * pow(sin(k_m), -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 (k_m <= 2.4d-29) then
        tmp = 2.0d0 * (((l / k_m) * ((sqrt(cos(k_m)) / sin(k_m)) / sqrt(t_m))) ** 2.0d0)
    else
        tmp = 2.0d0 * ((((l / sqrt(t_m)) / k_m) ** 2.0d0) * (cos(k_m) * (sin(k_m) ** (-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 (k_m <= 2.4e-29) {
		tmp = 2.0 * Math.pow(((l / k_m) * ((Math.sqrt(Math.cos(k_m)) / Math.sin(k_m)) / Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * (Math.pow(((l / Math.sqrt(t_m)) / k_m), 2.0) * (Math.cos(k_m) * Math.pow(Math.sin(k_m), -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 k_m <= 2.4e-29:
		tmp = 2.0 * math.pow(((l / k_m) * ((math.sqrt(math.cos(k_m)) / math.sin(k_m)) / math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 * (math.pow(((l / math.sqrt(t_m)) / k_m), 2.0) * (math.cos(k_m) * math.pow(math.sin(k_m), -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 (k_m <= 2.4e-29)
		tmp = Float64(2.0 * (Float64(Float64(l / k_m) * Float64(Float64(sqrt(cos(k_m)) / sin(k_m)) / sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64((Float64(Float64(l / sqrt(t_m)) / k_m) ^ 2.0) * Float64(cos(k_m) * (sin(k_m) ^ -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 (k_m <= 2.4e-29)
		tmp = 2.0 * (((l / k_m) * ((sqrt(cos(k_m)) / sin(k_m)) / sqrt(t_m))) ^ 2.0);
	else
		tmp = 2.0 * ((((l / sqrt(t_m)) / k_m) ^ 2.0) * (cos(k_m) * (sin(k_m) ^ -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[k$95$m, 2.4e-29], N[(2.0 * N[Power[N[(N[(l / k$95$m), $MachinePrecision] * N[(N[(N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(N[(l / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.39999999999999992e-29

   1. Initial program 30.3%

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

      1. associate-*l* 30.8%

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

      2. associate-/r* 30.3%

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

      3. sub-neg 30.3%

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

      4. distribute-rgt-in 23.0%

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

      5. unpow2 23.0%

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

      6. times-frac 17.2%

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

      7. sqr-neg 17.2%

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

      8. times-frac 23.0%

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

      9. unpow2 23.0%

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

      10. distribute-rgt-in 30.3%

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

      11. +-commutative 30.3%

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

      12. associate-+l+ 36.6%

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

   3. Simplified 36.6%

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

   5. Taylor expanded in t around 0 72.4%

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

      1. associate-*r* 72.4%

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

      2. times-frac 72.5%

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

      3. *-commutative 72.5%

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

      4. associate-/r* 68.5%

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

   7. Simplified 68.5%

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

      1. add-sqr-sqrt 40.4%

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

      2. pow2 40.4%

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

      3. sqrt-div 37.3%

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

      4. sqrt-div 33.3%

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

      5. pow2 33.3%

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

      6. sqrt-prod 22.5%

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

      7. add-sqr-sqrt 39.1%

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

      8. unpow2 39.1%

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

      9. sqrt-prod 13.5%

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

      10. add-sqr-sqrt 41.4%

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

   9. Applied egg-rr 41.4%

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

       1. expm1-log1p-u 39.4%

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

       2. expm1-udef 36.8%

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

   11. Applied egg-rr 34.9%

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

       1. expm1-def 35.8%

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

       2. expm1-log1p 36.3%

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

       3. associate-*l/ 36.4%

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

       4. times-frac 36.7%

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

   13. Simplified 36.7%

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

   if 2.39999999999999992e-29 < k

   1. Initial program 34.3%

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

      1. associate-*l* 34.3%

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

      2. associate-/r* 34.3%

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

      3. sub-neg 34.3%

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

      4. distribute-rgt-in 34.3%

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

      5. unpow2 34.3%

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

      6. times-frac 32.7%

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

      7. sqr-neg 32.7%

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

      8. times-frac 34.3%

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

      9. unpow2 34.3%

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

      10. distribute-rgt-in 34.3%

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

      11. +-commutative 34.3%

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

      12. associate-+l+ 45.6%

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

   3. Simplified 45.6%

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

   5. Taylor expanded in t around 0 80.9%

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

      1. associate-*r* 80.8%

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

      2. times-frac 80.8%

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

      3. *-commutative 80.8%

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

      4. associate-/r* 84.0%

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

   7. Simplified 84.0%

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

      1. add-sqr-sqrt 64.5%

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

      2. pow2 64.5%

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

      3. sqrt-div 56.5%

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

      4. sqrt-div 42.3%

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

      5. pow2 42.3%

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

      6. sqrt-prod 20.7%

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

      7. add-sqr-sqrt 44.0%

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

      8. unpow2 44.0%

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

      9. sqrt-prod 45.8%

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

      10. add-sqr-sqrt 45.9%

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

   9. Applied egg-rr 45.9%

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

       1. expm1-log1p-u 30.1%

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

       2. expm1-udef 30.1%

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

       3. div-inv 30.1%

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

       4. pow-flip 30.1%

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

       5. metadata-eval 30.1%

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

   11. Applied egg-rr 30.1%

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

       1. expm1-def 30.1%

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

       2. expm1-log1p 45.9%

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

   13. Simplified 45.9%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-29}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k} \cdot \frac{\frac{\sqrt{\cos k}}{\sin k}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\frac{\ell}{\sqrt{t}}}{k}\right)}^{2} \cdot \left(\cos k \cdot {\sin k}^{-2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 6.4 \cdot 10^{-47}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k\_m} \cdot \frac{\frac{\sqrt{\cos k\_m}}{\sin k\_m}}{\sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \frac{-\cos k\_m}{{\sin k\_m}^{2} \cdot \left(-t\_m\right)}\right)\\ \end{array} \end{array} \]

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 6.4e-47)
    (*
     2.0
     (pow (* (/ l k_m) (/ (/ (sqrt (cos k_m)) (sin k_m)) (sqrt t_m))) 2.0))
    (*
     2.0
     (*
      (pow (/ l k_m) 2.0)
      (/ (- (cos k_m)) (* (pow (sin k_m) 2.0) (- t_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 tmp;
	if (k_m <= 6.4e-47) {
		tmp = 2.0 * pow(((l / k_m) * ((sqrt(cos(k_m)) / sin(k_m)) / sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * (pow((l / k_m), 2.0) * (-cos(k_m) / (pow(sin(k_m), 2.0) * -t_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) :: tmp
    if (k_m <= 6.4d-47) then
        tmp = 2.0d0 * (((l / k_m) * ((sqrt(cos(k_m)) / sin(k_m)) / sqrt(t_m))) ** 2.0d0)
    else
        tmp = 2.0d0 * (((l / k_m) ** 2.0d0) * (-cos(k_m) / ((sin(k_m) ** 2.0d0) * -t_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 tmp;
	if (k_m <= 6.4e-47) {
		tmp = 2.0 * Math.pow(((l / k_m) * ((Math.sqrt(Math.cos(k_m)) / Math.sin(k_m)) / Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * (Math.pow((l / k_m), 2.0) * (-Math.cos(k_m) / (Math.pow(Math.sin(k_m), 2.0) * -t_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):
	tmp = 0
	if k_m <= 6.4e-47:
		tmp = 2.0 * math.pow(((l / k_m) * ((math.sqrt(math.cos(k_m)) / math.sin(k_m)) / math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 * (math.pow((l / k_m), 2.0) * (-math.cos(k_m) / (math.pow(math.sin(k_m), 2.0) * -t_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)
	tmp = 0.0
	if (k_m <= 6.4e-47)
		tmp = Float64(2.0 * (Float64(Float64(l / k_m) * Float64(Float64(sqrt(cos(k_m)) / sin(k_m)) / sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64((Float64(l / k_m) ^ 2.0) * Float64(Float64(-cos(k_m)) / Float64((sin(k_m) ^ 2.0) * Float64(-t_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)
	tmp = 0.0;
	if (k_m <= 6.4e-47)
		tmp = 2.0 * (((l / k_m) * ((sqrt(cos(k_m)) / sin(k_m)) / sqrt(t_m))) ^ 2.0);
	else
		tmp = 2.0 * (((l / k_m) ^ 2.0) * (-cos(k_m) / ((sin(k_m) ^ 2.0) * -t_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_] := N[(t$95$s * If[LessEqual[k$95$m, 6.4e-47], N[(2.0 * N[Power[N[(N[(l / k$95$m), $MachinePrecision] * N[(N[(N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[((-N[Cos[k$95$m], $MachinePrecision]) / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * (-t$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.3999999999999998e-47

   1. Initial program 30.7%

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

      1. associate-*l* 31.1%

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

      2. associate-/r* 30.7%

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

      3. sub-neg 30.7%

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

      4. distribute-rgt-in 23.3%

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

      5. unpow2 23.3%

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

      6. times-frac 17.4%

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

      7. sqr-neg 17.4%

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

      8. times-frac 23.3%

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

      9. unpow2 23.3%

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

      10. distribute-rgt-in 30.7%

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

      11. +-commutative 30.7%

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

      12. associate-+l+ 36.9%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in t around 0 72.6%

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

      1. associate-*r* 72.6%

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

      2. times-frac 73.2%

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

      3. *-commutative 73.2%

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

      4. associate-/r* 69.1%

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

   7. Simplified 69.1%

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

      1. add-sqr-sqrt 40.8%

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

      2. pow2 40.8%

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

      3. sqrt-div 37.6%

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

      4. sqrt-div 33.6%

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

      5. pow2 33.6%

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

      6. sqrt-prod 22.7%

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

      7. add-sqr-sqrt 39.0%

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

      8. unpow2 39.0%

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

      9. sqrt-prod 13.1%

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

      10. add-sqr-sqrt 41.3%

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

   9. Applied egg-rr 41.3%

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

       1. expm1-log1p-u 39.3%

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

       2. expm1-udef 37.1%

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

   11. Applied egg-rr 35.2%

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

       1. expm1-def 35.2%

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

       2. expm1-log1p 35.7%

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

       3. associate-*l/ 35.7%

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

       4. times-frac 36.0%

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

   13. Simplified 36.0%

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

   if 6.3999999999999998e-47 < k

   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* 33.2%

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

      2. associate-/r* 33.2%

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

      3. sub-neg 33.2%

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

      4. distribute-rgt-in 33.2%

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

      5. unpow2 33.2%

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

      6. times-frac 31.7%

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

      7. sqr-neg 31.7%

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

      8. times-frac 33.2%

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

      9. unpow2 33.2%

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

      10. distribute-rgt-in 33.2%

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

      11. +-commutative 33.2%

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

      12. associate-+l+ 44.3%

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

   3. Simplified 44.3%

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

   5. Taylor expanded in t around 0 80.0%

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

      1. associate-*r* 79.9%

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

      2. times-frac 78.5%

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

      3. *-commutative 78.5%

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

      4. associate-/r* 81.6%

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

   7. Simplified 81.6%

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

      1. add-sqr-sqrt 62.7%

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

      2. pow2 62.7%

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

      3. sqrt-div 55.0%

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

      4. sqrt-div 41.2%

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

      5. pow2 41.2%

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

      6. sqrt-prod 20.2%

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

      7. add-sqr-sqrt 44.3%

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

      8. unpow2 44.3%

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

      9. sqrt-prod 46.1%

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

      10. add-sqr-sqrt 46.1%

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

   9. Applied egg-rr 46.1%

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

   10. Taylor expanded in t around -inf 0.0%

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

       1. mul-1-neg 0.0%

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

       2. times-frac 0.0%

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

       3. unpow2 0.0%

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

       4. unpow2 0.0%

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

       5. times-frac 0.0%

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

       6. unpow2 0.0%

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

       7. distribute-lft-neg-in 0.0%

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

       8. associate-*r* 0.0%

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

       9. unpow2 0.0%

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

       10. rem-square-sqrt 93.3%

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

       11. *-commutative 93.3%

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

       12. associate-*l* 93.3%

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

   12. Simplified 93.3%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.4 \cdot 10^{-47}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k} \cdot \frac{\frac{\sqrt{\cos k}}{\sin k}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{-\cos k}{{\sin k}^{2} \cdot \left(-t\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.2 \cdot 10^{-31}:\\ \;\;\;\;{\left(\frac{\frac{\ell \cdot \sqrt{2}}{k\_m}}{k\_m} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \frac{-\cos k\_m}{{\sin k\_m}^{2} \cdot \left(-t\_m\right)}\right)\\ \end{array} \end{array} \]

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 4.2e-31)
    (pow (* (/ (/ (* l (sqrt 2.0)) k_m) k_m) (sqrt (/ 1.0 t_m))) 2.0)
    (*
     2.0
     (*
      (pow (/ l k_m) 2.0)
      (/ (- (cos k_m)) (* (pow (sin k_m) 2.0) (- t_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 tmp;
	if (k_m <= 4.2e-31) {
		tmp = pow(((((l * sqrt(2.0)) / k_m) / k_m) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 * (pow((l / k_m), 2.0) * (-cos(k_m) / (pow(sin(k_m), 2.0) * -t_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) :: tmp
    if (k_m <= 4.2d-31) then
        tmp = ((((l * sqrt(2.0d0)) / k_m) / k_m) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = 2.0d0 * (((l / k_m) ** 2.0d0) * (-cos(k_m) / ((sin(k_m) ** 2.0d0) * -t_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 tmp;
	if (k_m <= 4.2e-31) {
		tmp = Math.pow(((((l * Math.sqrt(2.0)) / k_m) / k_m) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 * (Math.pow((l / k_m), 2.0) * (-Math.cos(k_m) / (Math.pow(Math.sin(k_m), 2.0) * -t_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):
	tmp = 0
	if k_m <= 4.2e-31:
		tmp = math.pow(((((l * math.sqrt(2.0)) / k_m) / k_m) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = 2.0 * (math.pow((l / k_m), 2.0) * (-math.cos(k_m) / (math.pow(math.sin(k_m), 2.0) * -t_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)
	tmp = 0.0
	if (k_m <= 4.2e-31)
		tmp = Float64(Float64(Float64(Float64(l * sqrt(2.0)) / k_m) / k_m) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64((Float64(l / k_m) ^ 2.0) * Float64(Float64(-cos(k_m)) / Float64((sin(k_m) ^ 2.0) * Float64(-t_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)
	tmp = 0.0;
	if (k_m <= 4.2e-31)
		tmp = ((((l * sqrt(2.0)) / k_m) / k_m) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = 2.0 * (((l / k_m) ^ 2.0) * (-cos(k_m) / ((sin(k_m) ^ 2.0) * -t_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_] := N[(t$95$s * If[LessEqual[k$95$m, 4.2e-31], N[Power[N[(N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[((-N[Cos[k$95$m], $MachinePrecision]) / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * (-t$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.19999999999999982e-31

   1. Initial program 30.5%

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

      1. associate-*l* 30.9%

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

      2. associate-/r* 30.5%

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

      3. sub-neg 30.5%

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

      4. distribute-rgt-in 23.2%

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

      5. unpow2 23.2%

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

      6. times-frac 17.3%

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

      7. sqr-neg 17.3%

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

      8. times-frac 23.2%

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

      9. unpow2 23.2%

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

      10. distribute-rgt-in 30.5%

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

      11. +-commutative 30.5%

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

      12. associate-+l+ 36.7%

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

   3. Simplified 36.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 36.7%

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

      2. associate-/r* 37.2%

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

      3. associate-/r* 37.2%

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

      4. associate-/l/ 37.2%

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

      5. *-commutative 37.2%

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

      6. associate-/r* 37.2%

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

      7. associate-/r/ 37.8%

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

      8. add-sqr-sqrt 22.1%

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

   6. Applied egg-rr 26.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{t}^{3}}} \cdot \frac{\ell}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}} \cdot \frac{\sqrt{\frac{2}{{t}^{3}}} \cdot \frac{\ell}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}} \]
   7. Step-by-step derivation

      1. unpow2 26.7%

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

      2. associate-/r/ 26.9%

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

      3. *-commutative 26.9%

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

      4. associate-/l* 26.9%

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

   8. Simplified 26.9%

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

   9. Taylor expanded in k around 0 37.3%

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

       1. add-sqr-sqrt 25.8%

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

       2. sqrt-div 21.6%

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

       3. unpow2 21.6%

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

       4. sqrt-prod 7.8%

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

       5. add-sqr-sqrt 21.6%

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

       6. sqrt-div 21.6%

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

       7. unpow2 21.6%

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

       8. sqrt-prod 7.8%

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

       9. add-sqr-sqrt 22.1%

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

   11. Applied egg-rr 22.1%

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

       1. associate-*l/ 22.1%

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

       2. associate-*r/ 22.1%

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

       3. rem-square-sqrt 38.2%

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

   13. Simplified 38.2%

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

   if 4.19999999999999982e-31 < k

   1. Initial program 33.7%

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

      1. associate-*l* 33.7%

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

      2. associate-/r* 33.7%

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

      3. sub-neg 33.7%

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

      4. distribute-rgt-in 33.7%

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

      5. unpow2 33.7%

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

      6. times-frac 32.2%

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

      7. sqr-neg 32.2%

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

      8. times-frac 33.7%

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

      9. unpow2 33.7%

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

      10. distribute-rgt-in 33.7%

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

      11. +-commutative 33.7%

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

      12. associate-+l+ 45.0%

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

   3. Simplified 45.0%

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

   5. Taylor expanded in t around 0 79.7%

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

      1. associate-*r* 79.7%

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

      2. times-frac 79.6%

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

      3. *-commutative 79.6%

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

      4. associate-/r* 82.8%

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

   7. Simplified 82.8%

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

      1. add-sqr-sqrt 63.6%

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

      2. pow2 63.6%

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

      3. sqrt-div 55.7%

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

      4. sqrt-div 41.7%

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

      5. pow2 41.7%

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

      6. sqrt-prod 20.4%

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

      7. add-sqr-sqrt 44.8%

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

      8. unpow2 44.8%

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

      9. sqrt-prod 46.7%

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

      10. add-sqr-sqrt 46.7%

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

   9. Applied egg-rr 46.7%

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

   10. Taylor expanded in t around -inf 0.0%

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

       1. mul-1-neg 0.0%

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

       2. times-frac 0.0%

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

       3. unpow2 0.0%

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

       4. unpow2 0.0%

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

       5. times-frac 0.0%

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

       6. unpow2 0.0%

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

       7. distribute-lft-neg-in 0.0%

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

       8. associate-*r* 0.0%

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

       9. unpow2 0.0%

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

       10. rem-square-sqrt 93.2%

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

       11. *-commutative 93.2%

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

       12. associate-*l* 93.2%

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

   12. Simplified 93.2%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-31}:\\ \;\;\;\;{\left(\frac{\frac{\ell \cdot \sqrt{2}}{k}}{k} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{-\cos k}{{\sin k}^{2} \cdot \left(-t\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.2 \cdot 10^{-31}:\\ \;\;\;\;{\left(\frac{\frac{\ell \cdot \sqrt{2}}{k\_m}}{k\_m} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k\_m}{{\sin k\_m}^{2}} \cdot \frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m}\right)\\ \end{array} \end{array} \]

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 4.2e-31)
    (pow (* (/ (/ (* l (sqrt 2.0)) k_m) k_m) (sqrt (/ 1.0 t_m))) 2.0)
    (*
     2.0
     (* (/ (cos k_m) (pow (sin k_m) 2.0)) (/ (pow (/ l k_m) 2.0) t_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 tmp;
	if (k_m <= 4.2e-31) {
		tmp = pow(((((l * sqrt(2.0)) / k_m) / k_m) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 * ((cos(k_m) / pow(sin(k_m), 2.0)) * (pow((l / k_m), 2.0) / t_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) :: tmp
    if (k_m <= 4.2d-31) then
        tmp = ((((l * sqrt(2.0d0)) / k_m) / k_m) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = 2.0d0 * ((cos(k_m) / (sin(k_m) ** 2.0d0)) * (((l / k_m) ** 2.0d0) / t_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 tmp;
	if (k_m <= 4.2e-31) {
		tmp = Math.pow(((((l * Math.sqrt(2.0)) / k_m) / k_m) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 * ((Math.cos(k_m) / Math.pow(Math.sin(k_m), 2.0)) * (Math.pow((l / k_m), 2.0) / t_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):
	tmp = 0
	if k_m <= 4.2e-31:
		tmp = math.pow(((((l * math.sqrt(2.0)) / k_m) / k_m) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = 2.0 * ((math.cos(k_m) / math.pow(math.sin(k_m), 2.0)) * (math.pow((l / k_m), 2.0) / t_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)
	tmp = 0.0
	if (k_m <= 4.2e-31)
		tmp = Float64(Float64(Float64(Float64(l * sqrt(2.0)) / k_m) / k_m) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) / (sin(k_m) ^ 2.0)) * Float64((Float64(l / k_m) ^ 2.0) / t_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)
	tmp = 0.0;
	if (k_m <= 4.2e-31)
		tmp = ((((l * sqrt(2.0)) / k_m) / k_m) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = 2.0 * ((cos(k_m) / (sin(k_m) ^ 2.0)) * (((l / k_m) ^ 2.0) / t_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_] := N[(t$95$s * If[LessEqual[k$95$m, 4.2e-31], N[Power[N[(N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.19999999999999982e-31

   1. Initial program 30.5%

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

      1. associate-*l* 30.9%

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

      2. associate-/r* 30.5%

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

      3. sub-neg 30.5%

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

      4. distribute-rgt-in 23.2%

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

      5. unpow2 23.2%

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

      6. times-frac 17.3%

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

      7. sqr-neg 17.3%

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

      8. times-frac 23.2%

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

      9. unpow2 23.2%

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

      10. distribute-rgt-in 30.5%

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

      11. +-commutative 30.5%

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

      12. associate-+l+ 36.7%

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

   3. Simplified 36.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 36.7%

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

      2. associate-/r* 37.2%

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

      3. associate-/r* 37.2%

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

      4. associate-/l/ 37.2%

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

      5. *-commutative 37.2%

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

      6. associate-/r* 37.2%

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

      7. associate-/r/ 37.8%

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

      8. add-sqr-sqrt 22.1%

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

   6. Applied egg-rr 26.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{t}^{3}}} \cdot \frac{\ell}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}} \cdot \frac{\sqrt{\frac{2}{{t}^{3}}} \cdot \frac{\ell}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}} \]
   7. Step-by-step derivation

      1. unpow2 26.7%

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

      2. associate-/r/ 26.9%

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

      3. *-commutative 26.9%

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

      4. associate-/l* 26.9%

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

   8. Simplified 26.9%

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

   9. Taylor expanded in k around 0 37.3%

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

       1. add-sqr-sqrt 25.8%

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

       2. sqrt-div 21.6%

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

       3. unpow2 21.6%

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

       4. sqrt-prod 7.8%

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

       5. add-sqr-sqrt 21.6%

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

       6. sqrt-div 21.6%

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

       7. unpow2 21.6%

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

       8. sqrt-prod 7.8%

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

       9. add-sqr-sqrt 22.1%

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

   11. Applied egg-rr 22.1%

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

       1. associate-*l/ 22.1%

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

       2. associate-*r/ 22.1%

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

       3. rem-square-sqrt 38.2%

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

   13. Simplified 38.2%

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

   if 4.19999999999999982e-31 < k

   1. Initial program 33.7%

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

      1. associate-*l* 33.7%

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

      2. associate-/r* 33.7%

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

      3. sub-neg 33.7%

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

      4. distribute-rgt-in 33.7%

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

      5. unpow2 33.7%

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

      6. times-frac 32.2%

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

      7. sqr-neg 32.2%

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

      8. times-frac 33.7%

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

      9. unpow2 33.7%

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

      10. distribute-rgt-in 33.7%

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

      11. +-commutative 33.7%

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

      12. associate-+l+ 45.0%

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

   3. Simplified 45.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative 45.0%

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

      2. distribute-lft-in 45.0%

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

      3. mul0-rgt 45.0%

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

   6. Applied egg-rr 45.0%

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

      1. +-lft-identity 45.0%

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

   8. Simplified 45.0%

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

   9. Taylor expanded in t around 0 79.7%

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

       1. associate-*r* 79.7%

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

       2. times-frac 79.6%

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

       3. associate-/r* 82.8%

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

       4. unpow2 82.8%

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

       5. unpow2 82.8%

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

       6. times-frac 93.1%

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

       7. unpow2 93.1%

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

   11. Simplified 93.1%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-31}:\\ \;\;\;\;{\left(\frac{\frac{\ell \cdot \sqrt{2}}{k}}{k} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.1 \cdot 10^{+27}:\\ \;\;\;\;{\left(\frac{\frac{\ell \cdot \sqrt{2}}{k\_m}}{k\_m} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{t\_m} \cdot \frac{1}{\frac{k\_m}{\ell} \cdot \frac{k\_m}{\ell}}\right)\\ \end{array} \end{array} \]

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 2.1e+27)
    (pow (* (/ (/ (* l (sqrt 2.0)) k_m) k_m) (sqrt (/ 1.0 t_m))) 2.0)
    (* 2.0 (* (/ -0.16666666666666666 t_m) (/ 1.0 (* (/ k_m l) (/ k_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 <= 2.1e+27) {
		tmp = pow(((((l * sqrt(2.0)) / k_m) / k_m) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 2.1d+27) then
        tmp = ((((l * sqrt(2.0d0)) / k_m) / k_m) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = 2.0d0 * (((-0.16666666666666666d0) / t_m) * (1.0d0 / ((k_m / l) * (k_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 <= 2.1e+27) {
		tmp = Math.pow(((((l * Math.sqrt(2.0)) / k_m) / k_m) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 2.1e+27:
		tmp = math.pow(((((l * math.sqrt(2.0)) / k_m) / k_m) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 2.1e+27)
		tmp = Float64(Float64(Float64(Float64(l * sqrt(2.0)) / k_m) / k_m) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64(Float64(-0.16666666666666666 / t_m) * Float64(1.0 / Float64(Float64(k_m / l) * Float64(k_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 <= 2.1e+27)
		tmp = ((((l * sqrt(2.0)) / k_m) / k_m) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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, 2.1e+27], N[Power[N[(N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[(-0.16666666666666666 / t$95$m), $MachinePrecision] * N[(1.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.09999999999999995e27

   1. Initial program 30.6%

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

      1. associate-*l* 31.0%

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

      2. associate-/r* 30.6%

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

      3. sub-neg 30.6%

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

      4. distribute-rgt-in 23.8%

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

      5. unpow2 23.8%

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

      6. times-frac 18.3%

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

      7. sqr-neg 18.3%

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

      8. times-frac 23.8%

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

      9. unpow2 23.8%

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

      10. distribute-rgt-in 30.6%

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

      11. +-commutative 30.6%

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

      12. associate-+l+ 36.4%

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

   3. Simplified 36.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-rgt-identity 36.4%

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

      2. associate-/r* 36.9%

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

      3. associate-/r* 36.9%

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

      4. associate-/l/ 36.9%

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

      5. *-commutative 36.9%

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

      6. associate-/r* 36.9%

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

      7. associate-/r/ 37.4%

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

      8. add-sqr-sqrt 22.3%

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

   6. Applied egg-rr 26.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{2}{{t}^{3}}} \cdot \frac{\ell}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}} \cdot \frac{\sqrt{\frac{2}{{t}^{3}}} \cdot \frac{\ell}{\sqrt{\sin k \cdot \tan k}}}{\frac{k}{t}}} \]
   7. Step-by-step derivation

      1. unpow2 26.8%

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

      2. associate-/r/ 27.0%

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

      3. *-commutative 27.0%

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

      4. associate-/l* 27.0%

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

   8. Simplified 27.0%

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

   9. Taylor expanded in k around 0 36.3%

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

       1. add-sqr-sqrt 24.1%

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

       2. sqrt-div 20.2%

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

       3. unpow2 20.2%

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

       4. sqrt-prod 7.3%

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

       5. add-sqr-sqrt 20.2%

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

       6. sqrt-div 20.1%

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

       7. unpow2 20.1%

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

       8. sqrt-prod 7.3%

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

       9. add-sqr-sqrt 20.6%

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

   11. Applied egg-rr 20.6%

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

       1. associate-*l/ 20.6%

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

       2. associate-*r/ 20.6%

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

       3. rem-square-sqrt 37.2%

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

   13. Simplified 37.2%

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

   if 2.09999999999999995e27 < k

   1. Initial program 34.3%

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

      1. associate-*l* 34.3%

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

      2. associate-/r* 34.3%

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

      3. sub-neg 34.3%

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

      4. distribute-rgt-in 34.3%

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

      5. unpow2 34.3%

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

      6. times-frac 32.3%

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

      7. sqr-neg 32.3%

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

      8. times-frac 34.3%

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

      9. unpow2 34.3%

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

      10. distribute-rgt-in 34.3%

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

      11. +-commutative 34.3%

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

      12. associate-+l+ 48.5%

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

   3. Simplified 48.5%

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

   5. Taylor expanded in t around 0 79.8%

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

      1. times-frac 81.9%

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

   7. Simplified 81.9%

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

   8. Taylor expanded in k around 0 68.8%

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

   9. Taylor expanded in k around inf 68.9%

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

       1. associate-*r/ 68.9%

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

       2. *-commutative 68.9%

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

       3. times-frac 68.8%

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

       4. unpow2 68.8%

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

       5. unpow2 68.8%

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

       6. times-frac 70.3%

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

       7. unpow2 70.3%

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

   11. Simplified 70.3%

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

       1. unpow2 70.3%

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

       2. clear-num 70.3%

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

       3. clear-num 70.3%

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

       4. frac-times 70.3%

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

       5. metadata-eval 70.3%

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

   13. Applied egg-rr 70.3%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{+27}:\\ \;\;\;\;{\left(\frac{\frac{\ell \cdot \sqrt{2}}{k}}{k} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.1 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{\sqrt{t\_m}}}{{k\_m}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{t\_m} \cdot \frac{1}{\frac{k\_m}{\ell} \cdot \frac{k\_m}{\ell}}\right)\\ \end{array} \end{array} \]

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 2.1e+27)
    (* 2.0 (pow (/ (/ l (sqrt t_m)) (pow k_m 2.0)) 2.0))
    (* 2.0 (* (/ -0.16666666666666666 t_m) (/ 1.0 (* (/ k_m l) (/ k_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 <= 2.1e+27) {
		tmp = 2.0 * pow(((l / sqrt(t_m)) / pow(k_m, 2.0)), 2.0);
	} else {
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 2.1d+27) then
        tmp = 2.0d0 * (((l / sqrt(t_m)) / (k_m ** 2.0d0)) ** 2.0d0)
    else
        tmp = 2.0d0 * (((-0.16666666666666666d0) / t_m) * (1.0d0 / ((k_m / l) * (k_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 <= 2.1e+27) {
		tmp = 2.0 * Math.pow(((l / Math.sqrt(t_m)) / Math.pow(k_m, 2.0)), 2.0);
	} else {
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 2.1e+27:
		tmp = 2.0 * math.pow(((l / math.sqrt(t_m)) / math.pow(k_m, 2.0)), 2.0)
	else:
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 2.1e+27)
		tmp = Float64(2.0 * (Float64(Float64(l / sqrt(t_m)) / (k_m ^ 2.0)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(-0.16666666666666666 / t_m) * Float64(1.0 / Float64(Float64(k_m / l) * Float64(k_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 <= 2.1e+27)
		tmp = 2.0 * (((l / sqrt(t_m)) / (k_m ^ 2.0)) ^ 2.0);
	else
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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, 2.1e+27], N[(2.0 * N[Power[N[(N[(l / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(-0.16666666666666666 / t$95$m), $MachinePrecision] * N[(1.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.09999999999999995e27

   1. Initial program 30.6%

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

      1. associate-*l* 31.0%

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

      2. associate-/r* 30.6%

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

      3. sub-neg 30.6%

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

      4. distribute-rgt-in 23.8%

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

      5. unpow2 23.8%

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

      6. times-frac 18.3%

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

      7. sqr-neg 18.3%

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

      8. times-frac 23.8%

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

      9. unpow2 23.8%

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

      10. distribute-rgt-in 30.6%

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

      11. +-commutative 30.6%

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

      12. associate-+l+ 36.4%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in k around 0 62.5%

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

      1. *-un-lft-identity 62.5%

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

      2. associate-/r* 63.1%

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

   7. Applied egg-rr 63.1%

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

      1. add-sqr-sqrt 39.4%

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

      2. sqrt-div 29.9%

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

      3. sqrt-div 29.9%

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

      4. pow2 29.9%

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

      5. sqrt-prod 15.2%

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

      6. add-sqr-sqrt 21.6%

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

      7. metadata-eval 21.6%

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

      8. pow-prod-up 21.6%

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

      9. sqrt-prod 21.6%

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

      10. add-sqr-sqrt 21.6%

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

      11. sqrt-div 21.6%

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

      12. sqrt-div 21.6%

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

      13. pow2 21.6%

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

      14. sqrt-prod 17.9%

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

      15. add-sqr-sqrt 33.1%

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

      16. metadata-eval 33.1%

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

      17. pow-prod-up 33.1%

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

      18. sqrt-prod 36.3%

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

      19. add-sqr-sqrt 36.3%

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

   9. Applied egg-rr 36.3%

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

       1. unpow2 36.3%

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

       2. associate-/l/ 35.0%

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

       3. associate-/r* 36.3%

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

   11. Simplified 36.3%

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

   if 2.09999999999999995e27 < k

   1. Initial program 34.3%

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

      1. associate-*l* 34.3%

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

      2. associate-/r* 34.3%

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

      3. sub-neg 34.3%

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

      4. distribute-rgt-in 34.3%

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

      5. unpow2 34.3%

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

      6. times-frac 32.3%

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

      7. sqr-neg 32.3%

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

      8. times-frac 34.3%

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

      9. unpow2 34.3%

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

      10. distribute-rgt-in 34.3%

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

      11. +-commutative 34.3%

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

      12. associate-+l+ 48.5%

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

   3. Simplified 48.5%

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

   5. Taylor expanded in t around 0 79.8%

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

      1. times-frac 81.9%

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

   7. Simplified 81.9%

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

   8. Taylor expanded in k around 0 68.8%

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

   9. Taylor expanded in k around inf 68.9%

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

       1. associate-*r/ 68.9%

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

       2. *-commutative 68.9%

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

       3. times-frac 68.8%

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

       4. unpow2 68.8%

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

       5. unpow2 68.8%

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

       6. times-frac 70.3%

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

       7. unpow2 70.3%

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

   11. Simplified 70.3%

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

       1. unpow2 70.3%

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

       2. clear-num 70.3%

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

       3. clear-num 70.3%

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

       4. frac-times 70.3%

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

       5. metadata-eval 70.3%

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

   13. Applied egg-rr 70.3%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot {\left(\frac{\frac{\ell}{\sqrt{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{{k\_m}^{2}}\right)}^{2}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \left(\frac{1}{t\_m} \cdot \left({k\_m}^{-2} - 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= l 5e-93)
    (* 2.0 (/ (pow (/ l (pow k_m 2.0)) 2.0) t_m))
    (*
     2.0
     (*
      (pow (/ l k_m) 2.0)
      (* (/ 1.0 t_m) (- (pow k_m -2.0) 0.16666666666666666)))))))

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 <= 5e-93) {
		tmp = 2.0 * (pow((l / pow(k_m, 2.0)), 2.0) / t_m);
	} else {
		tmp = 2.0 * (pow((l / k_m), 2.0) * ((1.0 / t_m) * (pow(k_m, -2.0) - 0.16666666666666666)));
	}
	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 <= 5d-93) then
        tmp = 2.0d0 * (((l / (k_m ** 2.0d0)) ** 2.0d0) / t_m)
    else
        tmp = 2.0d0 * (((l / k_m) ** 2.0d0) * ((1.0d0 / t_m) * ((k_m ** (-2.0d0)) - 0.16666666666666666d0)))
    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 <= 5e-93) {
		tmp = 2.0 * (Math.pow((l / Math.pow(k_m, 2.0)), 2.0) / t_m);
	} else {
		tmp = 2.0 * (Math.pow((l / k_m), 2.0) * ((1.0 / t_m) * (Math.pow(k_m, -2.0) - 0.16666666666666666)));
	}
	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 <= 5e-93:
		tmp = 2.0 * (math.pow((l / math.pow(k_m, 2.0)), 2.0) / t_m)
	else:
		tmp = 2.0 * (math.pow((l / k_m), 2.0) * ((1.0 / t_m) * (math.pow(k_m, -2.0) - 0.16666666666666666)))
	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 (l <= 5e-93)
		tmp = Float64(2.0 * Float64((Float64(l / (k_m ^ 2.0)) ^ 2.0) / t_m));
	else
		tmp = Float64(2.0 * Float64((Float64(l / k_m) ^ 2.0) * Float64(Float64(1.0 / t_m) * Float64((k_m ^ -2.0) - 0.16666666666666666))));
	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 <= 5e-93)
		tmp = 2.0 * (((l / (k_m ^ 2.0)) ^ 2.0) / t_m);
	else
		tmp = 2.0 * (((l / k_m) ^ 2.0) * ((1.0 / t_m) * ((k_m ^ -2.0) - 0.16666666666666666)));
	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[l, 5e-93], N[(2.0 * N[(N[Power[N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(1.0 / t$95$m), $MachinePrecision] * N[(N[Power[k$95$m, -2.0], $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.99999999999999994e-93

   1. Initial program 31.3%

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

      1. associate-*l* 31.3%

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

      2. associate-/r* 31.3%

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

      3. sub-neg 31.3%

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

      4. distribute-rgt-in 28.0%

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

      5. unpow2 28.0%

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

      6. times-frac 23.5%

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

      7. sqr-neg 23.5%

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

      8. times-frac 28.0%

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

      9. unpow2 28.0%

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

      10. distribute-rgt-in 31.3%

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

      11. +-commutative 31.3%

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

      12. associate-+l+ 39.6%

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

   3. Simplified 39.6%

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

   5. Taylor expanded in k around 0 63.3%

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

      1. *-un-lft-identity 63.3%

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

      2. associate-/r* 63.9%

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

   7. Applied egg-rr 63.9%

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

      1. add-sqr-sqrt 63.9%

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

      2. sqrt-div 63.9%

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

      3. pow2 63.9%

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

      4. sqrt-prod 18.3%

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

      5. add-sqr-sqrt 41.4%

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

      6. metadata-eval 41.4%

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

      7. pow-prod-up 41.4%

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

      8. sqrt-prod 41.4%

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

      9. add-sqr-sqrt 41.4%

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

      10. sqrt-div 41.4%

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

      11. pow2 41.4%

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

      12. sqrt-prod 23.9%

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

      13. add-sqr-sqrt 76.2%

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

      14. metadata-eval 76.2%

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

      15. pow-prod-up 76.1%

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

      16. sqrt-prod 79.9%

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

      17. add-sqr-sqrt 79.9%

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

   9. Applied egg-rr 79.9%

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

       1. unpow2 79.9%

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

   11. Simplified 79.9%

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

   if 4.99999999999999994e-93 < l

   1. Initial program 31.3%

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

      1. associate-*l* 32.5%

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

      2. associate-/r* 31.3%

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

      3. sub-neg 31.3%

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

      4. distribute-rgt-in 20.5%

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

      5. unpow2 20.5%

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

      6. times-frac 15.0%

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

      7. sqr-neg 15.0%

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

      8. times-frac 20.5%

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

      9. unpow2 20.5%

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

      10. distribute-rgt-in 31.3%

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

      11. +-commutative 31.3%

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

      12. associate-+l+ 36.7%

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

   3. Simplified 36.7%

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

   5. Taylor expanded in t around 0 73.5%

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

      1. times-frac 74.9%

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

   7. Simplified 74.9%

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

   8. Taylor expanded in k around 0 64.9%

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

      1. expm1-log1p-u 37.8%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right)\right)} \]

      2. expm1-udef 36.4%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)\right)} - 1\right)} \]

   10. Applied egg-rr 39.0%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{\ell}{k}\right)}^{2} \cdot \left(\frac{{k}^{-2}}{t} - \frac{0.16666666666666666}{t}\right)\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 41.6%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k}\right)}^{2} \cdot \left(\frac{{k}^{-2}}{t} - \frac{0.16666666666666666}{t}\right)\right)\right)} \]

       2. expm1-log1p 69.1%

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

       3. *-lft-identity 69.1%

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

       4. associate-*l/ 69.1%

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

       5. metadata-eval 69.1%

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

       6. associate-*l/ 69.1%

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

       7. distribute-lft-out-- 69.1%

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

   12. Simplified 69.1%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{{k}^{2}}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \left(\frac{1}{t} \cdot \left({k}^{-2} - 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.1 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k\_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{t\_m} \cdot \frac{1}{\frac{k\_m}{\ell} \cdot \frac{k\_m}{\ell}}\right)\\ \end{array} \end{array} \]

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 1.1e+27)
    (* 2.0 (/ (pow l 2.0) (* t_m (pow k_m 4.0))))
    (* 2.0 (* (/ -0.16666666666666666 t_m) (/ 1.0 (* (/ k_m l) (/ k_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 <= 1.1e+27) {
		tmp = 2.0 * (pow(l, 2.0) / (t_m * pow(k_m, 4.0)));
	} else {
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 1.1d+27) then
        tmp = 2.0d0 * ((l ** 2.0d0) / (t_m * (k_m ** 4.0d0)))
    else
        tmp = 2.0d0 * (((-0.16666666666666666d0) / t_m) * (1.0d0 / ((k_m / l) * (k_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 <= 1.1e+27) {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k_m, 4.0)));
	} else {
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 1.1e+27:
		tmp = 2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k_m, 4.0)))
	else:
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 1.1e+27)
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k_m ^ 4.0))));
	else
		tmp = Float64(2.0 * Float64(Float64(-0.16666666666666666 / t_m) * Float64(1.0 / Float64(Float64(k_m / l) * Float64(k_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 <= 1.1e+27)
		tmp = 2.0 * ((l ^ 2.0) / (t_m * (k_m ^ 4.0)));
	else
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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, 1.1e+27], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(-0.16666666666666666 / t$95$m), $MachinePrecision] * N[(1.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.0999999999999999e27

   1. Initial program 30.6%

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

      1. associate-*l* 31.0%

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

      2. associate-/r* 30.6%

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

      3. sub-neg 30.6%

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

      4. distribute-rgt-in 23.8%

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

      5. unpow2 23.8%

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

      6. times-frac 18.3%

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

      7. sqr-neg 18.3%

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

      8. times-frac 23.8%

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

      9. unpow2 23.8%

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

      10. distribute-rgt-in 30.6%

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

      11. +-commutative 30.6%

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

      12. associate-+l+ 36.4%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in k around 0 62.5%

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

   if 1.0999999999999999e27 < k

   1. Initial program 34.3%

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

      1. associate-*l* 34.3%

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

      2. associate-/r* 34.3%

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

      3. sub-neg 34.3%

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

      4. distribute-rgt-in 34.3%

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

      5. unpow2 34.3%

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

      6. times-frac 32.3%

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

      7. sqr-neg 32.3%

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

      8. times-frac 34.3%

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

      9. unpow2 34.3%

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

      10. distribute-rgt-in 34.3%

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

      11. +-commutative 34.3%

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

      12. associate-+l+ 48.5%

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

   3. Simplified 48.5%

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

   5. Taylor expanded in t around 0 79.8%

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

      1. times-frac 81.9%

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

   7. Simplified 81.9%

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

   8. Taylor expanded in k around 0 68.8%

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

   9. Taylor expanded in k around inf 68.9%

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

       1. associate-*r/ 68.9%

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

       2. *-commutative 68.9%

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

       3. times-frac 68.8%

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

       4. unpow2 68.8%

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

       5. unpow2 68.8%

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

       6. times-frac 70.3%

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

       7. unpow2 70.3%

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

   11. Simplified 70.3%

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

       1. unpow2 70.3%

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

       2. clear-num 70.3%

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

       3. clear-num 70.3%

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

       4. frac-times 70.3%

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

       5. metadata-eval 70.3%

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

   13. Applied egg-rr 70.3%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.1 \cdot 10^{+27}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k\_m}^{4}} \cdot \frac{2}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{t\_m} \cdot \frac{1}{\frac{k\_m}{\ell} \cdot \frac{k\_m}{\ell}}\right)\\ \end{array} \end{array} \]

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 2.1e+27)
    (* (/ (pow l 2.0) (pow k_m 4.0)) (/ 2.0 t_m))
    (* 2.0 (* (/ -0.16666666666666666 t_m) (/ 1.0 (* (/ k_m l) (/ k_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 <= 2.1e+27) {
		tmp = (pow(l, 2.0) / pow(k_m, 4.0)) * (2.0 / t_m);
	} else {
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 2.1d+27) then
        tmp = ((l ** 2.0d0) / (k_m ** 4.0d0)) * (2.0d0 / t_m)
    else
        tmp = 2.0d0 * (((-0.16666666666666666d0) / t_m) * (1.0d0 / ((k_m / l) * (k_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 <= 2.1e+27) {
		tmp = (Math.pow(l, 2.0) / Math.pow(k_m, 4.0)) * (2.0 / t_m);
	} else {
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 2.1e+27:
		tmp = (math.pow(l, 2.0) / math.pow(k_m, 4.0)) * (2.0 / t_m)
	else:
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 2.1e+27)
		tmp = Float64(Float64((l ^ 2.0) / (k_m ^ 4.0)) * Float64(2.0 / t_m));
	else
		tmp = Float64(2.0 * Float64(Float64(-0.16666666666666666 / t_m) * Float64(1.0 / Float64(Float64(k_m / l) * Float64(k_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 <= 2.1e+27)
		tmp = ((l ^ 2.0) / (k_m ^ 4.0)) * (2.0 / t_m);
	else
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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, 2.1e+27], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(-0.16666666666666666 / t$95$m), $MachinePrecision] * N[(1.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.09999999999999995e27

   1. Initial program 30.6%

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

      1. associate-*l* 31.0%

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

      2. associate-/r* 30.6%

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

      3. sub-neg 30.6%

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

      4. distribute-rgt-in 23.8%

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

      5. unpow2 23.8%

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

      6. times-frac 18.3%

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

      7. sqr-neg 18.3%

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

      8. times-frac 23.8%

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

      9. unpow2 23.8%

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

      10. distribute-rgt-in 30.6%

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

      11. +-commutative 30.6%

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

      12. associate-+l+ 36.4%

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

   3. Simplified 36.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative 36.4%

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

      2. distribute-lft-in 36.4%

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

      3. mul0-rgt 31.1%

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

   6. Applied egg-rr 31.1%

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

      1. +-lft-identity 36.4%

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

   8. Simplified 36.4%

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

   9. Taylor expanded in k around 0 62.5%

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

       1. associate-*r/ 62.5%

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

       2. *-commutative 62.5%

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

       3. times-frac 63.0%

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

   11. Simplified 63.0%

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

   if 2.09999999999999995e27 < k

   1. Initial program 34.3%

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

      1. associate-*l* 34.3%

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

      2. associate-/r* 34.3%

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

      3. sub-neg 34.3%

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

      4. distribute-rgt-in 34.3%

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

      5. unpow2 34.3%

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

      6. times-frac 32.3%

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

      7. sqr-neg 32.3%

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

      8. times-frac 34.3%

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

      9. unpow2 34.3%

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

      10. distribute-rgt-in 34.3%

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

      11. +-commutative 34.3%

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

      12. associate-+l+ 48.5%

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

   3. Simplified 48.5%

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

   5. Taylor expanded in t around 0 79.8%

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

      1. times-frac 81.9%

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

   7. Simplified 81.9%

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

   8. Taylor expanded in k around 0 68.8%

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

   9. Taylor expanded in k around inf 68.9%

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

       1. associate-*r/ 68.9%

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

       2. *-commutative 68.9%

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

       3. times-frac 68.8%

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

       4. unpow2 68.8%

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

       5. unpow2 68.8%

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

       6. times-frac 70.3%

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

       7. unpow2 70.3%

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

   11. Simplified 70.3%

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

       1. unpow2 70.3%

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

       2. clear-num 70.3%

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

       3. clear-num 70.3%

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

       4. frac-times 70.3%

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

       5. metadata-eval 70.3%

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

   13. Applied egg-rr 70.3%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{+27}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{4}} \cdot \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.1 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{4}}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{t\_m} \cdot \frac{1}{\frac{k\_m}{\ell} \cdot \frac{k\_m}{\ell}}\right)\\ \end{array} \end{array} \]

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 2.1e+27)
    (* 2.0 (/ (/ (pow l 2.0) (pow k_m 4.0)) t_m))
    (* 2.0 (* (/ -0.16666666666666666 t_m) (/ 1.0 (* (/ k_m l) (/ k_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 <= 2.1e+27) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 4.0)) / t_m);
	} else {
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 2.1d+27) then
        tmp = 2.0d0 * (((l ** 2.0d0) / (k_m ** 4.0d0)) / t_m)
    else
        tmp = 2.0d0 * (((-0.16666666666666666d0) / t_m) * (1.0d0 / ((k_m / l) * (k_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 <= 2.1e+27) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 4.0)) / t_m);
	} else {
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 2.1e+27:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 4.0)) / t_m)
	else:
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 2.1e+27)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 4.0)) / t_m));
	else
		tmp = Float64(2.0 * Float64(Float64(-0.16666666666666666 / t_m) * Float64(1.0 / Float64(Float64(k_m / l) * Float64(k_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 <= 2.1e+27)
		tmp = 2.0 * (((l ^ 2.0) / (k_m ^ 4.0)) / t_m);
	else
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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, 2.1e+27], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(-0.16666666666666666 / t$95$m), $MachinePrecision] * N[(1.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.09999999999999995e27

   1. Initial program 30.6%

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

      1. associate-*l* 31.0%

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

      2. associate-/r* 30.6%

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

      3. sub-neg 30.6%

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

      4. distribute-rgt-in 23.8%

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

      5. unpow2 23.8%

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

      6. times-frac 18.3%

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

      7. sqr-neg 18.3%

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

      8. times-frac 23.8%

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

      9. unpow2 23.8%

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

      10. distribute-rgt-in 30.6%

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

      11. +-commutative 30.6%

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

      12. associate-+l+ 36.4%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in k around 0 62.5%

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

      1. *-un-lft-identity 62.5%

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

      2. associate-/r* 63.1%

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

   7. Applied egg-rr 63.1%

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

   if 2.09999999999999995e27 < k

   1. Initial program 34.3%

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

      1. associate-*l* 34.3%

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

      2. associate-/r* 34.3%

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

      3. sub-neg 34.3%

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

      4. distribute-rgt-in 34.3%

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

      5. unpow2 34.3%

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

      6. times-frac 32.3%

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

      7. sqr-neg 32.3%

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

      8. times-frac 34.3%

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

      9. unpow2 34.3%

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

      10. distribute-rgt-in 34.3%

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

      11. +-commutative 34.3%

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

      12. associate-+l+ 48.5%

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

   3. Simplified 48.5%

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

   5. Taylor expanded in t around 0 79.8%

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

      1. times-frac 81.9%

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

   7. Simplified 81.9%

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

   8. Taylor expanded in k around 0 68.8%

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

   9. Taylor expanded in k around inf 68.9%

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

       1. associate-*r/ 68.9%

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

       2. *-commutative 68.9%

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

       3. times-frac 68.8%

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

       4. unpow2 68.8%

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

       5. unpow2 68.8%

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

       6. times-frac 70.3%

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

       7. unpow2 70.3%

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

   11. Simplified 70.3%

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

       1. unpow2 70.3%

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

       2. clear-num 70.3%

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

       3. clear-num 70.3%

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

       4. frac-times 70.3%

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

       5. metadata-eval 70.3%

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

   13. Applied egg-rr 70.3%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.1 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{{k\_m}^{2}}\right)}^{2}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{t\_m} \cdot \frac{1}{\frac{k\_m}{\ell} \cdot \frac{k\_m}{\ell}}\right)\\ \end{array} \end{array} \]

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 2.1e+27)
    (* 2.0 (/ (pow (/ l (pow k_m 2.0)) 2.0) t_m))
    (* 2.0 (* (/ -0.16666666666666666 t_m) (/ 1.0 (* (/ k_m l) (/ k_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 <= 2.1e+27) {
		tmp = 2.0 * (pow((l / pow(k_m, 2.0)), 2.0) / t_m);
	} else {
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 2.1d+27) then
        tmp = 2.0d0 * (((l / (k_m ** 2.0d0)) ** 2.0d0) / t_m)
    else
        tmp = 2.0d0 * (((-0.16666666666666666d0) / t_m) * (1.0d0 / ((k_m / l) * (k_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 <= 2.1e+27) {
		tmp = 2.0 * (Math.pow((l / Math.pow(k_m, 2.0)), 2.0) / t_m);
	} else {
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 2.1e+27:
		tmp = 2.0 * (math.pow((l / math.pow(k_m, 2.0)), 2.0) / t_m)
	else:
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 2.1e+27)
		tmp = Float64(2.0 * Float64((Float64(l / (k_m ^ 2.0)) ^ 2.0) / t_m));
	else
		tmp = Float64(2.0 * Float64(Float64(-0.16666666666666666 / t_m) * Float64(1.0 / Float64(Float64(k_m / l) * Float64(k_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 <= 2.1e+27)
		tmp = 2.0 * (((l / (k_m ^ 2.0)) ^ 2.0) / t_m);
	else
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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, 2.1e+27], N[(2.0 * N[(N[Power[N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(-0.16666666666666666 / t$95$m), $MachinePrecision] * N[(1.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.09999999999999995e27

   1. Initial program 30.6%

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

      1. associate-*l* 31.0%

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

      2. associate-/r* 30.6%

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

      3. sub-neg 30.6%

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

      4. distribute-rgt-in 23.8%

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

      5. unpow2 23.8%

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

      6. times-frac 18.3%

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

      7. sqr-neg 18.3%

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

      8. times-frac 23.8%

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

      9. unpow2 23.8%

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

      10. distribute-rgt-in 30.6%

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

      11. +-commutative 30.6%

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

      12. associate-+l+ 36.4%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in k around 0 62.5%

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

      1. *-un-lft-identity 62.5%

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

      2. associate-/r* 63.1%

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

   7. Applied egg-rr 63.1%

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

      1. add-sqr-sqrt 63.0%

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

      2. sqrt-div 63.0%

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

      3. pow2 63.0%

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

      4. sqrt-prod 30.0%

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

      5. add-sqr-sqrt 42.7%

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

      6. metadata-eval 42.7%

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

      7. pow-prod-up 42.7%

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

      8. sqrt-prod 42.7%

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

      9. add-sqr-sqrt 42.7%

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

      10. sqrt-div 42.7%

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

      11. pow2 42.7%

          \[\leadsto 2 \cdot \left(1 \cdot \frac{\frac{\ell}{{k}^{2}} \cdot \frac{\sqrt{\color{blue}{\ell \cdot \ell}}}{\sqrt{{k}^{4}}}}{t}\right) \]

      12. sqrt-prod 35.7%

          \[\leadsto 2 \cdot \left(1 \cdot \frac{\frac{\ell}{{k}^{2}} \cdot \frac{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}{\sqrt{{k}^{4}}}}{t}\right) \]

      13. add-sqr-sqrt 74.7%

          \[\leadsto 2 \cdot \left(1 \cdot \frac{\frac{\ell}{{k}^{2}} \cdot \frac{\color{blue}{\ell}}{\sqrt{{k}^{4}}}}{t}\right) \]

      14. metadata-eval 74.7%

          \[\leadsto 2 \cdot \left(1 \cdot \frac{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{\sqrt{{k}^{\color{blue}{\left(2 + 2\right)}}}}}{t}\right) \]

      15. pow-prod-up 74.7%

          \[\leadsto 2 \cdot \left(1 \cdot \frac{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{\sqrt{\color{blue}{{k}^{2} \cdot {k}^{2}}}}}{t}\right) \]

      16. sqrt-prod 77.9%

          \[\leadsto 2 \cdot \left(1 \cdot \frac{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{\color{blue}{\sqrt{{k}^{2}} \cdot \sqrt{{k}^{2}}}}}{t}\right) \]

      17. add-sqr-sqrt 77.9%

          \[\leadsto 2 \cdot \left(1 \cdot \frac{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{\color{blue}{{k}^{2}}}}{t}\right) \]

   9. Applied egg-rr 77.9%

      \[\leadsto 2 \cdot \left(1 \cdot \frac{\color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{k}^{2}}}}{t}\right) \]
   10. Step-by-step derivation

       1. unpow2 77.9%

          \[\leadsto 2 \cdot \left(1 \cdot \frac{\color{blue}{{\left(\frac{\ell}{{k}^{2}}\right)}^{2}}}{t}\right) \]

   11. Simplified 77.9%

       \[\leadsto 2 \cdot \left(1 \cdot \frac{\color{blue}{{\left(\frac{\ell}{{k}^{2}}\right)}^{2}}}{t}\right) \]

   if 2.09999999999999995e27 < k

   1. Initial program 34.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 34.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 34.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 34.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 34.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 34.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 32.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 32.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 34.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 34.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 34.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 34.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 48.5%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 48.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 79.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. times-frac 81.9%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

   7. Simplified 81.9%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

   8. Taylor expanded in k around 0 68.8%

      \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]

   9. Taylor expanded in k around inf 68.9%

      \[\leadsto 2 \cdot \color{blue}{\left(-0.16666666666666666 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 68.9%

          \[\leadsto 2 \cdot \color{blue}{\frac{-0.16666666666666666 \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

       2. *-commutative 68.9%

          \[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

       3. times-frac 68.8%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{t} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)} \]

       4. unpow2 68.8%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \]

       5. unpow2 68.8%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \]

       6. times-frac 70.3%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right) \]

       7. unpow2 70.3%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}\right) \]

   11. Simplified 70.3%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{t} \cdot {\left(\frac{\ell}{k}\right)}^{2}\right)} \]
   12. Step-by-step derivation

       1. unpow2 70.3%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right) \]

       2. clear-num 70.3%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \left(\color{blue}{\frac{1}{\frac{k}{\ell}}} \cdot \frac{\ell}{k}\right)\right) \]

       3. clear-num 70.3%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \left(\frac{1}{\frac{k}{\ell}} \cdot \color{blue}{\frac{1}{\frac{k}{\ell}}}\right)\right) \]

       4. frac-times 70.3%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{\frac{1 \cdot 1}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}\right) \]

       5. metadata-eval 70.3%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{\color{blue}{1}}{\frac{k}{\ell} \cdot \frac{k}{\ell}}\right) \]

   13. Applied egg-rr 70.3%

       \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{{k}^{2}}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.1 \cdot 10^{+27}:\\ \;\;\;\;\left|{\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \frac{-0.3333333333333333}{t\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{t\_m} \cdot \frac{1}{\frac{k\_m}{\ell} \cdot \frac{k\_m}{\ell}}\right)\\ \end{array} \end{array} \]

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 2.1e+27)
    (fabs (* (pow (/ l k_m) 2.0) (/ -0.3333333333333333 t_m)))
    (* 2.0 (* (/ -0.16666666666666666 t_m) (/ 1.0 (* (/ k_m l) (/ k_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 <= 2.1e+27) {
		tmp = fabs((pow((l / k_m), 2.0) * (-0.3333333333333333 / t_m)));
	} else {
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 2.1d+27) then
        tmp = abs((((l / k_m) ** 2.0d0) * ((-0.3333333333333333d0) / t_m)))
    else
        tmp = 2.0d0 * (((-0.16666666666666666d0) / t_m) * (1.0d0 / ((k_m / l) * (k_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 <= 2.1e+27) {
		tmp = Math.abs((Math.pow((l / k_m), 2.0) * (-0.3333333333333333 / t_m)));
	} else {
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 2.1e+27:
		tmp = math.fabs((math.pow((l / k_m), 2.0) * (-0.3333333333333333 / t_m)))
	else:
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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 <= 2.1e+27)
		tmp = abs(Float64((Float64(l / k_m) ^ 2.0) * Float64(-0.3333333333333333 / t_m)));
	else
		tmp = Float64(2.0 * Float64(Float64(-0.16666666666666666 / t_m) * Float64(1.0 / Float64(Float64(k_m / l) * Float64(k_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 <= 2.1e+27)
		tmp = abs((((l / k_m) ^ 2.0) * (-0.3333333333333333 / t_m)));
	else
		tmp = 2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_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, 2.1e+27], N[Abs[N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(-0.3333333333333333 / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(2.0 * N[(N[(-0.16666666666666666 / t$95$m), $MachinePrecision] * N[(1.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.09999999999999995e27

   1. Initial program 30.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 31.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 30.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 30.6%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 23.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 23.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 18.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 18.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 23.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 23.8%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 30.6%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 30.6%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 36.4%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 36.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 73.2%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. times-frac 74.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

   7. Simplified 74.4%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

   8. Taylor expanded in k around 0 67.2%

      \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]

   9. Taylor expanded in k around inf 22.3%

      \[\leadsto 2 \cdot \color{blue}{\left(-0.16666666666666666 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 22.3%

          \[\leadsto 2 \cdot \color{blue}{\frac{-0.16666666666666666 \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

       2. *-commutative 22.3%

          \[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

       3. times-frac 22.4%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{t} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)} \]

       4. unpow2 22.4%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \]

       5. unpow2 22.4%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \]

       6. times-frac 24.2%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right) \]

       7. unpow2 24.2%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}\right) \]

   11. Simplified 24.2%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{t} \cdot {\left(\frac{\ell}{k}\right)}^{2}\right)} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 22.8%

          \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot {\left(\frac{\ell}{k}\right)}^{2}\right)} \cdot \sqrt{2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot {\left(\frac{\ell}{k}\right)}^{2}\right)}} \]

       2. sqrt-unprod 41.5%

          \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot {\left(\frac{\ell}{k}\right)}^{2}\right)\right) \cdot \left(2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot {\left(\frac{\ell}{k}\right)}^{2}\right)\right)}} \]

       3. pow2 41.5%

          \[\leadsto \sqrt{\color{blue}{{\left(2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot {\left(\frac{\ell}{k}\right)}^{2}\right)\right)}^{2}}} \]

       4. associate-*r* 41.5%

          \[\leadsto \sqrt{{\color{blue}{\left(\left(2 \cdot \frac{-0.16666666666666666}{t}\right) \cdot {\left(\frac{\ell}{k}\right)}^{2}\right)}}^{2}} \]

   13. Applied egg-rr 41.5%

       \[\leadsto \color{blue}{\sqrt{{\left(\left(2 \cdot \frac{-0.16666666666666666}{t}\right) \cdot {\left(\frac{\ell}{k}\right)}^{2}\right)}^{2}}} \]
   14. Step-by-step derivation

       1. unpow2 41.5%

          \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot \frac{-0.16666666666666666}{t}\right) \cdot {\left(\frac{\ell}{k}\right)}^{2}\right) \cdot \left(\left(2 \cdot \frac{-0.16666666666666666}{t}\right) \cdot {\left(\frac{\ell}{k}\right)}^{2}\right)}} \]

       2. rem-sqrt-square 37.1%

          \[\leadsto \color{blue}{\left|\left(2 \cdot \frac{-0.16666666666666666}{t}\right) \cdot {\left(\frac{\ell}{k}\right)}^{2}\right|} \]

       3. *-commutative 37.1%

          \[\leadsto \left|\color{blue}{{\left(\frac{\ell}{k}\right)}^{2} \cdot \left(2 \cdot \frac{-0.16666666666666666}{t}\right)}\right| \]

       4. associate-*r/ 37.1%

          \[\leadsto \left|{\left(\frac{\ell}{k}\right)}^{2} \cdot \color{blue}{\frac{2 \cdot -0.16666666666666666}{t}}\right| \]

       5. metadata-eval 37.1%

          \[\leadsto \left|{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\color{blue}{-0.3333333333333333}}{t}\right| \]

   15. Simplified 37.1%

       \[\leadsto \color{blue}{\left|{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{-0.3333333333333333}{t}\right|} \]

   if 2.09999999999999995e27 < k

   1. Initial program 34.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 34.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

      2. associate-/r* 34.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

      3. sub-neg 34.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      4. distribute-rgt-in 34.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

      5. unpow2 34.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      6. times-frac 32.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      7. sqr-neg 32.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      8. times-frac 34.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      9. unpow2 34.3%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

      10. distribute-rgt-in 34.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

      11. +-commutative 34.3%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

      12. associate-+l+ 48.5%

          \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

   3. Simplified 48.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 79.8%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. times-frac 81.9%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

   7. Simplified 81.9%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

   8. Taylor expanded in k around 0 68.8%

      \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]

   9. Taylor expanded in k around inf 68.9%

      \[\leadsto 2 \cdot \color{blue}{\left(-0.16666666666666666 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 68.9%

          \[\leadsto 2 \cdot \color{blue}{\frac{-0.16666666666666666 \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

       2. *-commutative 68.9%

          \[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

       3. times-frac 68.8%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{t} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)} \]

       4. unpow2 68.8%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \]

       5. unpow2 68.8%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \]

       6. times-frac 70.3%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right) \]

       7. unpow2 70.3%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}\right) \]

   11. Simplified 70.3%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{t} \cdot {\left(\frac{\ell}{k}\right)}^{2}\right)} \]
   12. Step-by-step derivation

       1. unpow2 70.3%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right) \]

       2. clear-num 70.3%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \left(\color{blue}{\frac{1}{\frac{k}{\ell}}} \cdot \frac{\ell}{k}\right)\right) \]

       3. clear-num 70.3%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \left(\frac{1}{\frac{k}{\ell}} \cdot \color{blue}{\frac{1}{\frac{k}{\ell}}}\right)\right) \]

       4. frac-times 70.3%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{\frac{1 \cdot 1}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}\right) \]

       5. metadata-eval 70.3%

          \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{\color{blue}{1}}{\frac{k}{\ell} \cdot \frac{k}{\ell}}\right) \]

   13. Applied egg-rr 70.3%

       \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{+27}:\\ \;\;\;\;\left|{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{-0.3333333333333333}{t}\right|\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 30.1% accurate, 28.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \left(\frac{-0.16666666666666666}{t\_m} \cdot \frac{1}{\frac{k\_m}{\ell} \cdot \frac{k\_m}{\ell}}\right)\right) \end{array} \]

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
  (* 2.0 (* (/ -0.16666666666666666 t_m) (/ 1.0 (* (/ k_m l) (/ k_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) {
	return t_s * (2.0 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_m / 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 * (((-0.16666666666666666d0) / t_m) * (1.0d0 / ((k_m / l) * (k_m / 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 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_m / 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 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_m / 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(-0.16666666666666666 / t_m) * Float64(1.0 / Float64(Float64(k_m / l) * Float64(k_m / 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 * ((-0.16666666666666666 / t_m) * (1.0 / ((k_m / l) * (k_m / 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[(-0.16666666666666666 / t$95$m), $MachinePrecision] * N[(1.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 31.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   2. associate-/r* 31.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   3. sub-neg 31.3%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

   4. distribute-rgt-in 25.8%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

   5. unpow2 25.8%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   6. times-frac 21.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   7. sqr-neg 21.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   8. times-frac 25.8%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   9. unpow2 25.8%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   10. distribute-rgt-in 31.3%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

   11. +-commutative 31.3%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

   12. associate-+l+ 38.8%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

3. Simplified 38.8%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 74.5%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
6. Step-by-step derivation

   1. times-frac 75.9%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

7. Simplified 75.9%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

8. Taylor expanded in k around 0 67.5%

   \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]

9. Taylor expanded in k around inf 31.4%

   \[\leadsto 2 \cdot \color{blue}{\left(-0.16666666666666666 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
10. Step-by-step derivation

    1. associate-*r/ 31.4%

       \[\leadsto 2 \cdot \color{blue}{\frac{-0.16666666666666666 \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

    2. *-commutative 31.4%

       \[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

    3. times-frac 31.5%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{t} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)} \]

    4. unpow2 31.5%

       \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \]

    5. unpow2 31.5%

       \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \]

    6. times-frac 33.2%

       \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right) \]

    7. unpow2 33.2%

       \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}\right) \]

11. Simplified 33.2%

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{t} \cdot {\left(\frac{\ell}{k}\right)}^{2}\right)} \]
12. Step-by-step derivation

    1. unpow2 33.2%

       \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right) \]

    2. clear-num 33.2%

       \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \left(\color{blue}{\frac{1}{\frac{k}{\ell}}} \cdot \frac{\ell}{k}\right)\right) \]

    3. clear-num 33.2%

       \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \left(\frac{1}{\frac{k}{\ell}} \cdot \color{blue}{\frac{1}{\frac{k}{\ell}}}\right)\right) \]

    4. frac-times 33.2%

       \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{\frac{1 \cdot 1}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}\right) \]

    5. metadata-eval 33.2%

       \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{\color{blue}{1}}{\frac{k}{\ell} \cdot \frac{k}{\ell}}\right) \]

13. Applied egg-rr 33.2%

    \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{\frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}\right) \]

14. Final simplification 33.2%

    \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell}}\right) \]
15. Add Preprocessing

Alternative 15: 30.1% accurate, 32.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \left(\frac{-0.16666666666666666}{t\_m} \cdot \left(\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}\right)\right)\right) \end{array} \]

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 (* 2.0 (* (/ -0.16666666666666666 t_m) (* (/ l k_m) (/ l 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) {
	return t_s * (2.0 * ((-0.16666666666666666 / t_m) * ((l / k_m) * (l / k_m))));
}

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 * (((-0.16666666666666666d0) / t_m) * ((l / k_m) * (l / k_m))))
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 * ((-0.16666666666666666 / t_m) * ((l / k_m) * (l / k_m))));
}

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 * ((-0.16666666666666666 / t_m) * ((l / k_m) * (l / k_m))))

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(-0.16666666666666666 / t_m) * Float64(Float64(l / k_m) * Float64(l / k_m)))))
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 * ((-0.16666666666666666 / t_m) * ((l / k_m) * (l / k_m))));
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[(-0.16666666666666666 / t$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 31.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]

   2. associate-/r* 31.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]

   3. sub-neg 31.3%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

   4. distribute-rgt-in 25.8%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + \left(-1\right) \cdot \tan k}} \]

   5. unpow2 25.8%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   6. times-frac 21.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   7. sqr-neg 21.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   8. times-frac 25.8%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   9. unpow2 25.8%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + \left(-1\right) \cdot \tan k} \]

   10. distribute-rgt-in 31.3%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\color{blue}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + \left(-1\right)\right)}} \]

   11. +-commutative 31.3%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 1\right)} + \left(-1\right)\right)} \]

   12. associate-+l+ 38.8%

       \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}} \]

3. Simplified 38.8%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 74.5%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
6. Step-by-step derivation

   1. times-frac 75.9%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

7. Simplified 75.9%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

8. Taylor expanded in k around 0 67.5%

   \[\leadsto 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \color{blue}{\left(\frac{1}{{k}^{2} \cdot t} - 0.16666666666666666 \cdot \frac{1}{t}\right)}\right) \]

9. Taylor expanded in k around inf 31.4%

   \[\leadsto 2 \cdot \color{blue}{\left(-0.16666666666666666 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]
10. Step-by-step derivation

    1. associate-*r/ 31.4%

       \[\leadsto 2 \cdot \color{blue}{\frac{-0.16666666666666666 \cdot {\ell}^{2}}{{k}^{2} \cdot t}} \]

    2. *-commutative 31.4%

       \[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{2}}} \]

    3. times-frac 31.5%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{t} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)} \]

    4. unpow2 31.5%

       \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \]

    5. unpow2 31.5%

       \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \]

    6. times-frac 33.2%

       \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right) \]

    7. unpow2 33.2%

       \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}\right) \]

11. Simplified 33.2%

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{t} \cdot {\left(\frac{\ell}{k}\right)}^{2}\right)} \]
12. Step-by-step derivation

    1. unpow2 33.2%

       \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right) \]

13. Applied egg-rr 33.2%

    \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right) \]

14. Final simplification 33.2%

    \[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(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))))