Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.0% → 99.5%

Time: 25.4s

Alternatives: 13

Speedup: 2.0×

• 39.4% 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: 36.0% 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: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 10^{-33}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k\_m \cdot \sqrt{t\_m}} \cdot \frac{\sqrt{\cos k\_m}}{\sin k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\frac{\ell}{k\_m}}{\sqrt{t\_m}}\right)}^{2} \cdot \frac{\cos k\_m}{{\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \]

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 1e-33)
    (*
     2.0
     (pow (* (/ l (* k_m (sqrt t_m))) (/ (sqrt (cos k_m)) (sin k_m))) 2.0))
    (*
     2.0
     (*
      (pow (/ (/ l k_m) (sqrt t_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 <= 1e-33) {
		tmp = 2.0 * pow(((l / (k_m * sqrt(t_m))) * (sqrt(cos(k_m)) / sin(k_m))), 2.0);
	} else {
		tmp = 2.0 * (pow(((l / k_m) / sqrt(t_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 <= 1d-33) then
        tmp = 2.0d0 * (((l / (k_m * sqrt(t_m))) * (sqrt(cos(k_m)) / sin(k_m))) ** 2.0d0)
    else
        tmp = 2.0d0 * ((((l / k_m) / sqrt(t_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 <= 1e-33) {
		tmp = 2.0 * Math.pow(((l / (k_m * Math.sqrt(t_m))) * (Math.sqrt(Math.cos(k_m)) / Math.sin(k_m))), 2.0);
	} else {
		tmp = 2.0 * (Math.pow(((l / k_m) / Math.sqrt(t_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 <= 1e-33:
		tmp = 2.0 * math.pow(((l / (k_m * math.sqrt(t_m))) * (math.sqrt(math.cos(k_m)) / math.sin(k_m))), 2.0)
	else:
		tmp = 2.0 * (math.pow(((l / k_m) / math.sqrt(t_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 <= 1e-33)
		tmp = Float64(2.0 * (Float64(Float64(l / Float64(k_m * sqrt(t_m))) * Float64(sqrt(cos(k_m)) / sin(k_m))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64((Float64(Float64(l / k_m) / sqrt(t_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 <= 1e-33)
		tmp = 2.0 * (((l / (k_m * sqrt(t_m))) * (sqrt(cos(k_m)) / sin(k_m))) ^ 2.0);
	else
		tmp = 2.0 * ((((l / k_m) / sqrt(t_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, 1e-33], N[(2.0 * N[Power[N[(N[(l / N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(N[(l / k$95$m), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $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 < 1.0000000000000001e-33

   1. Initial program 35.6%

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

      1. associate-*l* 35.6%

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

      2. associate-/r* 35.6%

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

      3. sub-neg 35.6%

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

      4. distribute-rgt-in 27.9%

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

      5. unpow2 27.9%

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

      6. times-frac 21.9%

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

      7. sqr-neg 21.9%

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

      8. times-frac 27.9%

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

      9. unpow2 27.9%

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

      10. distribute-rgt-in 35.6%

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

      11. +-commutative 35.6%

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

      12. associate-+l+ 42.5%

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

   3. Simplified 42.5%

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

   5. Taylor expanded in t around 0 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 75.7%

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

   7. Simplified 75.7%

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

   8. Taylor expanded in l around 0 73.2%

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

      1. associate-*r* 73.2%

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

      2. times-frac 76.1%

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

      3. *-commutative 76.1%

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

      4. associate-/r* 73.8%

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

   10. Simplified 73.8%

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

       1. add-sqr-sqrt 47.2%

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

       2. pow2 47.2%

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

   12. Applied egg-rr 43.9%

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

       1. associate-/l/ 44.9%

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

   14. Simplified 44.9%

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

   if 1.0000000000000001e-33 < k

   1. Initial program 33.8%

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

      1. associate-*l* 33.8%

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

      2. associate-/r* 33.8%

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

      3. sub-neg 33.8%

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

      4. distribute-rgt-in 33.8%

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

      5. unpow2 33.8%

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

      6. times-frac 26.7%

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

      7. sqr-neg 26.7%

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

      8. times-frac 33.8%

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

      9. unpow2 33.8%

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

      10. distribute-rgt-in 33.8%

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

      11. +-commutative 33.8%

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

      12. associate-+l+ 41.3%

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

   3. Simplified 41.3%

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

   5. Taylor expanded in t around 0 73.3%

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

      1. times-frac 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in l around 0 73.3%

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

      1. associate-*r* 73.3%

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

      2. times-frac 73.3%

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

      3. *-commutative 73.3%

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

      4. associate-/r* 75.4%

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

   10. Simplified 75.4%

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

       1. *-un-lft-identity 75.4%

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

       2. add-sqr-sqrt 58.5%

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

       3. pow2 58.5%

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

       4. sqrt-div 51.4%

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

       5. sqrt-div 37.4%

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

       6. unpow2 37.4%

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

       7. sqrt-prod 12.4%

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

       8. add-sqr-sqrt 38.8%

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

       9. unpow2 38.8%

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

       10. sqrt-prod 44.1%

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

       11. add-sqr-sqrt 44.2%

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

   12. Applied egg-rr 44.2%

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

       1. *-lft-identity 44.2%

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

       2. associate-/l/ 44.2%

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

       3. associate-/r* 44.2%

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

   14. Simplified 44.2%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-33}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k \cdot \sqrt{t}} \cdot \frac{\sqrt{\cos k}}{\sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\frac{\ell}{k}}{\sqrt{t}}\right)}^{2} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{k\_m \cdot \sqrt{t\_m}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.8 \cdot 10^{-64}:\\ \;\;\;\;2 \cdot {\left(t\_2 \cdot \frac{\sqrt{\cos k\_m}}{\sin k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k\_m}{{\sin k\_m}^{2}} \cdot {t\_2}^{2}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ l (* k_m (sqrt t_m)))))
   (*
    t_s
    (if (<= k_m 1.8e-64)
      (* 2.0 (pow (* t_2 (/ (sqrt (cos k_m)) (sin k_m))) 2.0))
      (* 2.0 (* (/ (cos k_m) (pow (sin k_m) 2.0)) (pow t_2 2.0)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = l / (k_m * sqrt(t_m));
	double tmp;
	if (k_m <= 1.8e-64) {
		tmp = 2.0 * pow((t_2 * (sqrt(cos(k_m)) / sin(k_m))), 2.0);
	} else {
		tmp = 2.0 * ((cos(k_m) / pow(sin(k_m), 2.0)) * pow(t_2, 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = l / (k_m * sqrt(t_m))
    if (k_m <= 1.8d-64) then
        tmp = 2.0d0 * ((t_2 * (sqrt(cos(k_m)) / sin(k_m))) ** 2.0d0)
    else
        tmp = 2.0d0 * ((cos(k_m) / (sin(k_m) ** 2.0d0)) * (t_2 ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = l / (k_m * Math.sqrt(t_m));
	double tmp;
	if (k_m <= 1.8e-64) {
		tmp = 2.0 * Math.pow((t_2 * (Math.sqrt(Math.cos(k_m)) / Math.sin(k_m))), 2.0);
	} else {
		tmp = 2.0 * ((Math.cos(k_m) / Math.pow(Math.sin(k_m), 2.0)) * Math.pow(t_2, 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = l / (k_m * math.sqrt(t_m))
	tmp = 0
	if k_m <= 1.8e-64:
		tmp = 2.0 * math.pow((t_2 * (math.sqrt(math.cos(k_m)) / math.sin(k_m))), 2.0)
	else:
		tmp = 2.0 * ((math.cos(k_m) / math.pow(math.sin(k_m), 2.0)) * math.pow(t_2, 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(l / Float64(k_m * sqrt(t_m)))
	tmp = 0.0
	if (k_m <= 1.8e-64)
		tmp = Float64(2.0 * (Float64(t_2 * Float64(sqrt(cos(k_m)) / sin(k_m))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) / (sin(k_m) ^ 2.0)) * (t_2 ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = l / (k_m * sqrt(t_m));
	tmp = 0.0;
	if (k_m <= 1.8e-64)
		tmp = 2.0 * ((t_2 * (sqrt(cos(k_m)) / sin(k_m))) ^ 2.0);
	else
		tmp = 2.0 * ((cos(k_m) / (sin(k_m) ^ 2.0)) * (t_2 ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(l / N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.8e-64], N[(2.0 * N[Power[N[(t$95$2 * N[(N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $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[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.7999999999999999e-64

   1. Initial program 36.4%

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

      1. associate-*l* 36.4%

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

      2. associate-/r* 36.4%

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

      3. sub-neg 36.4%

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

      4. distribute-rgt-in 28.2%

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

      5. unpow2 28.2%

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

      6. times-frac 21.9%

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

      7. sqr-neg 21.9%

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

      8. times-frac 28.2%

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

      9. unpow2 28.2%

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

      10. distribute-rgt-in 36.4%

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

      11. +-commutative 36.4%

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

      12. associate-+l+ 43.2%

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

   3. Simplified 43.2%

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

   5. Taylor expanded in t around 0 72.8%

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

      1. times-frac 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 l around 0 72.8%

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

      1. associate-*r* 72.8%

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

      2. times-frac 74.7%

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

      3. *-commutative 74.7%

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

      4. associate-/r* 72.3%

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

   10. Simplified 72.3%

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

       1. add-sqr-sqrt 48.2%

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

       2. pow2 48.2%

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

   12. Applied egg-rr 44.7%

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

       1. associate-/l/ 45.8%

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

   14. Simplified 45.8%

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

   if 1.7999999999999999e-64 < k

   1. Initial program 32.3%

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

      1. associate-*l* 32.3%

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

      2. associate-/r* 32.3%

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

      3. sub-neg 32.3%

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

      4. distribute-rgt-in 32.3%

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

      5. unpow2 32.3%

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

      6. times-frac 26.0%

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

      7. sqr-neg 26.0%

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

      8. times-frac 32.3%

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

      9. unpow2 32.3%

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

      10. distribute-rgt-in 32.3%

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

      11. +-commutative 32.3%

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

      12. associate-+l+ 40.1%

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

   3. Simplified 40.1%

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

   5. Taylor expanded in t around 0 74.2%

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

      1. times-frac 77.4%

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

   7. Simplified 77.4%

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

   8. Taylor expanded in l around 0 74.2%

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

      1. associate-*r* 74.1%

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

      2. times-frac 76.4%

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

      3. *-commutative 76.4%

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

      4. associate-/r* 78.3%

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

   10. Simplified 78.3%

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

       1. *-un-lft-identity 78.3%

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

       2. add-sqr-sqrt 55.0%

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

       3. pow2 55.0%

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

       4. sqrt-div 48.8%

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

       5. sqrt-div 36.5%

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

       6. unpow2 36.5%

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

       7. sqrt-prod 10.9%

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

       8. add-sqr-sqrt 37.7%

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

       9. unpow2 37.7%

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

       10. sqrt-prod 42.4%

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

       11. add-sqr-sqrt 42.4%

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

   12. Applied egg-rr 42.4%

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

       1. *-lft-identity 42.4%

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

       2. associate-/l/ 42.4%

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

   14. Simplified 42.4%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-64}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k \cdot \sqrt{t}} \cdot \frac{\sqrt{\cos k}}{\sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot {\left(\frac{\ell}{k \cdot \sqrt{t}}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.008:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k\_m \cdot \sqrt{t\_m}} \cdot \frac{\sqrt{\cos k\_m}}{\sin k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k\_m}{{\sin k\_m}^{2}} \cdot \frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m}\right)\\ \end{array} \end{array} \]

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 0.008)
    (*
     2.0
     (pow (* (/ l (* k_m (sqrt t_m))) (/ (sqrt (cos k_m)) (sin k_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 <= 0.008) {
		tmp = 2.0 * pow(((l / (k_m * sqrt(t_m))) * (sqrt(cos(k_m)) / sin(k_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 <= 0.008d0) then
        tmp = 2.0d0 * (((l / (k_m * sqrt(t_m))) * (sqrt(cos(k_m)) / sin(k_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 <= 0.008) {
		tmp = 2.0 * Math.pow(((l / (k_m * Math.sqrt(t_m))) * (Math.sqrt(Math.cos(k_m)) / Math.sin(k_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 <= 0.008:
		tmp = 2.0 * math.pow(((l / (k_m * math.sqrt(t_m))) * (math.sqrt(math.cos(k_m)) / math.sin(k_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 <= 0.008)
		tmp = Float64(2.0 * (Float64(Float64(l / Float64(k_m * sqrt(t_m))) * Float64(sqrt(cos(k_m)) / sin(k_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 <= 0.008)
		tmp = 2.0 * (((l / (k_m * sqrt(t_m))) * (sqrt(cos(k_m)) / sin(k_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, 0.008], N[(2.0 * N[Power[N[(N[(l / N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $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 < 0.0080000000000000002

   1. Initial program 35.4%

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

      1. associate-*l* 35.4%

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

      2. associate-/r* 35.4%

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

      3. sub-neg 35.4%

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

      4. distribute-rgt-in 27.9%

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

      5. unpow2 27.9%

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

      6. times-frac 21.6%

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

      7. sqr-neg 21.6%

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

      8. times-frac 27.9%

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

      9. unpow2 27.9%

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

      10. distribute-rgt-in 35.4%

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

      11. +-commutative 35.4%

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

      12. associate-+l+ 42.0%

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

   3. Simplified 42.0%

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

   5. Taylor expanded in t around 0 73.6%

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

      1. times-frac 76.0%

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

   7. Simplified 76.0%

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

   8. Taylor expanded in l around 0 73.6%

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

      1. associate-*r* 73.6%

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

      2. times-frac 76.4%

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

      3. *-commutative 76.4%

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

      4. associate-/r* 74.2%

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

   10. Simplified 74.2%

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

       1. add-sqr-sqrt 46.5%

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

       2. pow2 46.5%

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

   12. Applied egg-rr 43.3%

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

       1. associate-/l/ 44.3%

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

   14. Simplified 44.3%

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

   if 0.0080000000000000002 < k

   1. Initial program 34.4%

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

      1. associate-*l* 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* 34.4%

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

      3. sub-neg 34.4%

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

      4. distribute-rgt-in 34.4%

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

      5. unpow2 34.4%

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

      6. times-frac 28.1%

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

      7. sqr-neg 28.1%

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

      8. times-frac 34.4%

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

      9. unpow2 34.4%

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

      10. distribute-rgt-in 34.4%

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

      11. +-commutative 34.4%

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

      12. associate-+l+ 42.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in t around 0 72.0%

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

      1. times-frac 74.6%

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

   7. Simplified 74.6%

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

   8. Taylor expanded in l around 0 72.0%

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

      1. associate-*r* 71.9%

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

      2. times-frac 72.0%

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

      3. *-commutative 72.0%

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

      4. associate-/r* 74.4%

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

   10. Simplified 74.4%

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

       1. associate-*r/ 74.3%

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

       2. add-sqr-sqrt 61.8%

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

       3. pow2 61.8%

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

       4. sqrt-div 53.9%

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

       5. sqrt-div 38.4%

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

       6. unpow2 38.4%

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

       7. sqrt-prod 13.8%

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

       8. add-sqr-sqrt 39.9%

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

       9. unpow2 39.9%

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

       10. sqrt-prod 45.8%

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

       11. add-sqr-sqrt 45.9%

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

   12. Applied egg-rr 45.9%

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

       1. *-commutative 45.9%

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

       2. associate-/l* 45.9%

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

       3. associate-/l/ 45.9%

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

       4. associate-/r* 45.9%

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

   14. Simplified 45.9%

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

   15. Taylor expanded in k around inf 72.0%

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

       1. associate-*r* 71.9%

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

       2. times-frac 72.0%

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

       3. associate-/r* 74.6%

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

       4. unpow2 74.6%

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

       5. unpow2 74.6%

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

       6. times-frac 95.2%

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

       7. unpow2 95.2%

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

   17. Simplified 95.2%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.008:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k \cdot \sqrt{t}} \cdot \frac{\sqrt{\cos k}}{\sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 14:\\ \;\;\;\;2 \cdot {\left(\frac{\sqrt{\cos k\_m} \cdot \frac{\frac{\ell}{k\_m}}{\sqrt{t\_m}}}{\sin k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k\_m}{{\sin k\_m}^{2}} \cdot \frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m}\right)\\ \end{array} \end{array} \]

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 14.0)
    (*
     2.0
     (pow (/ (* (sqrt (cos k_m)) (/ (/ l k_m) (sqrt t_m))) (sin k_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 <= 14.0) {
		tmp = 2.0 * pow(((sqrt(cos(k_m)) * ((l / k_m) / sqrt(t_m))) / sin(k_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 <= 14.0d0) then
        tmp = 2.0d0 * (((sqrt(cos(k_m)) * ((l / k_m) / sqrt(t_m))) / sin(k_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 <= 14.0) {
		tmp = 2.0 * Math.pow(((Math.sqrt(Math.cos(k_m)) * ((l / k_m) / Math.sqrt(t_m))) / Math.sin(k_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 <= 14.0:
		tmp = 2.0 * math.pow(((math.sqrt(math.cos(k_m)) * ((l / k_m) / math.sqrt(t_m))) / math.sin(k_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 <= 14.0)
		tmp = Float64(2.0 * (Float64(Float64(sqrt(cos(k_m)) * Float64(Float64(l / k_m) / sqrt(t_m))) / sin(k_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 <= 14.0)
		tmp = 2.0 * (((sqrt(cos(k_m)) * ((l / k_m) / sqrt(t_m))) / sin(k_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, 14.0], N[(2.0 * N[Power[N[(N[(N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $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 < 14

   1. Initial program 35.2%

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

      1. associate-*l* 35.2%

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

      2. associate-/r* 35.2%

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

      3. sub-neg 35.2%

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

      4. distribute-rgt-in 27.8%

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

      5. unpow2 27.8%

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

      6. times-frac 21.5%

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

      7. sqr-neg 21.5%

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

      8. times-frac 27.8%

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

      9. unpow2 27.8%

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

      10. distribute-rgt-in 35.2%

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

      11. +-commutative 35.2%

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

      12. associate-+l+ 42.3%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in t around 0 73.8%

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

      1. times-frac 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in l around 0 73.8%

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

      1. associate-*r* 73.8%

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

      2. times-frac 76.5%

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

      3. *-commutative 76.5%

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

      4. associate-/r* 74.4%

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

   10. Simplified 74.4%

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

       1. add-sqr-sqrt 46.2%

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

       2. pow2 46.2%

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

   12. Applied egg-rr 43.1%

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

       1. associate-*r/ 43.1%

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

       2. associate-/l/ 44.1%

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

       3. associate-/r* 44.1%

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

   14. Simplified 44.1%

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

   if 14 < k

   1. Initial program 34.9%

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

      1. associate-*l* 34.8%

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

      2. associate-/r* 34.9%

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

      3. sub-neg 34.9%

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

      4. distribute-rgt-in 34.9%

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

      5. unpow2 34.9%

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

      6. times-frac 28.5%

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

      7. sqr-neg 28.5%

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

      8. times-frac 34.9%

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

      9. unpow2 34.9%

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

      10. distribute-rgt-in 34.9%

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

      11. +-commutative 34.9%

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

      12. associate-+l+ 41.8%

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

   3. Simplified 41.8%

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

   5. Taylor expanded in t around 0 71.6%

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

      1. times-frac 74.2%

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

   7. Simplified 74.2%

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

   8. Taylor expanded in l around 0 71.6%

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

      1. associate-*r* 71.5%

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

      2. times-frac 71.6%

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

      3. *-commutative 71.6%

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

      4. associate-/r* 74.0%

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

   10. Simplified 74.0%

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

       1. associate-*r/ 73.9%

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

       2. add-sqr-sqrt 62.7%

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

       3. pow2 62.7%

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

       4. sqrt-div 54.7%

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

       5. sqrt-div 39.0%

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

       6. unpow2 39.0%

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

       7. sqrt-prod 14.0%

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

       8. add-sqr-sqrt 40.6%

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

       9. unpow2 40.6%

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

       10. sqrt-prod 46.5%

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

       11. add-sqr-sqrt 46.6%

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

   12. Applied egg-rr 46.6%

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

       1. *-commutative 46.6%

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

       2. associate-/l* 46.6%

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

       3. associate-/l/ 46.6%

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

       4. associate-/r* 46.6%

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

   14. Simplified 46.6%

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

   15. Taylor expanded in k around inf 71.6%

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

       1. associate-*r* 71.5%

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

       2. times-frac 71.6%

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

       3. associate-/r* 74.2%

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

       4. unpow2 74.2%

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

       5. unpow2 74.2%

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

       6. times-frac 95.2%

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

       7. unpow2 95.2%

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

   17. Simplified 95.2%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 14:\\ \;\;\;\;2 \cdot {\left(\frac{\sqrt{\cos k} \cdot \frac{\frac{\ell}{k}}{\sqrt{t}}}{\sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.00048:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k\_m}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k\_m}{{\sin k\_m}^{2}} \cdot \frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m}\right)\\ \end{array} \end{array} \]

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 0.00048)
    (* 2.0 (pow (* (/ l (pow k_m 2.0)) (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 <= 0.00048) {
		tmp = 2.0 * pow(((l / pow(k_m, 2.0)) * 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 <= 0.00048d0) then
        tmp = 2.0d0 * (((l / (k_m ** 2.0d0)) * 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 <= 0.00048) {
		tmp = 2.0 * Math.pow(((l / Math.pow(k_m, 2.0)) * 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 <= 0.00048:
		tmp = 2.0 * math.pow(((l / math.pow(k_m, 2.0)) * 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 <= 0.00048)
		tmp = Float64(2.0 * (Float64(Float64(l / (k_m ^ 2.0)) * 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 <= 0.00048)
		tmp = 2.0 * (((l / (k_m ^ 2.0)) * 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, 0.00048], N[(2.0 * N[Power[N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $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 < 4.80000000000000012e-4

   1. Initial program 35.4%

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

      1. associate-*l* 35.4%

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

      2. associate-/r* 35.4%

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

      3. sub-neg 35.4%

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

      4. distribute-rgt-in 27.9%

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

      5. unpow2 27.9%

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

      6. times-frac 21.6%

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

      7. sqr-neg 21.6%

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

      8. times-frac 27.9%

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

      9. unpow2 27.9%

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

      10. distribute-rgt-in 35.4%

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

      11. +-commutative 35.4%

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

      12. associate-+l+ 42.0%

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

   3. Simplified 42.0%

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

   5. Taylor expanded in k around 0 65.3%

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

      1. *-commutative 65.3%

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

      2. associate-/r* 65.8%

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

   7. Simplified 65.8%

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

      1. add-sqr-sqrt 41.7%

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

      2. pow2 41.7%

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

      3. div-inv 41.7%

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

      4. pow-flip 41.7%

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

      5. metadata-eval 41.7%

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

   9. Applied egg-rr 41.7%

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

   10. Taylor expanded in l around 0 42.7%

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

   if 4.80000000000000012e-4 < k

   1. Initial program 34.4%

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

      1. associate-*l* 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* 34.4%

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

      3. sub-neg 34.4%

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

      4. distribute-rgt-in 34.4%

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

      5. unpow2 34.4%

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

      6. times-frac 28.1%

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

      7. sqr-neg 28.1%

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

      8. times-frac 34.4%

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

      9. unpow2 34.4%

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

      10. distribute-rgt-in 34.4%

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

      11. +-commutative 34.4%

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

      12. associate-+l+ 42.7%

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

   3. Simplified 42.7%

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

   5. Taylor expanded in t around 0 72.0%

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

      1. times-frac 74.6%

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

   7. Simplified 74.6%

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

   8. Taylor expanded in l around 0 72.0%

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

      1. associate-*r* 71.9%

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

      2. times-frac 72.0%

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

      3. *-commutative 72.0%

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

      4. associate-/r* 74.4%

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

   10. Simplified 74.4%

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

       1. associate-*r/ 74.3%

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

       2. add-sqr-sqrt 61.8%

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

       3. pow2 61.8%

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

       4. sqrt-div 53.9%

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

       5. sqrt-div 38.4%

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

       6. unpow2 38.4%

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

       7. sqrt-prod 13.8%

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

       8. add-sqr-sqrt 39.9%

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

       9. unpow2 39.9%

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

       10. sqrt-prod 45.8%

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

       11. add-sqr-sqrt 45.9%

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

   12. Applied egg-rr 45.9%

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

       1. *-commutative 45.9%

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

       2. associate-/l* 45.9%

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

       3. associate-/l/ 45.9%

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

       4. associate-/r* 45.9%

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

   14. Simplified 45.9%

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

   15. Taylor expanded in k around inf 72.0%

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

       1. associate-*r* 71.9%

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

       2. times-frac 72.0%

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

       3. associate-/r* 74.6%

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

       4. unpow2 74.6%

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

       5. unpow2 74.6%

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

       6. times-frac 95.2%

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

       7. unpow2 95.2%

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

   17. Simplified 95.2%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00048:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 27000000:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k\_m}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m} \cdot \left(-0.16666666666666666 + \frac{1}{{k\_m}^{2}}\right)\right)\\ \end{array} \end{array} \]

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 27000000.0)
    (* 2.0 (pow (* (/ l (pow k_m 2.0)) (sqrt (/ 1.0 t_m))) 2.0))
    (*
     2.0
     (*
      (/ (pow (/ l k_m) 2.0) t_m)
      (+ -0.16666666666666666 (/ 1.0 (pow 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 <= 27000000.0) {
		tmp = 2.0 * pow(((l / pow(k_m, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 * ((pow((l / k_m), 2.0) / t_m) * (-0.16666666666666666 + (1.0 / pow(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 <= 27000000.0d0) then
        tmp = 2.0d0 * (((l / (k_m ** 2.0d0)) * sqrt((1.0d0 / t_m))) ** 2.0d0)
    else
        tmp = 2.0d0 * ((((l / k_m) ** 2.0d0) / t_m) * ((-0.16666666666666666d0) + (1.0d0 / (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 <= 27000000.0) {
		tmp = 2.0 * Math.pow(((l / Math.pow(k_m, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 * ((Math.pow((l / k_m), 2.0) / t_m) * (-0.16666666666666666 + (1.0 / Math.pow(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 <= 27000000.0:
		tmp = 2.0 * math.pow(((l / math.pow(k_m, 2.0)) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = 2.0 * ((math.pow((l / k_m), 2.0) / t_m) * (-0.16666666666666666 + (1.0 / math.pow(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 <= 27000000.0)
		tmp = Float64(2.0 * (Float64(Float64(l / (k_m ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k_m) ^ 2.0) / t_m) * Float64(-0.16666666666666666 + Float64(1.0 / (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 <= 27000000.0)
		tmp = 2.0 * (((l / (k_m ^ 2.0)) * sqrt((1.0 / t_m))) ^ 2.0);
	else
		tmp = 2.0 * ((((l / k_m) ^ 2.0) / t_m) * (-0.16666666666666666 + (1.0 / (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, 27000000.0], N[(2.0 * N[Power[N[(N[(l / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(-0.16666666666666666 + N[(1.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.7e7

   1. Initial program 35.3%

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

      1. associate-*l* 35.3%

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

      2. associate-/r* 35.3%

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

      3. sub-neg 35.3%

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

      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 21.8%

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

      7. sqr-neg 21.8%

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

      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 35.3%

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

      11. +-commutative 35.3%

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

      12. associate-+l+ 42.4%

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

   3. Simplified 42.4%

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

   5. Taylor expanded in k around 0 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-/r* 65.5%

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

   7. Simplified 65.5%

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

      1. add-sqr-sqrt 41.5%

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

      2. pow2 41.5%

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

      3. div-inv 41.5%

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

      4. pow-flip 41.5%

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

      5. metadata-eval 41.5%

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

   9. Applied egg-rr 41.5%

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

   10. Taylor expanded in l around 0 42.0%

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

   if 2.7e7 < k

   1. Initial program 34.4%

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

      1. associate-*l* 34.4%

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

      2. associate-/r* 34.4%

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

      3. sub-neg 34.4%

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

      4. distribute-rgt-in 34.4%

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

      5. unpow2 34.4%

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

      6. times-frac 27.8%

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

      7. sqr-neg 27.8%

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

      8. times-frac 34.4%

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

      9. unpow2 34.4%

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

      10. distribute-rgt-in 34.4%

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

      11. +-commutative 34.4%

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

      12. associate-+l+ 41.5%

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

   3. Simplified 41.5%

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

   5. Taylor expanded in t around 0 70.6%

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

      1. times-frac 73.4%

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

   7. Simplified 73.4%

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

   8. Taylor expanded in k around 0 57.4%

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

      1. sub-neg 57.4%

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

      2. *-commutative 57.4%

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

      3. associate-/r* 57.4%

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

      4. associate-*r/ 57.4%

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

      5. metadata-eval 57.4%

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

      6. distribute-neg-frac 57.4%

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

      7. metadata-eval 57.4%

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

   10. Simplified 57.4%

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

   11. Taylor expanded in l around 0 57.2%

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

       1. associate-/l* 57.0%

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

       2. associate-/r* 57.0%

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

       3. associate-*r/ 57.0%

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

       4. metadata-eval 57.0%

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

       5. div-sub 57.0%

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

       6. associate-/l* 57.0%

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

       7. associate-/r/ 57.0%

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

       8. associate-/r* 57.4%

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

       9. unpow2 57.4%

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

       10. unpow2 57.4%

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

       11. times-frac 61.5%

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

       12. unpow2 61.5%

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

       13. sub-neg 61.5%

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

       14. metadata-eval 61.5%

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

       15. +-commutative 61.5%

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

   13. Simplified 61.5%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 27000000:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(-0.16666666666666666 + \frac{1}{{k}^{2}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 27000000:\\ \;\;\;\;2 \cdot {\left({k\_m}^{-2} \cdot \frac{\ell}{\sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m} \cdot \left(-0.16666666666666666 + \frac{1}{{k\_m}^{2}}\right)\right)\\ \end{array} \end{array} \]

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 27000000.0)
    (* 2.0 (pow (* (pow k_m -2.0) (/ l (sqrt t_m))) 2.0))
    (*
     2.0
     (*
      (/ (pow (/ l k_m) 2.0) t_m)
      (+ -0.16666666666666666 (/ 1.0 (pow 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 <= 27000000.0) {
		tmp = 2.0 * pow((pow(k_m, -2.0) * (l / sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * ((pow((l / k_m), 2.0) / t_m) * (-0.16666666666666666 + (1.0 / pow(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 <= 27000000.0d0) then
        tmp = 2.0d0 * (((k_m ** (-2.0d0)) * (l / sqrt(t_m))) ** 2.0d0)
    else
        tmp = 2.0d0 * ((((l / k_m) ** 2.0d0) / t_m) * ((-0.16666666666666666d0) + (1.0d0 / (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 <= 27000000.0) {
		tmp = 2.0 * Math.pow((Math.pow(k_m, -2.0) * (l / Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * ((Math.pow((l / k_m), 2.0) / t_m) * (-0.16666666666666666 + (1.0 / Math.pow(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 <= 27000000.0:
		tmp = 2.0 * math.pow((math.pow(k_m, -2.0) * (l / math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 * ((math.pow((l / k_m), 2.0) / t_m) * (-0.16666666666666666 + (1.0 / math.pow(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 <= 27000000.0)
		tmp = Float64(2.0 * (Float64((k_m ^ -2.0) * Float64(l / sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k_m) ^ 2.0) / t_m) * Float64(-0.16666666666666666 + Float64(1.0 / (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 <= 27000000.0)
		tmp = 2.0 * (((k_m ^ -2.0) * (l / sqrt(t_m))) ^ 2.0);
	else
		tmp = 2.0 * ((((l / k_m) ^ 2.0) / t_m) * (-0.16666666666666666 + (1.0 / (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, 27000000.0], N[(2.0 * N[Power[N[(N[Power[k$95$m, -2.0], $MachinePrecision] * N[(l / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(-0.16666666666666666 + N[(1.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.7e7

   1. Initial program 35.3%

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

      1. associate-*l* 35.3%

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

      2. associate-/r* 35.3%

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

      3. sub-neg 35.3%

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

      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 21.8%

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

      7. sqr-neg 21.8%

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

      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 35.3%

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

      11. +-commutative 35.3%

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

      12. associate-+l+ 42.4%

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

   3. Simplified 42.4%

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

   5. Taylor expanded in k around 0 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-/r* 65.5%

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

   7. Simplified 65.5%

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

      1. add-sqr-sqrt 41.5%

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

      2. pow2 41.5%

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

      3. div-inv 41.5%

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

      4. pow-flip 41.5%

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

      5. metadata-eval 41.5%

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

   9. Applied egg-rr 41.5%

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

       1. *-commutative 41.5%

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

       2. sqrt-prod 38.8%

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

       3. sqrt-pow1 40.5%

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

       4. metadata-eval 40.5%

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

       5. sqrt-div 36.5%

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

       6. unpow2 36.5%

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

       7. sqrt-prod 16.6%

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

       8. add-sqr-sqrt 42.0%

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

   11. Applied egg-rr 42.0%

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

   if 2.7e7 < k

   1. Initial program 34.4%

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

      1. associate-*l* 34.4%

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

      2. associate-/r* 34.4%

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

      3. sub-neg 34.4%

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

      4. distribute-rgt-in 34.4%

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

      5. unpow2 34.4%

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

      6. times-frac 27.8%

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

      7. sqr-neg 27.8%

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

      8. times-frac 34.4%

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

      9. unpow2 34.4%

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

      10. distribute-rgt-in 34.4%

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

      11. +-commutative 34.4%

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

      12. associate-+l+ 41.5%

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

   3. Simplified 41.5%

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

   5. Taylor expanded in t around 0 70.6%

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

      1. times-frac 73.4%

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

   7. Simplified 73.4%

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

   8. Taylor expanded in k around 0 57.4%

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

      1. sub-neg 57.4%

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

      2. *-commutative 57.4%

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

      3. associate-/r* 57.4%

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

      4. associate-*r/ 57.4%

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

      5. metadata-eval 57.4%

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

      6. distribute-neg-frac 57.4%

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

      7. metadata-eval 57.4%

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

   10. Simplified 57.4%

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

   11. Taylor expanded in l around 0 57.2%

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

       1. associate-/l* 57.0%

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

       2. associate-/r* 57.0%

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

       3. associate-*r/ 57.0%

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

       4. metadata-eval 57.0%

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

       5. div-sub 57.0%

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

       6. associate-/l* 57.0%

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

       7. associate-/r/ 57.0%

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

       8. associate-/r* 57.4%

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

       9. unpow2 57.4%

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

       10. unpow2 57.4%

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

       11. times-frac 61.5%

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

       12. unpow2 61.5%

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

       13. sub-neg 61.5%

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

       14. metadata-eval 61.5%

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

       15. +-commutative 61.5%

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

   13. Simplified 61.5%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 27000000:\\ \;\;\;\;2 \cdot {\left({k}^{-2} \cdot \frac{\ell}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(-0.16666666666666666 + \frac{1}{{k}^{2}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 27000000:\\ \;\;\;\;2 \cdot {\left(\frac{\ell \cdot {k\_m}^{-2}}{\sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m} \cdot \left(-0.16666666666666666 + \frac{1}{{k\_m}^{2}}\right)\right)\\ \end{array} \end{array} \]

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 27000000.0)
    (* 2.0 (pow (/ (* l (pow k_m -2.0)) (sqrt t_m)) 2.0))
    (*
     2.0
     (*
      (/ (pow (/ l k_m) 2.0) t_m)
      (+ -0.16666666666666666 (/ 1.0 (pow 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 <= 27000000.0) {
		tmp = 2.0 * pow(((l * pow(k_m, -2.0)) / sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 * ((pow((l / k_m), 2.0) / t_m) * (-0.16666666666666666 + (1.0 / pow(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 <= 27000000.0d0) then
        tmp = 2.0d0 * (((l * (k_m ** (-2.0d0))) / sqrt(t_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * ((((l / k_m) ** 2.0d0) / t_m) * ((-0.16666666666666666d0) + (1.0d0 / (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 <= 27000000.0) {
		tmp = 2.0 * Math.pow(((l * Math.pow(k_m, -2.0)) / Math.sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 * ((Math.pow((l / k_m), 2.0) / t_m) * (-0.16666666666666666 + (1.0 / Math.pow(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 <= 27000000.0:
		tmp = 2.0 * math.pow(((l * math.pow(k_m, -2.0)) / math.sqrt(t_m)), 2.0)
	else:
		tmp = 2.0 * ((math.pow((l / k_m), 2.0) / t_m) * (-0.16666666666666666 + (1.0 / math.pow(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 <= 27000000.0)
		tmp = Float64(2.0 * (Float64(Float64(l * (k_m ^ -2.0)) / sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k_m) ^ 2.0) / t_m) * Float64(-0.16666666666666666 + Float64(1.0 / (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 <= 27000000.0)
		tmp = 2.0 * (((l * (k_m ^ -2.0)) / sqrt(t_m)) ^ 2.0);
	else
		tmp = 2.0 * ((((l / k_m) ^ 2.0) / t_m) * (-0.16666666666666666 + (1.0 / (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, 27000000.0], N[(2.0 * N[Power[N[(N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(-0.16666666666666666 + N[(1.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.7e7

   1. Initial program 35.3%

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

      1. associate-*l* 35.3%

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

      2. associate-/r* 35.3%

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

      3. sub-neg 35.3%

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

      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 21.8%

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

      7. sqr-neg 21.8%

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

      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 35.3%

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

      11. +-commutative 35.3%

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

      12. associate-+l+ 42.4%

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

   3. Simplified 42.4%

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

   5. Taylor expanded in k around 0 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-/r* 65.5%

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

   7. Simplified 65.5%

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

      1. add-sqr-sqrt 41.5%

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

      2. pow2 41.5%

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

      3. div-inv 41.5%

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

      4. pow-flip 41.5%

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

      5. metadata-eval 41.5%

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

   9. Applied egg-rr 41.5%

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

       1. add-cbrt-cube 40.1%

          \[\leadsto 2 \cdot {\color{blue}{\left(\sqrt[3]{\left(\sqrt{\frac{{\ell}^{2}}{t} \cdot {k}^{-4}} \cdot \sqrt{\frac{{\ell}^{2}}{t} \cdot {k}^{-4}}\right) \cdot \sqrt{\frac{{\ell}^{2}}{t} \cdot {k}^{-4}}}\right)}}^{2} \]

       2. pow1/3 40.0%

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

       3. pow3 40.0%

          \[\leadsto 2 \cdot {\left({\color{blue}{\left({\left(\sqrt{\frac{{\ell}^{2}}{t} \cdot {k}^{-4}}\right)}^{3}\right)}}^{0.3333333333333333}\right)}^{2} \]

       4. sqrt-unprod 37.3%

          \[\leadsto 2 \cdot {\left({\left({\color{blue}{\left(\sqrt{\frac{{\ell}^{2}}{t}} \cdot \sqrt{{k}^{-4}}\right)}}^{3}\right)}^{0.3333333333333333}\right)}^{2} \]

       5. sqrt-div 32.8%

          \[\leadsto 2 \cdot {\left({\left({\left(\color{blue}{\frac{\sqrt{{\ell}^{2}}}{\sqrt{t}}} \cdot \sqrt{{k}^{-4}}\right)}^{3}\right)}^{0.3333333333333333}\right)}^{2} \]

       6. unpow2 32.8%

          \[\leadsto 2 \cdot {\left({\left({\left(\frac{\sqrt{\color{blue}{\ell \cdot \ell}}}{\sqrt{t}} \cdot \sqrt{{k}^{-4}}\right)}^{3}\right)}^{0.3333333333333333}\right)}^{2} \]

       7. sqrt-prod 14.6%

          \[\leadsto 2 \cdot {\left({\left({\left(\frac{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}{\sqrt{t}} \cdot \sqrt{{k}^{-4}}\right)}^{3}\right)}^{0.3333333333333333}\right)}^{2} \]

       8. add-sqr-sqrt 35.1%

          \[\leadsto 2 \cdot {\left({\left({\left(\frac{\color{blue}{\ell}}{\sqrt{t}} \cdot \sqrt{{k}^{-4}}\right)}^{3}\right)}^{0.3333333333333333}\right)}^{2} \]

       9. sqrt-pow1 37.2%

          \[\leadsto 2 \cdot {\left({\left({\left(\frac{\ell}{\sqrt{t}} \cdot \color{blue}{{k}^{\left(\frac{-4}{2}\right)}}\right)}^{3}\right)}^{0.3333333333333333}\right)}^{2} \]

       10. metadata-eval 37.2%

           \[\leadsto 2 \cdot {\left({\left({\left(\frac{\ell}{\sqrt{t}} \cdot {k}^{\color{blue}{-2}}\right)}^{3}\right)}^{0.3333333333333333}\right)}^{2} \]

   11. Applied egg-rr 37.2%

       \[\leadsto 2 \cdot {\color{blue}{\left({\left({\left(\frac{\ell}{\sqrt{t}} \cdot {k}^{-2}\right)}^{3}\right)}^{0.3333333333333333}\right)}}^{2} \]
   12. Step-by-step derivation

       1. pow-pow 42.0%

          \[\leadsto 2 \cdot {\color{blue}{\left({\left(\frac{\ell}{\sqrt{t}} \cdot {k}^{-2}\right)}^{\left(3 \cdot 0.3333333333333333\right)}\right)}}^{2} \]

       2. metadata-eval 42.0%

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

       3. pow1 42.0%

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

       4. associate-*l/ 41.6%

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

   13. Applied egg-rr 41.6%

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

   if 2.7e7 < k

   1. Initial program 34.4%

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

      1. associate-*l* 34.4%

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

      2. associate-/r* 34.4%

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

      3. sub-neg 34.4%

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

      4. distribute-rgt-in 34.4%

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

      5. unpow2 34.4%

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

      6. times-frac 27.8%

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

      7. sqr-neg 27.8%

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

      8. times-frac 34.4%

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

      9. unpow2 34.4%

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

      10. distribute-rgt-in 34.4%

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

      11. +-commutative 34.4%

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

      12. associate-+l+ 41.5%

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

   3. Simplified 41.5%

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

   5. Taylor expanded in t around 0 70.6%

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

      1. times-frac 73.4%

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

   7. Simplified 73.4%

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

   8. Taylor expanded in k around 0 57.4%

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

      1. sub-neg 57.4%

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

      2. *-commutative 57.4%

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

      3. associate-/r* 57.4%

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

      4. associate-*r/ 57.4%

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

      5. metadata-eval 57.4%

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

      6. distribute-neg-frac 57.4%

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

      7. metadata-eval 57.4%

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

   10. Simplified 57.4%

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

   11. Taylor expanded in l around 0 57.2%

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

       1. associate-/l* 57.0%

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

       2. associate-/r* 57.0%

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

       3. associate-*r/ 57.0%

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

       4. metadata-eval 57.0%

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

       5. div-sub 57.0%

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

       6. associate-/l* 57.0%

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

       7. associate-/r/ 57.0%

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

       8. associate-/r* 57.4%

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

       9. unpow2 57.4%

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

       10. unpow2 57.4%

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

       11. times-frac 61.5%

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

       12. unpow2 61.5%

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

       13. sub-neg 61.5%

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

       14. metadata-eval 61.5%

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

       15. +-commutative 61.5%

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

   13. Simplified 61.5%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 27000000:\\ \;\;\;\;2 \cdot {\left(\frac{\ell \cdot {k}^{-2}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(-0.16666666666666666 + \frac{1}{{k}^{2}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \left({\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \frac{\left(-0.16666666666666666\right) - \frac{-1}{{k\_m}^{2}}}{t\_m}\right)\right) \end{array} \]

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
   (*
    (pow (/ l k_m) 2.0)
    (/ (- (- 0.16666666666666666) (/ -1.0 (pow 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) {
	return t_s * (2.0 * (pow((l / k_m), 2.0) * ((-0.16666666666666666 - (-1.0 / pow(k_m, 2.0))) / t_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 * (((l / k_m) ** 2.0d0) * ((-0.16666666666666666d0 - ((-1.0d0) / (k_m ** 2.0d0))) / t_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 * (Math.pow((l / k_m), 2.0) * ((-0.16666666666666666 - (-1.0 / Math.pow(k_m, 2.0))) / t_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 * (math.pow((l / k_m), 2.0) * ((-0.16666666666666666 - (-1.0 / math.pow(k_m, 2.0))) / t_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(l / k_m) ^ 2.0) * Float64(Float64(Float64(-0.16666666666666666) - Float64(-1.0 / (k_m ^ 2.0))) / t_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 * (((l / k_m) ^ 2.0) * ((-0.16666666666666666 - (-1.0 / (k_m ^ 2.0))) / t_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[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[((-0.16666666666666666) - N[(-1.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-/r* 35.1%

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

   3. sub-neg 35.1%

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

   4. distribute-rgt-in 29.5%

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

   5. unpow2 29.5%

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

   6. times-frac 23.2%

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

   7. sqr-neg 23.2%

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

   8. times-frac 29.5%

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

   9. unpow2 29.5%

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

   10. distribute-rgt-in 35.1%

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

   11. +-commutative 35.1%

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

   12. associate-+l+ 42.2%

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

3. Simplified 42.2%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
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 75.7%

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

7. Simplified 75.7%

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

8. Taylor expanded in k around 0 68.0%

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

   1. sub-neg 68.0%

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

   2. *-commutative 68.0%

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

   3. associate-/r* 68.0%

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

   4. associate-*r/ 68.0%

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

   5. metadata-eval 68.0%

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

   6. distribute-neg-frac 68.0%

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

   7. metadata-eval 68.0%

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

10. Simplified 68.0%

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

11. Taylor expanded in t around -inf 68.2%

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

    1. mul-1-neg 68.2%

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

    2. times-frac 68.0%

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

    3. distribute-rgt-neg-in 68.0%

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

    4. unpow2 68.0%

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

    5. unpow2 68.0%

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

    6. times-frac 74.6%

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

    7. unpow2 74.6%

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

    8. sub-neg 74.6%

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

    9. distribute-neg-frac 74.6%

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

    10. metadata-eval 74.6%

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

13. Simplified 74.6%

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

14. Final simplification 74.6%

    \[\leadsto 2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\left(-0.16666666666666666\right) - \frac{-1}{{k}^{2}}}{t}\right) \]
15. Add Preprocessing

Alternative 10: 74.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \left(\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m} \cdot \left(-0.16666666666666666 + \frac{1}{{k\_m}^{2}}\right)\right)\right) \end{array} \]

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
   (*
    (/ (pow (/ l k_m) 2.0) t_m)
    (+ -0.16666666666666666 (/ 1.0 (pow 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) {
	return t_s * (2.0 * ((pow((l / k_m), 2.0) / t_m) * (-0.16666666666666666 + (1.0 / pow(k_m, 2.0)))));
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 * ((((l / k_m) ** 2.0d0) / t_m) * ((-0.16666666666666666d0) + (1.0d0 / (k_m ** 2.0d0)))))
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * ((Math.pow((l / k_m), 2.0) / t_m) * (-0.16666666666666666 + (1.0 / Math.pow(k_m, 2.0)))));
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 * ((math.pow((l / k_m), 2.0) / t_m) * (-0.16666666666666666 + (1.0 / math.pow(k_m, 2.0)))))

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 * Float64(Float64((Float64(l / k_m) ^ 2.0) / t_m) * Float64(-0.16666666666666666 + Float64(1.0 / (k_m ^ 2.0))))))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 * ((((l / k_m) ^ 2.0) / t_m) * (-0.16666666666666666 + (1.0 / (k_m ^ 2.0)))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 * N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(-0.16666666666666666 + N[(1.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-/r* 35.1%

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

   3. sub-neg 35.1%

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

   4. distribute-rgt-in 29.5%

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

   5. unpow2 29.5%

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

   6. times-frac 23.2%

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

   7. sqr-neg 23.2%

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

   8. times-frac 29.5%

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

   9. unpow2 29.5%

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

   10. distribute-rgt-in 35.1%

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

   11. +-commutative 35.1%

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

   12. associate-+l+ 42.2%

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

3. Simplified 42.2%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
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 75.7%

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

7. Simplified 75.7%

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

8. Taylor expanded in k around 0 68.0%

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

   1. sub-neg 68.0%

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

   2. *-commutative 68.0%

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

   3. associate-/r* 68.0%

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

   4. associate-*r/ 68.0%

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

   5. metadata-eval 68.0%

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

   6. distribute-neg-frac 68.0%

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

   7. metadata-eval 68.0%

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

10. Simplified 68.0%

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

11. Taylor expanded in l around 0 67.9%

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

    1. associate-/l* 66.1%

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

    2. associate-/r* 66.1%

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

    3. associate-*r/ 66.1%

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

    4. metadata-eval 66.1%

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

    5. div-sub 66.1%

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

    6. associate-/l* 66.1%

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

    7. associate-/r/ 68.2%

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

    8. associate-/r* 68.7%

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

    9. unpow2 68.7%

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

    10. unpow2 68.7%

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

    11. times-frac 74.5%

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

    12. unpow2 74.5%

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

    13. sub-neg 74.5%

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

    14. metadata-eval 74.5%

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

    15. +-commutative 74.5%

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

13. Simplified 74.5%

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

14. Final simplification 74.5%

    \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \left(-0.16666666666666666 + \frac{1}{{k}^{2}}\right)\right) \]
15. Add Preprocessing

Alternative 11: 73.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \left({\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \left(\frac{1}{t\_m} \cdot \left(-0.16666666666666666 + {k\_m}^{-2}\right)\right)\right)\right) \end{array} \]

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
   (*
    (pow (/ l k_m) 2.0)
    (* (/ 1.0 t_m) (+ -0.16666666666666666 (pow 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) {
	return t_s * (2.0 * (pow((l / k_m), 2.0) * ((1.0 / t_m) * (-0.16666666666666666 + pow(k_m, -2.0)))));
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 * (((l / k_m) ** 2.0d0) * ((1.0d0 / t_m) * ((-0.16666666666666666d0) + (k_m ** (-2.0d0))))))
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * (Math.pow((l / k_m), 2.0) * ((1.0 / t_m) * (-0.16666666666666666 + Math.pow(k_m, -2.0)))));
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 * (math.pow((l / k_m), 2.0) * ((1.0 / t_m) * (-0.16666666666666666 + math.pow(k_m, -2.0)))))

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 * Float64((Float64(l / k_m) ^ 2.0) * Float64(Float64(1.0 / t_m) * Float64(-0.16666666666666666 + (k_m ^ -2.0))))))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 * (((l / k_m) ^ 2.0) * ((1.0 / t_m) * (-0.16666666666666666 + (k_m ^ -2.0)))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(1.0 / t$95$m), $MachinePrecision] * N[(-0.16666666666666666 + N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-/r* 35.1%

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

   3. sub-neg 35.1%

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

   4. distribute-rgt-in 29.5%

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

   5. unpow2 29.5%

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

   6. times-frac 23.2%

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

   7. sqr-neg 23.2%

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

   8. times-frac 29.5%

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

   9. unpow2 29.5%

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

   10. distribute-rgt-in 35.1%

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

   11. +-commutative 35.1%

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

   12. associate-+l+ 42.2%

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

3. Simplified 42.2%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
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 75.7%

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

7. Simplified 75.7%

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

8. Taylor expanded in k around 0 68.0%

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

   1. sub-neg 68.0%

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

   2. *-commutative 68.0%

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

   3. associate-/r* 68.0%

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

   4. associate-*r/ 68.0%

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

   5. metadata-eval 68.0%

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

   6. distribute-neg-frac 68.0%

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

   7. metadata-eval 68.0%

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

10. Simplified 68.0%

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

    1. associate-*l/ 67.9%

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

    2. div-inv 67.9%

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

    3. fma-define 67.9%

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

    4. pow-flip 67.9%

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

    5. metadata-eval 67.9%

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

12. Applied egg-rr 67.9%

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

    1. associate-/l* 66.1%

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

    2. associate-/r/ 68.0%

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

    3. unpow2 68.0%

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

    4. unpow2 68.0%

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

    5. times-frac 74.6%

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

    6. unpow2 74.6%

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

    7. fma-undefine 74.6%

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

    8. metadata-eval 74.6%

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

    9. associate-*l/ 74.6%

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

    10. distribute-lft-out 74.6%

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

14. Simplified 74.6%

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

15. Final simplification 74.6%

    \[\leadsto 2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \left(\frac{1}{t} \cdot \left(-0.16666666666666666 + {k}^{-2}\right)\right)\right) \]
16. Add Preprocessing

Alternative 12: 64.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 5 \cdot 10^{+56}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t\_m} \cdot {k\_m}^{-4}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m} \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

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 5e+56)
    (* 2.0 (* (/ (pow l 2.0) t_m) (pow k_m -4.0)))
    (* 2.0 (* (/ (pow (/ l k_m) 2.0) t_m) -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 (k_m <= 5e+56) {
		tmp = 2.0 * ((pow(l, 2.0) / t_m) * pow(k_m, -4.0));
	} else {
		tmp = 2.0 * ((pow((l / k_m), 2.0) / t_m) * -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 (k_m <= 5d+56) then
        tmp = 2.0d0 * (((l ** 2.0d0) / t_m) * (k_m ** (-4.0d0)))
    else
        tmp = 2.0d0 * ((((l / k_m) ** 2.0d0) / t_m) * (-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 (k_m <= 5e+56) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / t_m) * Math.pow(k_m, -4.0));
	} else {
		tmp = 2.0 * ((Math.pow((l / k_m), 2.0) / t_m) * -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 k_m <= 5e+56:
		tmp = 2.0 * ((math.pow(l, 2.0) / t_m) * math.pow(k_m, -4.0))
	else:
		tmp = 2.0 * ((math.pow((l / k_m), 2.0) / t_m) * -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 (k_m <= 5e+56)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / t_m) * (k_m ^ -4.0)));
	else
		tmp = Float64(2.0 * Float64(Float64((Float64(l / k_m) ^ 2.0) / t_m) * -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 (k_m <= 5e+56)
		tmp = 2.0 * (((l ^ 2.0) / t_m) * (k_m ^ -4.0));
	else
		tmp = 2.0 * ((((l / k_m) ^ 2.0) / t_m) * -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[k$95$m, 5e+56], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.00000000000000024e56

   1. Initial program 35.6%

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

      1. associate-*l* 35.6%

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

      2. associate-/r* 35.6%

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

      3. sub-neg 35.6%

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

      4. distribute-rgt-in 28.5%

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

      5. unpow2 28.5%

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

      6. times-frac 22.5%

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

      7. sqr-neg 22.5%

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

      8. times-frac 28.5%

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

      9. unpow2 28.5%

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

      10. distribute-rgt-in 35.6%

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

      11. +-commutative 35.6%

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

      12. associate-+l+ 42.4%

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

   3. Simplified 42.4%

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

   5. Taylor expanded in k around 0 64.4%

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

      1. *-commutative 64.4%

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

      2. associate-/r* 65.3%

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

   7. Simplified 65.3%

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

      1. div-inv 65.3%

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

      2. pow-flip 65.3%

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

      3. metadata-eval 65.3%

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

   9. Applied egg-rr 65.3%

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

   if 5.00000000000000024e56 < k

   1. Initial program 33.3%

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

      1. associate-*l* 33.3%

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

      2. associate-/r* 33.3%

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

      3. sub-neg 33.3%

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

      4. distribute-rgt-in 33.3%

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

      5. unpow2 33.3%

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

      6. times-frac 25.9%

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

      7. sqr-neg 25.9%

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

      8. times-frac 33.3%

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

      9. unpow2 33.3%

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

      10. distribute-rgt-in 33.3%

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

      11. +-commutative 33.3%

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

      12. associate-+l+ 41.4%

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

   3. Simplified 41.4%

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

   5. Taylor expanded in t around 0 66.9%

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

      1. times-frac 70.0%

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

   7. Simplified 70.0%

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

   8. Taylor expanded in k around 0 58.9%

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

      1. sub-neg 58.9%

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

      2. *-commutative 58.9%

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

      3. associate-/r* 58.9%

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

      4. associate-*r/ 58.9%

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

      5. metadata-eval 58.9%

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

      6. distribute-neg-frac 58.9%

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

      7. metadata-eval 58.9%

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

   10. Simplified 58.9%

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

   11. Taylor expanded in k around inf 58.4%

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

       1. *-commutative 58.4%

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

       2. associate-/r* 58.9%

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

       3. unpow2 58.9%

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

       4. unpow2 58.9%

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

       5. times-frac 63.5%

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

       6. unpow2 63.5%

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

   13. Simplified 63.5%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+56}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 31.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \left(\frac{{\left(\frac{\ell}{k\_m}\right)}^{2}}{t\_m} \cdot -0.16666666666666666\right)\right) \end{array} \]

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 (* (/ (pow (/ l k_m) 2.0) t_m) -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) {
	return t_s * (2.0 * ((pow((l / k_m), 2.0) / t_m) * -0.16666666666666666));
}

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 * ((((l / k_m) ** 2.0d0) / t_m) * (-0.16666666666666666d0)))
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * ((Math.pow((l / k_m), 2.0) / t_m) * -0.16666666666666666));
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 * ((math.pow((l / k_m), 2.0) / t_m) * -0.16666666666666666))

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((Float64(l / k_m) ^ 2.0) / t_m) * -0.16666666666666666)))
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 * ((((l / k_m) ^ 2.0) / t_m) * -0.16666666666666666));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 * N[(N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.1%

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

   1. associate-*l* 35.1%

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

   2. associate-/r* 35.1%

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

   3. sub-neg 35.1%

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

   4. distribute-rgt-in 29.5%

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

   5. unpow2 29.5%

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

   6. times-frac 23.2%

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

   7. sqr-neg 23.2%

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

   8. times-frac 29.5%

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

   9. unpow2 29.5%

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

   10. distribute-rgt-in 35.1%

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

   11. +-commutative 35.1%

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

   12. associate-+l+ 42.2%

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

3. Simplified 42.2%

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]
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 75.7%

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

7. Simplified 75.7%

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

8. Taylor expanded in k around 0 68.0%

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

   1. sub-neg 68.0%

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

   2. *-commutative 68.0%

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

   3. associate-/r* 68.0%

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

   4. associate-*r/ 68.0%

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

   5. metadata-eval 68.0%

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

   6. distribute-neg-frac 68.0%

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

   7. metadata-eval 68.0%

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

10. Simplified 68.0%

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

11. Taylor expanded in k around inf 31.7%

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

    1. *-commutative 31.7%

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

    2. associate-/r* 31.8%

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

    3. unpow2 31.8%

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

    4. unpow2 31.8%

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

    5. times-frac 33.4%

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

    6. unpow2 33.4%

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

13. Simplified 33.4%

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

14. Final simplification 33.4%

    \[\leadsto 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot -0.16666666666666666\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(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))))