Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.7% → 92.5%

Time: 32.3s

Alternatives: 19

Speedup: 3.8×

• 40.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.7% 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: 92.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-314}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(\sqrt{t\_m} \cdot \sin k\right)}{{\ell}^{2}} \cdot \frac{\sqrt{t\_m} \cdot \left(k \cdot \sin k\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t\_m \cdot \sin k\right)}^{-2}\right) \cdot \left(t\_m \cdot \frac{\ell}{k}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1e-314)
    (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t_m)) 2.0))
    (if (<= (* l l) 2e+299)
      (/
       2.0
       (*
        (/ (* k (* (sqrt t_m) (sin k))) (pow l 2.0))
        (/ (* (sqrt t_m) (* k (sin k))) (cos k))))
      (*
       (/ l k)
       (* 2.0 (* (* (cos k) (pow (* t_m (sin k)) -2.0)) (* t_m (/ l k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-314) {
		tmp = 2.0 / pow(((pow(k, 2.0) / l) * sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+299) {
		tmp = 2.0 / (((k * (sqrt(t_m) * sin(k))) / pow(l, 2.0)) * ((sqrt(t_m) * (k * sin(k))) / cos(k)));
	} else {
		tmp = (l / k) * (2.0 * ((cos(k) * pow((t_m * sin(k)), -2.0)) * (t_m * (l / k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d-314) then
        tmp = 2.0d0 / ((((k ** 2.0d0) / l) * sqrt(t_m)) ** 2.0d0)
    else if ((l * l) <= 2d+299) then
        tmp = 2.0d0 / (((k * (sqrt(t_m) * sin(k))) / (l ** 2.0d0)) * ((sqrt(t_m) * (k * sin(k))) / cos(k)))
    else
        tmp = (l / k) * (2.0d0 * ((cos(k) * ((t_m * sin(k)) ** (-2.0d0))) * (t_m * (l / k))))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if ((l * l) <= 1e-314) {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+299) {
		tmp = 2.0 / (((k * (Math.sqrt(t_m) * Math.sin(k))) / Math.pow(l, 2.0)) * ((Math.sqrt(t_m) * (k * Math.sin(k))) / Math.cos(k)));
	} else {
		tmp = (l / k) * (2.0 * ((Math.cos(k) * Math.pow((t_m * Math.sin(k)), -2.0)) * (t_m * (l / k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 1e-314:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) / l) * math.sqrt(t_m)), 2.0)
	elif (l * l) <= 2e+299:
		tmp = 2.0 / (((k * (math.sqrt(t_m) * math.sin(k))) / math.pow(l, 2.0)) * ((math.sqrt(t_m) * (k * math.sin(k))) / math.cos(k)))
	else:
		tmp = (l / k) * (2.0 * ((math.cos(k) * math.pow((t_m * math.sin(k)), -2.0)) * (t_m * (l / k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-314)
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	elseif (Float64(l * l) <= 2e+299)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(sqrt(t_m) * sin(k))) / (l ^ 2.0)) * Float64(Float64(sqrt(t_m) * Float64(k * sin(k))) / cos(k))));
	else
		tmp = Float64(Float64(l / k) * Float64(2.0 * Float64(Float64(cos(k) * (Float64(t_m * sin(k)) ^ -2.0)) * Float64(t_m * Float64(l / k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-314)
		tmp = 2.0 / ((((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0);
	elseif ((l * l) <= 2e+299)
		tmp = 2.0 / (((k * (sqrt(t_m) * sin(k))) / (l ^ 2.0)) * ((sqrt(t_m) * (k * sin(k))) / cos(k)));
	else
		tmp = (l / k) * (2.0 * ((cos(k) * ((t_m * sin(k)) ^ -2.0)) * (t_m * (l / k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-314], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+299], N[(2.0 / N[(N[(N[(k * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.9999999996e-315

   1. Initial program 29.6%

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

   3. Taylor expanded in t around 0 68.9%

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

      1. associate-/l* 68.9%

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

   5. Simplified 68.9%

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

      1. add-sqr-sqrt 30.1%

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

      2. pow2 30.1%

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

   7. Applied egg-rr 32.7%

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

      1. associate-/r* 32.7%

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

   9. Simplified 32.7%

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

   10. Taylor expanded in k around 0 52.9%

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

   if 9.9999999996e-315 < (*.f64 l l) < 2.0000000000000001e299

   1. Initial program 39.5%

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

   3. Taylor expanded in t around 0 84.1%

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

      1. add-sqr-sqrt 42.8%

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

      2. times-frac 42.8%

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

      3. sqrt-prod 42.8%

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

      4. sqrt-pow1 36.9%

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

      5. metadata-eval 36.9%

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

      6. pow1 36.9%

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

      7. *-commutative 36.9%

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

      8. sqrt-prod 29.3%

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

      9. sqrt-pow1 37.7%

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

      10. metadata-eval 37.7%

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

      11. pow1 37.7%

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

   5. Applied egg-rr 49.8%

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

   6. Taylor expanded in k around inf 49.8%

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

   if 2.0000000000000001e299 < (*.f64 l l)

   1. Initial program 44.5%

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

      1. associate-/r* 44.5%

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

      2. associate-/l/ 44.5%

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

      3. associate-*l/ 44.5%

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

      4. associate-/r/ 44.5%

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

      5. +-commutative 44.5%

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

      6. unpow2 44.5%

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

      7. sqr-neg 44.5%

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

      8. distribute-frac-neg2 44.5%

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

      9. distribute-frac-neg2 44.5%

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

      10. unpow2 44.5%

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

      11. +-rgt-identity 44.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 44.5%

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

      13. associate--l+ 44.5%

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

      14. +-commutative 44.5%

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

      15. associate--l+ 44.5%

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

   3. Simplified 44.5%

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

      1. associate-*r* 52.4%

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

      2. unpow2 52.4%

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

      3. times-frac 67.4%

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

      4. div-inv 67.4%

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

      5. frac-times 67.4%

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

      6. metadata-eval 67.4%

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

   6. Applied egg-rr 67.4%

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

   7. Taylor expanded in k around inf 63.8%

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

      1. times-frac 72.5%

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

      2. *-commutative 72.5%

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

   9. Simplified 72.5%

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

       1. associate-*r/ 71.0%

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

       2. associate-*r* 71.0%

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

       3. div-inv 71.0%

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

       4. pow-prod-down 74.7%

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

       5. pow-flip 74.8%

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

       6. metadata-eval 74.8%

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

   11. Applied egg-rr 74.8%

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

       1. associate-/l* 76.3%

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

       2. associate-/r/ 81.7%

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

       3. *-commutative 81.7%

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

       4. associate-*l* 83.4%

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

       5. *-commutative 83.4%

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

       6. associate-*l* 83.4%

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

       7. *-commutative 83.4%

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

   13. Simplified 83.4%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-314}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(\sqrt{t} \cdot \sin k\right)}{{\ell}^{2}} \cdot \frac{\sqrt{t} \cdot \left(k \cdot \sin k\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t \cdot \sin k\right)}^{-2}\right) \cdot \left(t \cdot \frac{\ell}{k}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 1.5 \cdot 10^{-210}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(\sin k \cdot \frac{\sqrt{t\_m}}{{\ell}^{2}}\right)\right) \cdot \left(k \cdot \frac{\sqrt{t\_m} \cdot \sin k}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t\_m \cdot \sin k\right)}^{-2}\right) \cdot \left(t\_m \cdot \frac{\ell}{k}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1.5e-210)
    (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t_m)) 2.0))
    (if (<= (* l l) 2e+299)
      (/
       2.0
       (*
        (* k (* (sin k) (/ (sqrt t_m) (pow l 2.0))))
        (* k (/ (* (sqrt t_m) (sin k)) (cos k)))))
      (*
       (/ l k)
       (* 2.0 (* (* (cos k) (pow (* t_m (sin k)) -2.0)) (* t_m (/ l k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1.5e-210) {
		tmp = 2.0 / pow(((pow(k, 2.0) / l) * sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+299) {
		tmp = 2.0 / ((k * (sin(k) * (sqrt(t_m) / pow(l, 2.0)))) * (k * ((sqrt(t_m) * sin(k)) / cos(k))));
	} else {
		tmp = (l / k) * (2.0 * ((cos(k) * pow((t_m * sin(k)), -2.0)) * (t_m * (l / k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1.5d-210) then
        tmp = 2.0d0 / ((((k ** 2.0d0) / l) * sqrt(t_m)) ** 2.0d0)
    else if ((l * l) <= 2d+299) then
        tmp = 2.0d0 / ((k * (sin(k) * (sqrt(t_m) / (l ** 2.0d0)))) * (k * ((sqrt(t_m) * sin(k)) / cos(k))))
    else
        tmp = (l / k) * (2.0d0 * ((cos(k) * ((t_m * sin(k)) ** (-2.0d0))) * (t_m * (l / k))))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if ((l * l) <= 1.5e-210) {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+299) {
		tmp = 2.0 / ((k * (Math.sin(k) * (Math.sqrt(t_m) / Math.pow(l, 2.0)))) * (k * ((Math.sqrt(t_m) * Math.sin(k)) / Math.cos(k))));
	} else {
		tmp = (l / k) * (2.0 * ((Math.cos(k) * Math.pow((t_m * Math.sin(k)), -2.0)) * (t_m * (l / k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 1.5e-210:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) / l) * math.sqrt(t_m)), 2.0)
	elif (l * l) <= 2e+299:
		tmp = 2.0 / ((k * (math.sin(k) * (math.sqrt(t_m) / math.pow(l, 2.0)))) * (k * ((math.sqrt(t_m) * math.sin(k)) / math.cos(k))))
	else:
		tmp = (l / k) * (2.0 * ((math.cos(k) * math.pow((t_m * math.sin(k)), -2.0)) * (t_m * (l / k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1.5e-210)
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	elseif (Float64(l * l) <= 2e+299)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(sin(k) * Float64(sqrt(t_m) / (l ^ 2.0)))) * Float64(k * Float64(Float64(sqrt(t_m) * sin(k)) / cos(k)))));
	else
		tmp = Float64(Float64(l / k) * Float64(2.0 * Float64(Float64(cos(k) * (Float64(t_m * sin(k)) ^ -2.0)) * Float64(t_m * Float64(l / k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 1.5e-210)
		tmp = 2.0 / ((((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0);
	elseif ((l * l) <= 2e+299)
		tmp = 2.0 / ((k * (sin(k) * (sqrt(t_m) / (l ^ 2.0)))) * (k * ((sqrt(t_m) * sin(k)) / cos(k))));
	else
		tmp = (l / k) * (2.0 * ((cos(k) * ((t_m * sin(k)) ^ -2.0)) * (t_m * (l / k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1.5e-210], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+299], N[(2.0 / N[(N[(k * N[(N[Sin[k], $MachinePrecision] * N[(N[Sqrt[t$95$m], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[(N[Sqrt[t$95$m], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1.5000000000000001e-210

   1. Initial program 31.9%

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

   3. Taylor expanded in t around 0 74.9%

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

      1. associate-/l* 74.9%

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

   5. Simplified 74.9%

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

      1. add-sqr-sqrt 35.7%

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

      2. pow2 35.7%

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

   7. Applied egg-rr 36.7%

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

      1. associate-/r* 36.7%

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

   9. Simplified 36.7%

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

   10. Taylor expanded in k around 0 56.4%

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

   if 1.5000000000000001e-210 < (*.f64 l l) < 2.0000000000000001e299

   1. Initial program 39.3%

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

   3. Taylor expanded in t around 0 81.8%

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

      1. add-sqr-sqrt 37.8%

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

      2. times-frac 37.8%

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

      3. sqrt-prod 37.8%

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

      4. sqrt-pow1 31.0%

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

      5. metadata-eval 31.0%

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

      6. pow1 31.0%

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

      7. *-commutative 31.0%

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

      8. sqrt-prod 23.2%

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

      9. sqrt-pow1 32.8%

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

      10. metadata-eval 32.8%

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

      11. pow1 32.8%

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

   5. Applied egg-rr 45.8%

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

      1. associate-/l* 44.2%

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

      2. associate-/l* 44.2%

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

      3. associate-/l* 44.2%

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

   7. Simplified 44.2%

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

   if 2.0000000000000001e299 < (*.f64 l l)

   1. Initial program 44.5%

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

      1. associate-/r* 44.5%

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

      2. associate-/l/ 44.5%

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

      3. associate-*l/ 44.5%

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

      4. associate-/r/ 44.5%

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

      5. +-commutative 44.5%

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

      6. unpow2 44.5%

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

      7. sqr-neg 44.5%

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

      8. distribute-frac-neg2 44.5%

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

      9. distribute-frac-neg2 44.5%

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

      10. unpow2 44.5%

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

      11. +-rgt-identity 44.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 44.5%

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

      13. associate--l+ 44.5%

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

      14. +-commutative 44.5%

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

      15. associate--l+ 44.5%

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

   3. Simplified 44.5%

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

      1. associate-*r* 52.4%

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

      2. unpow2 52.4%

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

      3. times-frac 67.4%

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

      4. div-inv 67.4%

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

      5. frac-times 67.4%

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

      6. metadata-eval 67.4%

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

   6. Applied egg-rr 67.4%

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

   7. Taylor expanded in k around inf 63.8%

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

      1. times-frac 72.5%

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

      2. *-commutative 72.5%

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

   9. Simplified 72.5%

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

       1. associate-*r/ 71.0%

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

       2. associate-*r* 71.0%

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

       3. div-inv 71.0%

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

       4. pow-prod-down 74.7%

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

       5. pow-flip 74.8%

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

       6. metadata-eval 74.8%

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

   11. Applied egg-rr 74.8%

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

       1. associate-/l* 76.3%

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

       2. associate-/r/ 81.7%

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

       3. *-commutative 81.7%

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

       4. associate-*l* 83.4%

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

       5. *-commutative 83.4%

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

       6. associate-*l* 83.4%

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

       7. *-commutative 83.4%

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

   13. Simplified 83.4%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 1.5 \cdot 10^{-210}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(\sin k \cdot \frac{\sqrt{t}}{{\ell}^{2}}\right)\right) \cdot \left(k \cdot \frac{\sqrt{t} \cdot \sin k}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t \cdot \sin k\right)}^{-2}\right) \cdot \left(t \cdot \frac{\ell}{k}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-314}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{t\_m} \cdot \left(k \cdot \sin k\right)}{\cos k} \cdot \left(\left(k \cdot \left(\sqrt{t\_m} \cdot \sin k\right)\right) \cdot {\ell}^{-2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t\_m \cdot \sin k\right)}^{-2}\right) \cdot \left(t\_m \cdot \frac{\ell}{k}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1e-314)
    (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t_m)) 2.0))
    (if (<= (* l l) 2e+299)
      (/
       2.0
       (*
        (/ (* (sqrt t_m) (* k (sin k))) (cos k))
        (* (* k (* (sqrt t_m) (sin k))) (pow l -2.0))))
      (*
       (/ l k)
       (* 2.0 (* (* (cos k) (pow (* t_m (sin k)) -2.0)) (* t_m (/ l k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-314) {
		tmp = 2.0 / pow(((pow(k, 2.0) / l) * sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+299) {
		tmp = 2.0 / (((sqrt(t_m) * (k * sin(k))) / cos(k)) * ((k * (sqrt(t_m) * sin(k))) * pow(l, -2.0)));
	} else {
		tmp = (l / k) * (2.0 * ((cos(k) * pow((t_m * sin(k)), -2.0)) * (t_m * (l / k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d-314) then
        tmp = 2.0d0 / ((((k ** 2.0d0) / l) * sqrt(t_m)) ** 2.0d0)
    else if ((l * l) <= 2d+299) then
        tmp = 2.0d0 / (((sqrt(t_m) * (k * sin(k))) / cos(k)) * ((k * (sqrt(t_m) * sin(k))) * (l ** (-2.0d0))))
    else
        tmp = (l / k) * (2.0d0 * ((cos(k) * ((t_m * sin(k)) ** (-2.0d0))) * (t_m * (l / k))))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if ((l * l) <= 1e-314) {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+299) {
		tmp = 2.0 / (((Math.sqrt(t_m) * (k * Math.sin(k))) / Math.cos(k)) * ((k * (Math.sqrt(t_m) * Math.sin(k))) * Math.pow(l, -2.0)));
	} else {
		tmp = (l / k) * (2.0 * ((Math.cos(k) * Math.pow((t_m * Math.sin(k)), -2.0)) * (t_m * (l / k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 1e-314:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) / l) * math.sqrt(t_m)), 2.0)
	elif (l * l) <= 2e+299:
		tmp = 2.0 / (((math.sqrt(t_m) * (k * math.sin(k))) / math.cos(k)) * ((k * (math.sqrt(t_m) * math.sin(k))) * math.pow(l, -2.0)))
	else:
		tmp = (l / k) * (2.0 * ((math.cos(k) * math.pow((t_m * math.sin(k)), -2.0)) * (t_m * (l / k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-314)
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	elseif (Float64(l * l) <= 2e+299)
		tmp = Float64(2.0 / Float64(Float64(Float64(sqrt(t_m) * Float64(k * sin(k))) / cos(k)) * Float64(Float64(k * Float64(sqrt(t_m) * sin(k))) * (l ^ -2.0))));
	else
		tmp = Float64(Float64(l / k) * Float64(2.0 * Float64(Float64(cos(k) * (Float64(t_m * sin(k)) ^ -2.0)) * Float64(t_m * Float64(l / k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-314)
		tmp = 2.0 / ((((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0);
	elseif ((l * l) <= 2e+299)
		tmp = 2.0 / (((sqrt(t_m) * (k * sin(k))) / cos(k)) * ((k * (sqrt(t_m) * sin(k))) * (l ^ -2.0)));
	else
		tmp = (l / k) * (2.0 * ((cos(k) * ((t_m * sin(k)) ^ -2.0)) * (t_m * (l / k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-314], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+299], N[(2.0 / N[(N[(N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.9999999996e-315

   1. Initial program 29.6%

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

   3. Taylor expanded in t around 0 68.9%

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

      1. associate-/l* 68.9%

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

   5. Simplified 68.9%

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

      1. add-sqr-sqrt 30.1%

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

      2. pow2 30.1%

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

   7. Applied egg-rr 32.7%

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

      1. associate-/r* 32.7%

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

   9. Simplified 32.7%

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

   10. Taylor expanded in k around 0 52.9%

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

   if 9.9999999996e-315 < (*.f64 l l) < 2.0000000000000001e299

   1. Initial program 39.5%

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

   3. Taylor expanded in t around 0 84.1%

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

      1. add-sqr-sqrt 42.8%

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

      2. times-frac 42.8%

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

      3. sqrt-prod 42.8%

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

      4. sqrt-pow1 36.9%

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

      5. metadata-eval 36.9%

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

      6. pow1 36.9%

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

      7. *-commutative 36.9%

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

      8. sqrt-prod 29.3%

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

      9. sqrt-pow1 37.7%

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

      10. metadata-eval 37.7%

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

      11. pow1 37.7%

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

   5. Applied egg-rr 49.8%

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

   6. Taylor expanded in k around inf 49.8%

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

      1. pow2 49.8%

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

      2. div-inv 49.8%

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

      3. pow2 49.8%

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

      4. pow-flip 49.8%

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

      5. metadata-eval 49.8%

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

   8. Applied egg-rr 49.8%

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

   if 2.0000000000000001e299 < (*.f64 l l)

   1. Initial program 44.5%

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

      1. associate-/r* 44.5%

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

      2. associate-/l/ 44.5%

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

      3. associate-*l/ 44.5%

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

      4. associate-/r/ 44.5%

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

      5. +-commutative 44.5%

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

      6. unpow2 44.5%

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

      7. sqr-neg 44.5%

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

      8. distribute-frac-neg2 44.5%

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

      9. distribute-frac-neg2 44.5%

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

      10. unpow2 44.5%

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

      11. +-rgt-identity 44.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 44.5%

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

      13. associate--l+ 44.5%

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

      14. +-commutative 44.5%

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

      15. associate--l+ 44.5%

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

   3. Simplified 44.5%

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

      1. associate-*r* 52.4%

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

      2. unpow2 52.4%

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

      3. times-frac 67.4%

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

      4. div-inv 67.4%

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

      5. frac-times 67.4%

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

      6. metadata-eval 67.4%

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

   6. Applied egg-rr 67.4%

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

   7. Taylor expanded in k around inf 63.8%

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

      1. times-frac 72.5%

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

      2. *-commutative 72.5%

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

   9. Simplified 72.5%

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

       1. associate-*r/ 71.0%

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

       2. associate-*r* 71.0%

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

       3. div-inv 71.0%

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

       4. pow-prod-down 74.7%

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

       5. pow-flip 74.8%

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

       6. metadata-eval 74.8%

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

   11. Applied egg-rr 74.8%

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

       1. associate-/l* 76.3%

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

       2. associate-/r/ 81.7%

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

       3. *-commutative 81.7%

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

       4. associate-*l* 83.4%

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

       5. *-commutative 83.4%

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

       6. associate-*l* 83.4%

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

       7. *-commutative 83.4%

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

   13. Simplified 83.4%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-314}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{t} \cdot \left(k \cdot \sin k\right)}{\cos k} \cdot \left(\left(k \cdot \left(\sqrt{t} \cdot \sin k\right)\right) \cdot {\ell}^{-2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t \cdot \sin k\right)}^{-2}\right) \cdot \left(t \cdot \frac{\ell}{k}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := k \cdot \left(\sqrt{t\_m} \cdot \sin k\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-314}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\frac{2}{\frac{t\_2}{{\ell}^{2}} \cdot \frac{t\_2}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t\_m \cdot \sin k\right)}^{-2}\right) \cdot \left(t\_m \cdot \frac{\ell}{k}\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* k (* (sqrt t_m) (sin k)))))
   (*
    t_s
    (if (<= (* l l) 1e-314)
      (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t_m)) 2.0))
      (if (<= (* l l) 2e+299)
        (/ 2.0 (* (/ t_2 (pow l 2.0)) (/ t_2 (cos k))))
        (*
         (/ l k)
         (*
          2.0
          (* (* (cos k) (pow (* t_m (sin k)) -2.0)) (* t_m (/ l k))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = k * (sqrt(t_m) * sin(k));
	double tmp;
	if ((l * l) <= 1e-314) {
		tmp = 2.0 / pow(((pow(k, 2.0) / l) * sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+299) {
		tmp = 2.0 / ((t_2 / pow(l, 2.0)) * (t_2 / cos(k)));
	} else {
		tmp = (l / k) * (2.0 * ((cos(k) * pow((t_m * sin(k)), -2.0)) * (t_m * (l / k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = k * (sqrt(t_m) * sin(k))
    if ((l * l) <= 1d-314) then
        tmp = 2.0d0 / ((((k ** 2.0d0) / l) * sqrt(t_m)) ** 2.0d0)
    else if ((l * l) <= 2d+299) then
        tmp = 2.0d0 / ((t_2 / (l ** 2.0d0)) * (t_2 / cos(k)))
    else
        tmp = (l / k) * (2.0d0 * ((cos(k) * ((t_m * sin(k)) ** (-2.0d0))) * (t_m * (l / k))))
    end if
    code = t_s * tmp
end function

Java

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) {
	double t_2 = k * (Math.sqrt(t_m) * Math.sin(k));
	double tmp;
	if ((l * l) <= 1e-314) {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+299) {
		tmp = 2.0 / ((t_2 / Math.pow(l, 2.0)) * (t_2 / Math.cos(k)));
	} else {
		tmp = (l / k) * (2.0 * ((Math.cos(k) * Math.pow((t_m * Math.sin(k)), -2.0)) * (t_m * (l / k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = k * (math.sqrt(t_m) * math.sin(k))
	tmp = 0
	if (l * l) <= 1e-314:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) / l) * math.sqrt(t_m)), 2.0)
	elif (l * l) <= 2e+299:
		tmp = 2.0 / ((t_2 / math.pow(l, 2.0)) * (t_2 / math.cos(k)))
	else:
		tmp = (l / k) * (2.0 * ((math.cos(k) * math.pow((t_m * math.sin(k)), -2.0)) * (t_m * (l / k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k * Float64(sqrt(t_m) * sin(k)))
	tmp = 0.0
	if (Float64(l * l) <= 1e-314)
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	elseif (Float64(l * l) <= 2e+299)
		tmp = Float64(2.0 / Float64(Float64(t_2 / (l ^ 2.0)) * Float64(t_2 / cos(k))));
	else
		tmp = Float64(Float64(l / k) * Float64(2.0 * Float64(Float64(cos(k) * (Float64(t_m * sin(k)) ^ -2.0)) * Float64(t_m * Float64(l / k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = k * (sqrt(t_m) * sin(k));
	tmp = 0.0;
	if ((l * l) <= 1e-314)
		tmp = 2.0 / ((((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0);
	elseif ((l * l) <= 2e+299)
		tmp = 2.0 / ((t_2 / (l ^ 2.0)) * (t_2 / cos(k)));
	else
		tmp = (l / k) * (2.0 * ((cos(k) * ((t_m * sin(k)) ^ -2.0)) * (t_m * (l / k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := Block[{t$95$2 = N[(k * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-314], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+299], N[(2.0 / N[(N[(t$95$2 / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.9999999996e-315

   1. Initial program 29.6%

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

   3. Taylor expanded in t around 0 68.9%

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

      1. associate-/l* 68.9%

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

   5. Simplified 68.9%

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

      1. add-sqr-sqrt 30.1%

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

      2. pow2 30.1%

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

   7. Applied egg-rr 32.7%

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

      1. associate-/r* 32.7%

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

   9. Simplified 32.7%

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

   10. Taylor expanded in k around 0 52.9%

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

   if 9.9999999996e-315 < (*.f64 l l) < 2.0000000000000001e299

   1. Initial program 39.5%

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

   3. Taylor expanded in t around 0 84.1%

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

      1. add-sqr-sqrt 42.8%

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

      2. times-frac 42.8%

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

      3. sqrt-prod 42.8%

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

      4. sqrt-pow1 36.9%

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

      5. metadata-eval 36.9%

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

      6. pow1 36.9%

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

      7. *-commutative 36.9%

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

      8. sqrt-prod 29.3%

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

      9. sqrt-pow1 37.7%

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

      10. metadata-eval 37.7%

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

      11. pow1 37.7%

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

   5. Applied egg-rr 49.8%

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

   if 2.0000000000000001e299 < (*.f64 l l)

   1. Initial program 44.5%

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

      1. associate-/r* 44.5%

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

      2. associate-/l/ 44.5%

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

      3. associate-*l/ 44.5%

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

      4. associate-/r/ 44.5%

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

      5. +-commutative 44.5%

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

      6. unpow2 44.5%

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

      7. sqr-neg 44.5%

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

      8. distribute-frac-neg2 44.5%

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

      9. distribute-frac-neg2 44.5%

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

      10. unpow2 44.5%

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

      11. +-rgt-identity 44.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 44.5%

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

      13. associate--l+ 44.5%

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

      14. +-commutative 44.5%

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

      15. associate--l+ 44.5%

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

   3. Simplified 44.5%

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

      1. associate-*r* 52.4%

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

      2. unpow2 52.4%

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

      3. times-frac 67.4%

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

      4. div-inv 67.4%

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

      5. frac-times 67.4%

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

      6. metadata-eval 67.4%

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

   6. Applied egg-rr 67.4%

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

   7. Taylor expanded in k around inf 63.8%

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

      1. times-frac 72.5%

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

      2. *-commutative 72.5%

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

   9. Simplified 72.5%

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

       1. associate-*r/ 71.0%

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

       2. associate-*r* 71.0%

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

       3. div-inv 71.0%

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

       4. pow-prod-down 74.7%

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

       5. pow-flip 74.8%

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

       6. metadata-eval 74.8%

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

   11. Applied egg-rr 74.8%

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

       1. associate-/l* 76.3%

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

       2. associate-/r/ 81.7%

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

       3. *-commutative 81.7%

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

       4. associate-*l* 83.4%

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

       5. *-commutative 83.4%

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

       6. associate-*l* 83.4%

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

       7. *-commutative 83.4%

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

   13. Simplified 83.4%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-314}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(\sqrt{t} \cdot \sin k\right)}{{\ell}^{2}} \cdot \frac{k \cdot \left(\sqrt{t} \cdot \sin k\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t \cdot \sin k\right)}^{-2}\right) \cdot \left(t \cdot \frac{\ell}{k}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-314}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\frac{{\ell}^{-2} \cdot {\left(\sqrt{t\_m} \cdot \left(k \cdot \sin k\right)\right)}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t\_m \cdot \sin k\right)}^{-2}\right) \cdot \left(t\_m \cdot \frac{\ell}{k}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1e-314)
    (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t_m)) 2.0))
    (if (<= (* l l) 2e+105)
      (/
       2.0
       (/ (* (pow l -2.0) (pow (* (sqrt t_m) (* k (sin k))) 2.0)) (cos k)))
      (*
       (/ l k)
       (* 2.0 (* (* (cos k) (pow (* t_m (sin k)) -2.0)) (* t_m (/ l k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-314) {
		tmp = 2.0 / pow(((pow(k, 2.0) / l) * sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+105) {
		tmp = 2.0 / ((pow(l, -2.0) * pow((sqrt(t_m) * (k * sin(k))), 2.0)) / cos(k));
	} else {
		tmp = (l / k) * (2.0 * ((cos(k) * pow((t_m * sin(k)), -2.0)) * (t_m * (l / k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d-314) then
        tmp = 2.0d0 / ((((k ** 2.0d0) / l) * sqrt(t_m)) ** 2.0d0)
    else if ((l * l) <= 2d+105) then
        tmp = 2.0d0 / (((l ** (-2.0d0)) * ((sqrt(t_m) * (k * sin(k))) ** 2.0d0)) / cos(k))
    else
        tmp = (l / k) * (2.0d0 * ((cos(k) * ((t_m * sin(k)) ** (-2.0d0))) * (t_m * (l / k))))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if ((l * l) <= 1e-314) {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+105) {
		tmp = 2.0 / ((Math.pow(l, -2.0) * Math.pow((Math.sqrt(t_m) * (k * Math.sin(k))), 2.0)) / Math.cos(k));
	} else {
		tmp = (l / k) * (2.0 * ((Math.cos(k) * Math.pow((t_m * Math.sin(k)), -2.0)) * (t_m * (l / k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 1e-314:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) / l) * math.sqrt(t_m)), 2.0)
	elif (l * l) <= 2e+105:
		tmp = 2.0 / ((math.pow(l, -2.0) * math.pow((math.sqrt(t_m) * (k * math.sin(k))), 2.0)) / math.cos(k))
	else:
		tmp = (l / k) * (2.0 * ((math.cos(k) * math.pow((t_m * math.sin(k)), -2.0)) * (t_m * (l / k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-314)
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	elseif (Float64(l * l) <= 2e+105)
		tmp = Float64(2.0 / Float64(Float64((l ^ -2.0) * (Float64(sqrt(t_m) * Float64(k * sin(k))) ^ 2.0)) / cos(k)));
	else
		tmp = Float64(Float64(l / k) * Float64(2.0 * Float64(Float64(cos(k) * (Float64(t_m * sin(k)) ^ -2.0)) * Float64(t_m * Float64(l / k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-314)
		tmp = 2.0 / ((((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0);
	elseif ((l * l) <= 2e+105)
		tmp = 2.0 / (((l ^ -2.0) * ((sqrt(t_m) * (k * sin(k))) ^ 2.0)) / cos(k));
	else
		tmp = (l / k) * (2.0 * ((cos(k) * ((t_m * sin(k)) ^ -2.0)) * (t_m * (l / k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-314], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+105], N[(2.0 / N[(N[(N[Power[l, -2.0], $MachinePrecision] * N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.9999999996e-315

   1. Initial program 29.6%

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

   3. Taylor expanded in t around 0 68.9%

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

      1. associate-/l* 68.9%

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

   5. Simplified 68.9%

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

      1. add-sqr-sqrt 30.1%

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

      2. pow2 30.1%

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

   7. Applied egg-rr 32.7%

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

      1. associate-/r* 32.7%

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

   9. Simplified 32.7%

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

   10. Taylor expanded in k around 0 52.9%

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

   if 9.9999999996e-315 < (*.f64 l l) < 1.9999999999999999e105

   1. Initial program 36.0%

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

   3. Taylor expanded in t around 0 91.5%

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

      1. add-sqr-sqrt 48.0%

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

      2. times-frac 48.0%

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

      3. sqrt-prod 47.9%

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

      4. sqrt-pow1 43.8%

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

      5. metadata-eval 43.8%

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

      6. pow1 43.8%

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

      7. *-commutative 43.8%

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

      8. sqrt-prod 36.1%

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

      9. sqrt-pow1 42.8%

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

      10. metadata-eval 42.8%

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

      11. pow1 42.8%

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

   5. Applied egg-rr 50.9%

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

   6. Taylor expanded in k around inf 50.9%

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

      1. associate-*r* 50.9%

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

      2. associate-*r/ 50.9%

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

      3. div-inv 50.9%

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

      4. associate-*r* 50.9%

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

      5. *-commutative 50.9%

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

      6. pow-flip 50.9%

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

      7. metadata-eval 50.9%

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

      8. associate-*r* 50.9%

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

      9. *-commutative 50.9%

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

   8. Applied egg-rr 50.9%

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

      1. *-commutative 50.9%

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

      2. associate-*r* 50.0%

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

      3. unpow2 50.0%

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

   10. Simplified 50.0%

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

   if 1.9999999999999999e105 < (*.f64 l l)

   1. Initial program 45.9%

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

      1. associate-/r* 45.8%

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

      2. associate-/l/ 45.8%

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

      3. associate-*l/ 45.8%

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

      4. associate-/r/ 45.8%

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

      5. +-commutative 45.8%

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

      6. unpow2 45.8%

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

      7. sqr-neg 45.8%

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

      8. distribute-frac-neg2 45.8%

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

      9. distribute-frac-neg2 45.8%

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

      10. unpow2 45.8%

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

      11. +-rgt-identity 45.8%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 45.8%

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

      13. associate--l+ 45.8%

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

      14. +-commutative 45.8%

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

      15. associate--l+ 45.8%

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

   3. Simplified 46.9%

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

      1. associate-*r* 51.5%

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

      2. unpow2 51.5%

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

      3. times-frac 64.4%

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

      4. div-inv 64.4%

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

      5. frac-times 64.4%

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

      6. metadata-eval 64.4%

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

   6. Applied egg-rr 64.4%

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

   7. Taylor expanded in k around inf 67.7%

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

      1. times-frac 72.7%

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

      2. *-commutative 72.7%

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

   9. Simplified 72.7%

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

       1. associate-*r/ 71.9%

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

       2. associate-*r* 71.9%

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

       3. div-inv 72.0%

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

       4. pow-prod-down 75.1%

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

       5. pow-flip 75.9%

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

       6. metadata-eval 75.9%

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

   11. Applied egg-rr 75.9%

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

       1. associate-/l* 76.7%

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

       2. associate-/r/ 81.9%

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

       3. *-commutative 81.9%

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

       4. associate-*l* 82.9%

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

       5. *-commutative 82.9%

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

       6. associate-*l* 82.9%

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

       7. *-commutative 82.9%

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

   13. Simplified 82.9%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-314}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\frac{{\ell}^{-2} \cdot {\left(\sqrt{t} \cdot \left(k \cdot \sin k\right)\right)}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t \cdot \sin k\right)}^{-2}\right) \cdot \left(t \cdot \frac{\ell}{k}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-314}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(k \cdot \left(\sqrt{t\_m} \cdot \sin k\right)\right)}^{2}}{{\ell}^{2}}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t\_m \cdot \sin k\right)}^{-2}\right) \cdot \left(t\_m \cdot \frac{\ell}{k}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1e-314)
    (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t_m)) 2.0))
    (if (<= (* l l) 2e+105)
      (/
       2.0
       (/ (/ (pow (* k (* (sqrt t_m) (sin k))) 2.0) (pow l 2.0)) (cos k)))
      (*
       (/ l k)
       (* 2.0 (* (* (cos k) (pow (* t_m (sin k)) -2.0)) (* t_m (/ l k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-314) {
		tmp = 2.0 / pow(((pow(k, 2.0) / l) * sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+105) {
		tmp = 2.0 / ((pow((k * (sqrt(t_m) * sin(k))), 2.0) / pow(l, 2.0)) / cos(k));
	} else {
		tmp = (l / k) * (2.0 * ((cos(k) * pow((t_m * sin(k)), -2.0)) * (t_m * (l / k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d-314) then
        tmp = 2.0d0 / ((((k ** 2.0d0) / l) * sqrt(t_m)) ** 2.0d0)
    else if ((l * l) <= 2d+105) then
        tmp = 2.0d0 / ((((k * (sqrt(t_m) * sin(k))) ** 2.0d0) / (l ** 2.0d0)) / cos(k))
    else
        tmp = (l / k) * (2.0d0 * ((cos(k) * ((t_m * sin(k)) ** (-2.0d0))) * (t_m * (l / k))))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if ((l * l) <= 1e-314) {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+105) {
		tmp = 2.0 / ((Math.pow((k * (Math.sqrt(t_m) * Math.sin(k))), 2.0) / Math.pow(l, 2.0)) / Math.cos(k));
	} else {
		tmp = (l / k) * (2.0 * ((Math.cos(k) * Math.pow((t_m * Math.sin(k)), -2.0)) * (t_m * (l / k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 1e-314:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) / l) * math.sqrt(t_m)), 2.0)
	elif (l * l) <= 2e+105:
		tmp = 2.0 / ((math.pow((k * (math.sqrt(t_m) * math.sin(k))), 2.0) / math.pow(l, 2.0)) / math.cos(k))
	else:
		tmp = (l / k) * (2.0 * ((math.cos(k) * math.pow((t_m * math.sin(k)), -2.0)) * (t_m * (l / k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-314)
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	elseif (Float64(l * l) <= 2e+105)
		tmp = Float64(2.0 / Float64(Float64((Float64(k * Float64(sqrt(t_m) * sin(k))) ^ 2.0) / (l ^ 2.0)) / cos(k)));
	else
		tmp = Float64(Float64(l / k) * Float64(2.0 * Float64(Float64(cos(k) * (Float64(t_m * sin(k)) ^ -2.0)) * Float64(t_m * Float64(l / k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-314)
		tmp = 2.0 / ((((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0);
	elseif ((l * l) <= 2e+105)
		tmp = 2.0 / ((((k * (sqrt(t_m) * sin(k))) ^ 2.0) / (l ^ 2.0)) / cos(k));
	else
		tmp = (l / k) * (2.0 * ((cos(k) * ((t_m * sin(k)) ^ -2.0)) * (t_m * (l / k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-314], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+105], N[(2.0 / N[(N[(N[Power[N[(k * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.9999999996e-315

   1. Initial program 29.6%

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

   3. Taylor expanded in t around 0 68.9%

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

      1. associate-/l* 68.9%

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

   5. Simplified 68.9%

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

      1. add-sqr-sqrt 30.1%

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

      2. pow2 30.1%

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

   7. Applied egg-rr 32.7%

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

      1. associate-/r* 32.7%

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

   9. Simplified 32.7%

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

   10. Taylor expanded in k around 0 52.9%

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

   if 9.9999999996e-315 < (*.f64 l l) < 1.9999999999999999e105

   1. Initial program 36.0%

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

   3. Taylor expanded in t around 0 91.5%

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

      1. associate-/l* 94.3%

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

   5. Simplified 94.3%

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

      1. associate-*r/ 91.5%

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

      2. associate-/r* 91.6%

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

      3. add-sqr-sqrt 47.9%

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

      4. pow2 47.9%

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

      5. sqrt-prod 47.9%

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

      6. sqrt-pow1 50.0%

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

      7. metadata-eval 50.0%

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

      8. pow1 50.0%

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

      9. *-commutative 50.0%

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

      10. sqrt-prod 50.0%

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

      11. sqrt-pow1 50.0%

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

      12. metadata-eval 50.0%

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

      13. pow1 50.0%

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

   7. Applied egg-rr 50.0%

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

   if 1.9999999999999999e105 < (*.f64 l l)

   1. Initial program 45.9%

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

      1. associate-/r* 45.8%

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

      2. associate-/l/ 45.8%

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

      3. associate-*l/ 45.8%

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

      4. associate-/r/ 45.8%

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

      5. +-commutative 45.8%

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

      6. unpow2 45.8%

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

      7. sqr-neg 45.8%

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

      8. distribute-frac-neg2 45.8%

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

      9. distribute-frac-neg2 45.8%

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

      10. unpow2 45.8%

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

      11. +-rgt-identity 45.8%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 45.8%

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

      13. associate--l+ 45.8%

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

      14. +-commutative 45.8%

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

      15. associate--l+ 45.8%

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

   3. Simplified 46.9%

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

      1. associate-*r* 51.5%

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

      2. unpow2 51.5%

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

      3. times-frac 64.4%

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

      4. div-inv 64.4%

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

      5. frac-times 64.4%

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

      6. metadata-eval 64.4%

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

   6. Applied egg-rr 64.4%

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

   7. Taylor expanded in k around inf 67.7%

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

      1. times-frac 72.7%

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

      2. *-commutative 72.7%

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

   9. Simplified 72.7%

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

       1. associate-*r/ 71.9%

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

       2. associate-*r* 71.9%

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

       3. div-inv 72.0%

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

       4. pow-prod-down 75.1%

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

       5. pow-flip 75.9%

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

       6. metadata-eval 75.9%

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

   11. Applied egg-rr 75.9%

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

       1. associate-/l* 76.7%

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

       2. associate-/r/ 81.9%

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

       3. *-commutative 81.9%

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

       4. associate-*l* 82.9%

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

       5. *-commutative 82.9%

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

       6. associate-*l* 82.9%

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

       7. *-commutative 82.9%

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

   13. Simplified 82.9%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-314}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\left(k \cdot \left(\sqrt{t} \cdot \sin k\right)\right)}^{2}}{{\ell}^{2}}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t \cdot \sin k\right)}^{-2}\right) \cdot \left(t \cdot \frac{\ell}{k}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-314}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\left({k}^{2} \cdot t\_m\right) \cdot \left(\tan k \cdot \frac{\sin k}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t\_m \cdot \sin k\right)}^{-2}\right) \cdot \left(t\_m \cdot \frac{\ell}{k}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1e-314)
    (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t_m)) 2.0))
    (if (<= (* l l) 2e+105)
      (/ 2.0 (* (* (pow k 2.0) t_m) (* (tan k) (/ (sin k) (pow l 2.0)))))
      (*
       (/ l k)
       (* 2.0 (* (* (cos k) (pow (* t_m (sin k)) -2.0)) (* t_m (/ l k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-314) {
		tmp = 2.0 / pow(((pow(k, 2.0) / l) * sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+105) {
		tmp = 2.0 / ((pow(k, 2.0) * t_m) * (tan(k) * (sin(k) / pow(l, 2.0))));
	} else {
		tmp = (l / k) * (2.0 * ((cos(k) * pow((t_m * sin(k)), -2.0)) * (t_m * (l / k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d-314) then
        tmp = 2.0d0 / ((((k ** 2.0d0) / l) * sqrt(t_m)) ** 2.0d0)
    else if ((l * l) <= 2d+105) then
        tmp = 2.0d0 / (((k ** 2.0d0) * t_m) * (tan(k) * (sin(k) / (l ** 2.0d0))))
    else
        tmp = (l / k) * (2.0d0 * ((cos(k) * ((t_m * sin(k)) ** (-2.0d0))) * (t_m * (l / k))))
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if ((l * l) <= 1e-314) {
		tmp = 2.0 / Math.pow(((Math.pow(k, 2.0) / l) * Math.sqrt(t_m)), 2.0);
	} else if ((l * l) <= 2e+105) {
		tmp = 2.0 / ((Math.pow(k, 2.0) * t_m) * (Math.tan(k) * (Math.sin(k) / Math.pow(l, 2.0))));
	} else {
		tmp = (l / k) * (2.0 * ((Math.cos(k) * Math.pow((t_m * Math.sin(k)), -2.0)) * (t_m * (l / k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 1e-314:
		tmp = 2.0 / math.pow(((math.pow(k, 2.0) / l) * math.sqrt(t_m)), 2.0)
	elif (l * l) <= 2e+105:
		tmp = 2.0 / ((math.pow(k, 2.0) * t_m) * (math.tan(k) * (math.sin(k) / math.pow(l, 2.0))))
	else:
		tmp = (l / k) * (2.0 * ((math.cos(k) * math.pow((t_m * math.sin(k)), -2.0)) * (t_m * (l / k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-314)
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	elseif (Float64(l * l) <= 2e+105)
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * t_m) * Float64(tan(k) * Float64(sin(k) / (l ^ 2.0)))));
	else
		tmp = Float64(Float64(l / k) * Float64(2.0 * Float64(Float64(cos(k) * (Float64(t_m * sin(k)) ^ -2.0)) * Float64(t_m * Float64(l / k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-314)
		tmp = 2.0 / ((((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0);
	elseif ((l * l) <= 2e+105)
		tmp = 2.0 / (((k ^ 2.0) * t_m) * (tan(k) * (sin(k) / (l ^ 2.0))));
	else
		tmp = (l / k) * (2.0 * ((cos(k) * ((t_m * sin(k)) ^ -2.0)) * (t_m * (l / k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-314], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+105], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 9.9999999996e-315

   1. Initial program 29.6%

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

   3. Taylor expanded in t around 0 68.9%

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

      1. associate-/l* 68.9%

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

   5. Simplified 68.9%

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

      1. add-sqr-sqrt 30.1%

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

      2. pow2 30.1%

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

   7. Applied egg-rr 32.7%

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

      1. associate-/r* 32.7%

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

   9. Simplified 32.7%

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

   10. Taylor expanded in k around 0 52.9%

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

   if 9.9999999996e-315 < (*.f64 l l) < 1.9999999999999999e105

   1. Initial program 36.0%

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

   3. Taylor expanded in t around 0 91.5%

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

      1. associate-/l* 94.3%

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

   5. Simplified 94.3%

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

      1. pow1 94.3%

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

      2. *-commutative 94.3%

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

      3. associate-/l* 94.3%

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

      4. unpow2 94.3%

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

      5. *-commutative 94.3%

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

      6. times-frac 94.2%

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

      7. tan-quot 94.3%

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

   7. Applied egg-rr 94.3%

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

      1. unpow1 94.3%

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

      2. *-commutative 94.3%

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

      3. associate-*r* 91.6%

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

   9. Simplified 91.6%

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

   if 1.9999999999999999e105 < (*.f64 l l)

   1. Initial program 45.9%

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

      1. associate-/r* 45.8%

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

      2. associate-/l/ 45.8%

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

      3. associate-*l/ 45.8%

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

      4. associate-/r/ 45.8%

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

      5. +-commutative 45.8%

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

      6. unpow2 45.8%

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

      7. sqr-neg 45.8%

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

      8. distribute-frac-neg2 45.8%

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

      9. distribute-frac-neg2 45.8%

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

      10. unpow2 45.8%

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

      11. +-rgt-identity 45.8%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 45.8%

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

      13. associate--l+ 45.8%

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

      14. +-commutative 45.8%

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

      15. associate--l+ 45.8%

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

   3. Simplified 46.9%

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

      1. associate-*r* 51.5%

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

      2. unpow2 51.5%

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

      3. times-frac 64.4%

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

      4. div-inv 64.4%

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

      5. frac-times 64.4%

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

      6. metadata-eval 64.4%

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

   6. Applied egg-rr 64.4%

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

   7. Taylor expanded in k around inf 67.7%

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

      1. times-frac 72.7%

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

      2. *-commutative 72.7%

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

   9. Simplified 72.7%

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

       1. associate-*r/ 71.9%

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

       2. associate-*r* 71.9%

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

       3. div-inv 72.0%

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

       4. pow-prod-down 75.1%

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

       5. pow-flip 75.9%

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

       6. metadata-eval 75.9%

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

   11. Applied egg-rr 75.9%

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

       1. associate-/l* 76.7%

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

       2. associate-/r/ 81.9%

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

       3. *-commutative 81.9%

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

       4. associate-*l* 82.9%

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

       5. *-commutative 82.9%

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

       6. associate-*l* 82.9%

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

       7. *-commutative 82.9%

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

   13. Simplified 82.9%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-314}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\left({k}^{2} \cdot t\right) \cdot \left(\tan k \cdot \frac{\sin k}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t \cdot \sin k\right)}^{-2}\right) \cdot \left(t \cdot \frac{\ell}{k}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\sqrt{\frac{t\_m}{\cos k}} \cdot \frac{\sin k}{\ell}\right)\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 1.35 \cdot 10^{+219}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t\_m \cdot \sin k\right)}^{-2}\right) \cdot \left(t\_m \cdot \frac{\ell}{k}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t\_m}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.5e-162)
    (/ 2.0 (pow (* k (* (sqrt (/ t_m (cos k))) (/ (sin k) l))) 2.0))
    (if (<= t_m 1.35e+219)
      (*
       (/ l k)
       (* 2.0 (* (* (cos k) (pow (* t_m (sin k)) -2.0)) (* t_m (/ l k)))))
      (* 2.0 (pow (/ l (* (pow k 2.0) (sqrt t_m))) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.5e-162) {
		tmp = 2.0 / pow((k * (sqrt((t_m / cos(k))) * (sin(k) / l))), 2.0);
	} else if (t_m <= 1.35e+219) {
		tmp = (l / k) * (2.0 * ((cos(k) * pow((t_m * sin(k)), -2.0)) * (t_m * (l / k))));
	} else {
		tmp = 2.0 * pow((l / (pow(k, 2.0) * sqrt(t_m))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.5d-162) then
        tmp = 2.0d0 / ((k * (sqrt((t_m / cos(k))) * (sin(k) / l))) ** 2.0d0)
    else if (t_m <= 1.35d+219) then
        tmp = (l / k) * (2.0d0 * ((cos(k) * ((t_m * sin(k)) ** (-2.0d0))) * (t_m * (l / k))))
    else
        tmp = 2.0d0 * ((l / ((k ** 2.0d0) * sqrt(t_m))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (t_m <= 1.5e-162) {
		tmp = 2.0 / Math.pow((k * (Math.sqrt((t_m / Math.cos(k))) * (Math.sin(k) / l))), 2.0);
	} else if (t_m <= 1.35e+219) {
		tmp = (l / k) * (2.0 * ((Math.cos(k) * Math.pow((t_m * Math.sin(k)), -2.0)) * (t_m * (l / k))));
	} else {
		tmp = 2.0 * Math.pow((l / (Math.pow(k, 2.0) * Math.sqrt(t_m))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.5e-162:
		tmp = 2.0 / math.pow((k * (math.sqrt((t_m / math.cos(k))) * (math.sin(k) / l))), 2.0)
	elif t_m <= 1.35e+219:
		tmp = (l / k) * (2.0 * ((math.cos(k) * math.pow((t_m * math.sin(k)), -2.0)) * (t_m * (l / k))))
	else:
		tmp = 2.0 * math.pow((l / (math.pow(k, 2.0) * math.sqrt(t_m))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.5e-162)
		tmp = Float64(2.0 / (Float64(k * Float64(sqrt(Float64(t_m / cos(k))) * Float64(sin(k) / l))) ^ 2.0));
	elseif (t_m <= 1.35e+219)
		tmp = Float64(Float64(l / k) * Float64(2.0 * Float64(Float64(cos(k) * (Float64(t_m * sin(k)) ^ -2.0)) * Float64(t_m * Float64(l / k)))));
	else
		tmp = Float64(2.0 * (Float64(l / Float64((k ^ 2.0) * sqrt(t_m))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.5e-162)
		tmp = 2.0 / ((k * (sqrt((t_m / cos(k))) * (sin(k) / l))) ^ 2.0);
	elseif (t_m <= 1.35e+219)
		tmp = (l / k) * (2.0 * ((cos(k) * ((t_m * sin(k)) ^ -2.0)) * (t_m * (l / k))));
	else
		tmp = 2.0 * ((l / ((k ^ 2.0) * sqrt(t_m))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.5e-162], N[(2.0 / N[Power[N[(k * N[(N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.35e+219], N[(N[(l / k), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.49999999999999999e-162

   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. Add Preprocessing

   3. Taylor expanded in t around 0 71.2%

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

      1. associate-/l* 72.4%

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

   5. Simplified 72.4%

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

      1. add-sqr-sqrt 24.1%

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

      2. pow2 24.1%

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

   7. Applied egg-rr 11.0%

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

      1. associate-/r* 11.0%

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

   9. Simplified 11.0%

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

   10. Taylor expanded in k around inf 36.0%

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

   if 1.49999999999999999e-162 < t < 1.3499999999999999e219

   1. Initial program 50.8%

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

      1. associate-/r* 50.8%

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

      2. associate-/l/ 50.8%

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

      3. associate-*l/ 53.3%

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

      4. associate-/r/ 53.3%

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

      5. +-commutative 53.3%

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

      6. unpow2 53.3%

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

      7. sqr-neg 53.3%

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

      8. distribute-frac-neg2 53.3%

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

      9. distribute-frac-neg2 53.3%

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

      10. unpow2 53.3%

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

      11. +-rgt-identity 53.3%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 53.3%

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

      13. associate--l+ 53.3%

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

      14. +-commutative 53.3%

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

      15. associate--l+ 53.3%

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

   3. Simplified 63.5%

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

      1. associate-*r* 71.8%

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

      2. unpow2 71.8%

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

      3. times-frac 76.7%

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

      4. div-inv 76.7%

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

      5. frac-times 76.8%

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

      6. metadata-eval 76.8%

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

   6. Applied egg-rr 76.8%

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

   7. Taylor expanded in k around inf 90.0%

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

      1. times-frac 91.1%

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

      2. *-commutative 91.1%

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

   9. Simplified 91.1%

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

       1. associate-*r/ 91.0%

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

       2. associate-*r* 91.0%

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

       3. div-inv 91.0%

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

       4. pow-prod-down 92.1%

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

       5. pow-flip 92.8%

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

       6. metadata-eval 92.8%

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

   11. Applied egg-rr 92.8%

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

       1. associate-/l* 93.0%

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

       2. associate-/r/ 94.2%

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

       3. *-commutative 94.2%

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

       4. associate-*l* 95.3%

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

       5. *-commutative 95.3%

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

       6. associate-*l* 95.3%

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

       7. *-commutative 95.3%

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

   13. Simplified 95.3%

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

   if 1.3499999999999999e219 < t

   1. Initial program 0.0%

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

      1. associate-/r* 0.0%

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

      2. associate-/l/ 0.0%

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

      3. associate-*l/ 0.0%

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

      4. associate-/r/ 0.0%

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

      5. +-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. sqr-neg 0.0%

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

      8. distribute-frac-neg2 0.0%

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

      9. distribute-frac-neg2 0.0%

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

      10. unpow2 0.0%

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

      11. +-rgt-identity 0.0%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 0.0%

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

      13. associate--l+ 0.0%

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

      14. +-commutative 0.0%

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

      15. associate--l+ 0.0%

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

   3. Simplified 11.1%

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

   5. Taylor expanded in k around 0 73.3%

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

      1. add-sqr-sqrt 73.3%

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

      2. sqrt-div 73.3%

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

      3. sqrt-pow1 62.3%

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

      4. metadata-eval 62.3%

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

      5. pow1 62.3%

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

      6. sqrt-prod 62.3%

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

      7. sqrt-pow1 62.3%

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

      8. metadata-eval 62.3%

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

      9. sqrt-div 62.3%

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

      10. sqrt-pow1 79.0%

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

      11. metadata-eval 79.0%

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

      12. pow1 79.0%

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

      13. sqrt-prod 79.0%

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

      14. sqrt-pow1 84.4%

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

      15. metadata-eval 84.4%

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

   7. Applied egg-rr 84.4%

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

      1. unpow2 84.4%

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

   9. Simplified 84.4%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\sqrt{\frac{t}{\cos k}} \cdot \frac{\sin k}{\ell}\right)\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+219}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t \cdot \sin k\right)}^{-2}\right) \cdot \left(t \cdot \frac{\ell}{k}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t\_m}{\cos k}} \cdot \left(k \cdot \frac{\sin k}{\ell}\right)\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 1.5 \cdot 10^{+219}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t\_m \cdot \sin k\right)}^{-2}\right) \cdot \left(t\_m \cdot \frac{\ell}{k}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t\_m}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.5e-162)
    (/ 2.0 (pow (* (sqrt (/ t_m (cos k))) (* k (/ (sin k) l))) 2.0))
    (if (<= t_m 1.5e+219)
      (*
       (/ l k)
       (* 2.0 (* (* (cos k) (pow (* t_m (sin k)) -2.0)) (* t_m (/ l k)))))
      (* 2.0 (pow (/ l (* (pow k 2.0) (sqrt t_m))) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.5e-162) {
		tmp = 2.0 / pow((sqrt((t_m / cos(k))) * (k * (sin(k) / l))), 2.0);
	} else if (t_m <= 1.5e+219) {
		tmp = (l / k) * (2.0 * ((cos(k) * pow((t_m * sin(k)), -2.0)) * (t_m * (l / k))));
	} else {
		tmp = 2.0 * pow((l / (pow(k, 2.0) * sqrt(t_m))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.5d-162) then
        tmp = 2.0d0 / ((sqrt((t_m / cos(k))) * (k * (sin(k) / l))) ** 2.0d0)
    else if (t_m <= 1.5d+219) then
        tmp = (l / k) * (2.0d0 * ((cos(k) * ((t_m * sin(k)) ** (-2.0d0))) * (t_m * (l / k))))
    else
        tmp = 2.0d0 * ((l / ((k ** 2.0d0) * sqrt(t_m))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (t_m <= 1.5e-162) {
		tmp = 2.0 / Math.pow((Math.sqrt((t_m / Math.cos(k))) * (k * (Math.sin(k) / l))), 2.0);
	} else if (t_m <= 1.5e+219) {
		tmp = (l / k) * (2.0 * ((Math.cos(k) * Math.pow((t_m * Math.sin(k)), -2.0)) * (t_m * (l / k))));
	} else {
		tmp = 2.0 * Math.pow((l / (Math.pow(k, 2.0) * Math.sqrt(t_m))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.5e-162:
		tmp = 2.0 / math.pow((math.sqrt((t_m / math.cos(k))) * (k * (math.sin(k) / l))), 2.0)
	elif t_m <= 1.5e+219:
		tmp = (l / k) * (2.0 * ((math.cos(k) * math.pow((t_m * math.sin(k)), -2.0)) * (t_m * (l / k))))
	else:
		tmp = 2.0 * math.pow((l / (math.pow(k, 2.0) * math.sqrt(t_m))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.5e-162)
		tmp = Float64(2.0 / (Float64(sqrt(Float64(t_m / cos(k))) * Float64(k * Float64(sin(k) / l))) ^ 2.0));
	elseif (t_m <= 1.5e+219)
		tmp = Float64(Float64(l / k) * Float64(2.0 * Float64(Float64(cos(k) * (Float64(t_m * sin(k)) ^ -2.0)) * Float64(t_m * Float64(l / k)))));
	else
		tmp = Float64(2.0 * (Float64(l / Float64((k ^ 2.0) * sqrt(t_m))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.5e-162)
		tmp = 2.0 / ((sqrt((t_m / cos(k))) * (k * (sin(k) / l))) ^ 2.0);
	elseif (t_m <= 1.5e+219)
		tmp = (l / k) * (2.0 * ((cos(k) * ((t_m * sin(k)) ^ -2.0)) * (t_m * (l / k))));
	else
		tmp = 2.0 * ((l / ((k ^ 2.0) * sqrt(t_m))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.5e-162], N[(2.0 / N[Power[N[(N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.5e+219], N[(N[(l / k), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.49999999999999999e-162

   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. Add Preprocessing

   3. Taylor expanded in t around 0 71.2%

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

      1. associate-/l* 72.4%

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

   5. Simplified 72.4%

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

      1. add-sqr-sqrt 24.1%

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

      2. pow2 24.1%

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

   7. Applied egg-rr 11.0%

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

      1. associate-/r* 11.0%

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

   9. Simplified 11.0%

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

   10. Taylor expanded in k around inf 36.7%

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

       1. associate-/l* 36.7%

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

   12. Simplified 36.7%

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

   if 1.49999999999999999e-162 < t < 1.4999999999999999e219

   1. Initial program 50.8%

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

      1. associate-/r* 50.8%

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

      2. associate-/l/ 50.8%

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

      3. associate-*l/ 53.3%

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

      4. associate-/r/ 53.3%

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

      5. +-commutative 53.3%

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

      6. unpow2 53.3%

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

      7. sqr-neg 53.3%

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

      8. distribute-frac-neg2 53.3%

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

      9. distribute-frac-neg2 53.3%

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

      10. unpow2 53.3%

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

      11. +-rgt-identity 53.3%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 53.3%

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

      13. associate--l+ 53.3%

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

      14. +-commutative 53.3%

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

      15. associate--l+ 53.3%

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

   3. Simplified 63.5%

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

      1. associate-*r* 71.8%

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

      2. unpow2 71.8%

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

      3. times-frac 76.7%

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

      4. div-inv 76.7%

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

      5. frac-times 76.8%

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

      6. metadata-eval 76.8%

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

   6. Applied egg-rr 76.8%

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

   7. Taylor expanded in k around inf 90.0%

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

      1. times-frac 91.1%

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

      2. *-commutative 91.1%

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

   9. Simplified 91.1%

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

       1. associate-*r/ 91.0%

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

       2. associate-*r* 91.0%

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

       3. div-inv 91.0%

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

       4. pow-prod-down 92.1%

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

       5. pow-flip 92.8%

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

       6. metadata-eval 92.8%

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

   11. Applied egg-rr 92.8%

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

       1. associate-/l* 93.0%

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

       2. associate-/r/ 94.2%

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

       3. *-commutative 94.2%

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

       4. associate-*l* 95.3%

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

       5. *-commutative 95.3%

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

       6. associate-*l* 95.3%

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

       7. *-commutative 95.3%

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

   13. Simplified 95.3%

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

   if 1.4999999999999999e219 < t

   1. Initial program 0.0%

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

      1. associate-/r* 0.0%

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

      2. associate-/l/ 0.0%

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

      3. associate-*l/ 0.0%

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

      4. associate-/r/ 0.0%

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

      5. +-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. sqr-neg 0.0%

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

      8. distribute-frac-neg2 0.0%

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

      9. distribute-frac-neg2 0.0%

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

      10. unpow2 0.0%

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

      11. +-rgt-identity 0.0%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 0.0%

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

      13. associate--l+ 0.0%

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

      14. +-commutative 0.0%

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

      15. associate--l+ 0.0%

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

   3. Simplified 11.1%

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

   5. Taylor expanded in k around 0 73.3%

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

      1. add-sqr-sqrt 73.3%

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

      2. sqrt-div 73.3%

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

      3. sqrt-pow1 62.3%

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

      4. metadata-eval 62.3%

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

      5. pow1 62.3%

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

      6. sqrt-prod 62.3%

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

      7. sqrt-pow1 62.3%

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

      8. metadata-eval 62.3%

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

      9. sqrt-div 62.3%

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

      10. sqrt-pow1 79.0%

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

      11. metadata-eval 79.0%

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

      12. pow1 79.0%

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

      13. sqrt-prod 79.0%

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

      14. sqrt-pow1 84.4%

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

      15. metadata-eval 84.4%

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

   7. Applied egg-rr 84.4%

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

      1. unpow2 84.4%

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

   9. Simplified 84.4%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t}{\cos k}} \cdot \left(k \cdot \frac{\sin k}{\ell}\right)\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+219}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t \cdot \sin k\right)}^{-2}\right) \cdot \left(t \cdot \frac{\ell}{k}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t\_m}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t\_m \cdot \sin k\right)}^{-2}\right) \cdot \left(t\_m \cdot \frac{\ell}{k}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t\_m}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.5e-162)
    (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt (/ t_m (cos k)))) 2.0))
    (if (<= t_m 2e+220)
      (*
       (/ l k)
       (* 2.0 (* (* (cos k) (pow (* t_m (sin k)) -2.0)) (* t_m (/ l k)))))
      (* 2.0 (pow (/ l (* (pow k 2.0) (sqrt t_m))) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.5e-162) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt((t_m / cos(k)))), 2.0);
	} else if (t_m <= 2e+220) {
		tmp = (l / k) * (2.0 * ((cos(k) * pow((t_m * sin(k)), -2.0)) * (t_m * (l / k))));
	} else {
		tmp = 2.0 * pow((l / (pow(k, 2.0) * sqrt(t_m))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.5d-162) then
        tmp = 2.0d0 / ((((k * sin(k)) / l) * sqrt((t_m / cos(k)))) ** 2.0d0)
    else if (t_m <= 2d+220) then
        tmp = (l / k) * (2.0d0 * ((cos(k) * ((t_m * sin(k)) ** (-2.0d0))) * (t_m * (l / k))))
    else
        tmp = 2.0d0 * ((l / ((k ** 2.0d0) * sqrt(t_m))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (t_m <= 1.5e-162) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt((t_m / Math.cos(k)))), 2.0);
	} else if (t_m <= 2e+220) {
		tmp = (l / k) * (2.0 * ((Math.cos(k) * Math.pow((t_m * Math.sin(k)), -2.0)) * (t_m * (l / k))));
	} else {
		tmp = 2.0 * Math.pow((l / (Math.pow(k, 2.0) * Math.sqrt(t_m))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.5e-162:
		tmp = 2.0 / math.pow((((k * math.sin(k)) / l) * math.sqrt((t_m / math.cos(k)))), 2.0)
	elif t_m <= 2e+220:
		tmp = (l / k) * (2.0 * ((math.cos(k) * math.pow((t_m * math.sin(k)), -2.0)) * (t_m * (l / k))))
	else:
		tmp = 2.0 * math.pow((l / (math.pow(k, 2.0) * math.sqrt(t_m))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.5e-162)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(Float64(t_m / cos(k)))) ^ 2.0));
	elseif (t_m <= 2e+220)
		tmp = Float64(Float64(l / k) * Float64(2.0 * Float64(Float64(cos(k) * (Float64(t_m * sin(k)) ^ -2.0)) * Float64(t_m * Float64(l / k)))));
	else
		tmp = Float64(2.0 * (Float64(l / Float64((k ^ 2.0) * sqrt(t_m))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.5e-162)
		tmp = 2.0 / ((((k * sin(k)) / l) * sqrt((t_m / cos(k)))) ^ 2.0);
	elseif (t_m <= 2e+220)
		tmp = (l / k) * (2.0 * ((cos(k) * ((t_m * sin(k)) ^ -2.0)) * (t_m * (l / k))));
	else
		tmp = 2.0 * ((l / ((k ^ 2.0) * sqrt(t_m))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.5e-162], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e+220], N[(N[(l / k), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.49999999999999999e-162

   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. Add Preprocessing

   3. Taylor expanded in t around 0 71.2%

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

      1. associate-/l* 72.4%

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

   5. Simplified 72.4%

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

      1. add-sqr-sqrt 24.1%

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

      2. pow2 24.1%

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

   7. Applied egg-rr 11.0%

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

      1. associate-/r* 11.0%

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

   9. Simplified 11.0%

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

   10. Taylor expanded in k around inf 36.7%

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

   if 1.49999999999999999e-162 < t < 2e220

   1. Initial program 50.8%

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

      1. associate-/r* 50.8%

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

      2. associate-/l/ 50.8%

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

      3. associate-*l/ 53.3%

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

      4. associate-/r/ 53.3%

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

      5. +-commutative 53.3%

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

      6. unpow2 53.3%

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

      7. sqr-neg 53.3%

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

      8. distribute-frac-neg2 53.3%

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

      9. distribute-frac-neg2 53.3%

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

      10. unpow2 53.3%

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

      11. +-rgt-identity 53.3%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 53.3%

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

      13. associate--l+ 53.3%

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

      14. +-commutative 53.3%

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

      15. associate--l+ 53.3%

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

   3. Simplified 63.5%

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

      1. associate-*r* 71.8%

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

      2. unpow2 71.8%

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

      3. times-frac 76.7%

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

      4. div-inv 76.7%

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

      5. frac-times 76.8%

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

      6. metadata-eval 76.8%

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

   6. Applied egg-rr 76.8%

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

   7. Taylor expanded in k around inf 90.0%

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

      1. times-frac 91.1%

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

      2. *-commutative 91.1%

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

   9. Simplified 91.1%

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

       1. associate-*r/ 91.0%

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

       2. associate-*r* 91.0%

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

       3. div-inv 91.0%

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

       4. pow-prod-down 92.1%

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

       5. pow-flip 92.8%

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

       6. metadata-eval 92.8%

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

   11. Applied egg-rr 92.8%

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

       1. associate-/l* 93.0%

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

       2. associate-/r/ 94.2%

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

       3. *-commutative 94.2%

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

       4. associate-*l* 95.3%

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

       5. *-commutative 95.3%

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

       6. associate-*l* 95.3%

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

       7. *-commutative 95.3%

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

   13. Simplified 95.3%

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

   if 2e220 < t

   1. Initial program 0.0%

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

      1. associate-/r* 0.0%

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

      2. associate-/l/ 0.0%

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

      3. associate-*l/ 0.0%

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

      4. associate-/r/ 0.0%

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

      5. +-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. sqr-neg 0.0%

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

      8. distribute-frac-neg2 0.0%

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

      9. distribute-frac-neg2 0.0%

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

      10. unpow2 0.0%

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

      11. +-rgt-identity 0.0%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 0.0%

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

      13. associate--l+ 0.0%

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

      14. +-commutative 0.0%

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

      15. associate--l+ 0.0%

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

   3. Simplified 11.1%

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

   5. Taylor expanded in k around 0 73.3%

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

      1. add-sqr-sqrt 73.3%

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

      2. sqrt-div 73.3%

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

      3. sqrt-pow1 62.3%

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

      4. metadata-eval 62.3%

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

      5. pow1 62.3%

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

      6. sqrt-prod 62.3%

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

      7. sqrt-pow1 62.3%

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

      8. metadata-eval 62.3%

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

      9. sqrt-div 62.3%

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

      10. sqrt-pow1 79.0%

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

      11. metadata-eval 79.0%

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

      12. pow1 79.0%

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

      13. sqrt-prod 79.0%

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

      14. sqrt-pow1 84.4%

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

      15. metadata-eval 84.4%

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

   7. Applied egg-rr 84.4%

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

      1. unpow2 84.4%

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

   9. Simplified 84.4%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t \cdot \sin k\right)}^{-2}\right) \cdot \left(t \cdot \frac{\ell}{k}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.5 \cdot 10^{-162}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell \cdot \sqrt{\frac{1}{t\_m}}}{{k}^{2}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{+219}:\\ \;\;\;\;\left(2 \cdot \left(\cos k \cdot \left(\frac{\ell}{k} \cdot {\left(t\_m \cdot \sin k\right)}^{-2}\right)\right)\right) \cdot \frac{\ell}{\frac{k}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t\_m}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.5e-162)
    (* 2.0 (pow (/ (* l (sqrt (/ 1.0 t_m))) (pow k 2.0)) 2.0))
    (if (<= t_m 2e+219)
      (*
       (* 2.0 (* (cos k) (* (/ l k) (pow (* t_m (sin k)) -2.0))))
       (/ l (/ k t_m)))
      (* 2.0 (pow (/ l (* (pow k 2.0) (sqrt t_m))) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.5e-162) {
		tmp = 2.0 * pow(((l * sqrt((1.0 / t_m))) / pow(k, 2.0)), 2.0);
	} else if (t_m <= 2e+219) {
		tmp = (2.0 * (cos(k) * ((l / k) * pow((t_m * sin(k)), -2.0)))) * (l / (k / t_m));
	} else {
		tmp = 2.0 * pow((l / (pow(k, 2.0) * sqrt(t_m))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.5d-162) then
        tmp = 2.0d0 * (((l * sqrt((1.0d0 / t_m))) / (k ** 2.0d0)) ** 2.0d0)
    else if (t_m <= 2d+219) then
        tmp = (2.0d0 * (cos(k) * ((l / k) * ((t_m * sin(k)) ** (-2.0d0))))) * (l / (k / t_m))
    else
        tmp = 2.0d0 * ((l / ((k ** 2.0d0) * sqrt(t_m))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (t_m <= 1.5e-162) {
		tmp = 2.0 * Math.pow(((l * Math.sqrt((1.0 / t_m))) / Math.pow(k, 2.0)), 2.0);
	} else if (t_m <= 2e+219) {
		tmp = (2.0 * (Math.cos(k) * ((l / k) * Math.pow((t_m * Math.sin(k)), -2.0)))) * (l / (k / t_m));
	} else {
		tmp = 2.0 * Math.pow((l / (Math.pow(k, 2.0) * Math.sqrt(t_m))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.5e-162:
		tmp = 2.0 * math.pow(((l * math.sqrt((1.0 / t_m))) / math.pow(k, 2.0)), 2.0)
	elif t_m <= 2e+219:
		tmp = (2.0 * (math.cos(k) * ((l / k) * math.pow((t_m * math.sin(k)), -2.0)))) * (l / (k / t_m))
	else:
		tmp = 2.0 * math.pow((l / (math.pow(k, 2.0) * math.sqrt(t_m))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.5e-162)
		tmp = Float64(2.0 * (Float64(Float64(l * sqrt(Float64(1.0 / t_m))) / (k ^ 2.0)) ^ 2.0));
	elseif (t_m <= 2e+219)
		tmp = Float64(Float64(2.0 * Float64(cos(k) * Float64(Float64(l / k) * (Float64(t_m * sin(k)) ^ -2.0)))) * Float64(l / Float64(k / t_m)));
	else
		tmp = Float64(2.0 * (Float64(l / Float64((k ^ 2.0) * sqrt(t_m))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.5e-162)
		tmp = 2.0 * (((l * sqrt((1.0 / t_m))) / (k ^ 2.0)) ^ 2.0);
	elseif (t_m <= 2e+219)
		tmp = (2.0 * (cos(k) * ((l / k) * ((t_m * sin(k)) ^ -2.0)))) * (l / (k / t_m));
	else
		tmp = 2.0 * ((l / ((k ^ 2.0) * sqrt(t_m))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.5e-162], N[(2.0 * N[Power[N[(N[(l * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e+219], N[(N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[Power[N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.49999999999999999e-162

   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-/r* 35.2%

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

      2. associate-/l/ 35.2%

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

      3. associate-*l/ 35.8%

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

      4. associate-/r/ 35.1%

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

      5. +-commutative 35.1%

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

      6. unpow2 35.1%

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

      7. sqr-neg 35.1%

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

      8. distribute-frac-neg2 35.1%

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

      9. distribute-frac-neg2 35.1%

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

      10. unpow2 35.1%

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

      11. +-rgt-identity 35.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 35.1%

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

      13. associate--l+ 35.1%

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

      14. +-commutative 35.1%

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

      15. associate--l+ 35.1%

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

   3. Simplified 41.7%

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

   5. Taylor expanded in k around 0 54.9%

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

      1. add-sqr-sqrt 32.7%

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

      2. sqrt-div 7.4%

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

      3. sqrt-pow1 4.8%

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

      4. metadata-eval 4.8%

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

      5. pow1 4.8%

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

      6. sqrt-prod 4.8%

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

      7. sqrt-pow1 4.8%

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

      8. metadata-eval 4.8%

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

      9. sqrt-div 4.8%

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

      10. sqrt-pow1 9.4%

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

      11. metadata-eval 9.4%

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

      12. pow1 9.4%

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

      13. sqrt-prod 9.4%

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

      14. sqrt-pow1 10.0%

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

      15. metadata-eval 10.0%

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

   7. Applied egg-rr 10.0%

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

      1. unpow2 10.0%

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

   9. Simplified 10.0%

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

   10. Taylor expanded in l around 0 10.0%

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

       1. associate-*l/ 10.0%

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

   12. Simplified 10.0%

       \[\leadsto 2 \cdot {\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{t}}}{{k}^{2}}\right)}}^{2} \]

   if 1.49999999999999999e-162 < t < 1.99999999999999993e219

   1. Initial program 50.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 50.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-/l/ 50.8%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 53.3%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/r/ 53.3%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 53.3%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 53.3%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 53.3%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg2 53.3%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(-\frac{k}{t}\right) \cdot \color{blue}{\frac{k}{-t}} + 1\right) - 1} \]

      9. distribute-frac-neg2 53.3%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{-t}} \cdot \frac{k}{-t} + 1\right) - 1} \]

      10. unpow2 53.3%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) - 1} \]

      11. +-rgt-identity 53.3%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 53.3%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + \color{blue}{\left(1 - 1\right)}\right) + 1\right) - 1} \]

      13. associate--l+ 53.3%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + 1\right) - 1\right)} + 1\right) - 1} \]

      14. +-commutative 53.3%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} - 1\right) + 1\right) - 1} \]

      15. associate--l+ 53.3%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) - 1\right) + \left(1 - 1\right)}} \]

   3. Simplified 63.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 71.8%

         \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. unpow2 71.8%

         \[\leadsto \frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      3. times-frac 76.7%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

      4. div-inv 76.7%

         \[\leadsto \frac{\color{blue}{\left(\frac{2}{\tan k} \cdot \frac{1}{{t}^{3} \cdot \sin k}\right)} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

      5. frac-times 76.8%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

      6. metadata-eval 76.8%

         \[\leadsto \frac{\frac{\color{blue}{2}}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

   6. Applied egg-rr 76.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

   7. Taylor expanded in k around inf 90.0%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\ell \cdot \cos k}{k \cdot \left({t}^{2} \cdot {\sin k}^{2}\right)}\right)} \cdot \frac{\ell}{\frac{k}{t}} \]
   8. Step-by-step derivation

      1. times-frac 91.1%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\cos k}{{t}^{2} \cdot {\sin k}^{2}}\right)}\right) \cdot \frac{\ell}{\frac{k}{t}} \]

      2. *-commutative 91.1%

         \[\leadsto \left(2 \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{\color{blue}{{\sin k}^{2} \cdot {t}^{2}}}\right)\right) \cdot \frac{\ell}{\frac{k}{t}} \]

   9. Simplified 91.1%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot {t}^{2}}\right)\right)} \cdot \frac{\ell}{\frac{k}{t}} \]
   10. Step-by-step derivation

       1. associate-*l/ 90.0%

          \[\leadsto \left(2 \cdot \color{blue}{\frac{\ell \cdot \frac{\cos k}{{\sin k}^{2} \cdot {t}^{2}}}{k}}\right) \cdot \frac{\ell}{\frac{k}{t}} \]

       2. div-inv 90.0%

          \[\leadsto \left(2 \cdot \frac{\ell \cdot \color{blue}{\left(\cos k \cdot \frac{1}{{\sin k}^{2} \cdot {t}^{2}}\right)}}{k}\right) \cdot \frac{\ell}{\frac{k}{t}} \]

       3. pow-prod-down 91.2%

          \[\leadsto \left(2 \cdot \frac{\ell \cdot \left(\cos k \cdot \frac{1}{\color{blue}{{\left(\sin k \cdot t\right)}^{2}}}\right)}{k}\right) \cdot \frac{\ell}{\frac{k}{t}} \]

       4. pow-flip 91.9%

          \[\leadsto \left(2 \cdot \frac{\ell \cdot \left(\cos k \cdot \color{blue}{{\left(\sin k \cdot t\right)}^{\left(-2\right)}}\right)}{k}\right) \cdot \frac{\ell}{\frac{k}{t}} \]

       5. metadata-eval 91.9%

          \[\leadsto \left(2 \cdot \frac{\ell \cdot \left(\cos k \cdot {\left(\sin k \cdot t\right)}^{\color{blue}{-2}}\right)}{k}\right) \cdot \frac{\ell}{\frac{k}{t}} \]

   11. Applied egg-rr 91.9%

       \[\leadsto \left(2 \cdot \color{blue}{\frac{\ell \cdot \left(\cos k \cdot {\left(\sin k \cdot t\right)}^{-2}\right)}{k}}\right) \cdot \frac{\ell}{\frac{k}{t}} \]
   12. Step-by-step derivation

       1. associate-*l/ 93.0%

          \[\leadsto \left(2 \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \left(\cos k \cdot {\left(\sin k \cdot t\right)}^{-2}\right)\right)}\right) \cdot \frac{\ell}{\frac{k}{t}} \]

       2. associate-*r* 93.1%

          \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\frac{\ell}{k} \cdot \cos k\right) \cdot {\left(\sin k \cdot t\right)}^{-2}\right)}\right) \cdot \frac{\ell}{\frac{k}{t}} \]

       3. *-commutative 93.1%

          \[\leadsto \left(2 \cdot \left(\color{blue}{\left(\cos k \cdot \frac{\ell}{k}\right)} \cdot {\left(\sin k \cdot t\right)}^{-2}\right)\right) \cdot \frac{\ell}{\frac{k}{t}} \]

       4. associate-*l* 93.0%

          \[\leadsto \left(2 \cdot \color{blue}{\left(\cos k \cdot \left(\frac{\ell}{k} \cdot {\left(\sin k \cdot t\right)}^{-2}\right)\right)}\right) \cdot \frac{\ell}{\frac{k}{t}} \]

       5. *-commutative 93.0%

          \[\leadsto \left(2 \cdot \left(\cos k \cdot \left(\frac{\ell}{k} \cdot {\color{blue}{\left(t \cdot \sin k\right)}}^{-2}\right)\right)\right) \cdot \frac{\ell}{\frac{k}{t}} \]

   13. Simplified 93.0%

       \[\leadsto \left(2 \cdot \color{blue}{\left(\cos k \cdot \left(\frac{\ell}{k} \cdot {\left(t \cdot \sin k\right)}^{-2}\right)\right)}\right) \cdot \frac{\ell}{\frac{k}{t}} \]

   if 1.99999999999999993e219 < t

   1. Initial program 0.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 0.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-/l/ 0.0%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 0.0%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/r/ 0.0%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 0.0%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 0.0%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 0.0%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg2 0.0%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(-\frac{k}{t}\right) \cdot \color{blue}{\frac{k}{-t}} + 1\right) - 1} \]

      9. distribute-frac-neg2 0.0%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{-t}} \cdot \frac{k}{-t} + 1\right) - 1} \]

      10. unpow2 0.0%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) - 1} \]

      11. +-rgt-identity 0.0%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 0.0%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + \color{blue}{\left(1 - 1\right)}\right) + 1\right) - 1} \]

      13. associate--l+ 0.0%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + 1\right) - 1\right)} + 1\right) - 1} \]

      14. +-commutative 0.0%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} - 1\right) + 1\right) - 1} \]

      15. associate--l+ 0.0%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) - 1\right) + \left(1 - 1\right)}} \]

   3. Simplified 11.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 73.3%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 73.3%

         \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right)} \]

      2. sqrt-div 73.3%

         \[\leadsto 2 \cdot \left(\color{blue}{\frac{\sqrt{{\ell}^{2}}}{\sqrt{{k}^{4} \cdot t}}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      3. sqrt-pow1 62.3%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}}{\sqrt{{k}^{4} \cdot t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      4. metadata-eval 62.3%

         \[\leadsto 2 \cdot \left(\frac{{\ell}^{\color{blue}{1}}}{\sqrt{{k}^{4} \cdot t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      5. pow1 62.3%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell}}{\sqrt{{k}^{4} \cdot t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      6. sqrt-prod 62.3%

         \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\sqrt{{k}^{4}} \cdot \sqrt{t}}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      7. sqrt-pow1 62.3%

         \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{{k}^{\left(\frac{4}{2}\right)}} \cdot \sqrt{t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      8. metadata-eval 62.3%

         \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{\color{blue}{2}} \cdot \sqrt{t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      9. sqrt-div 62.3%

         \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \color{blue}{\frac{\sqrt{{\ell}^{2}}}{\sqrt{{k}^{4} \cdot t}}}\right) \]

      10. sqrt-pow1 79.0%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}}{\sqrt{{k}^{4} \cdot t}}\right) \]

      11. metadata-eval 79.0%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{{\ell}^{\color{blue}{1}}}{\sqrt{{k}^{4} \cdot t}}\right) \]

      12. pow1 79.0%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\color{blue}{\ell}}{\sqrt{{k}^{4} \cdot t}}\right) \]

      13. sqrt-prod 79.0%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\ell}{\color{blue}{\sqrt{{k}^{4}} \cdot \sqrt{t}}}\right) \]

      14. sqrt-pow1 84.4%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\ell}{\color{blue}{{k}^{\left(\frac{4}{2}\right)}} \cdot \sqrt{t}}\right) \]

      15. metadata-eval 84.4%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\ell}{{k}^{\color{blue}{2}} \cdot \sqrt{t}}\right) \]

   7. Applied egg-rr 84.4%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)} \]
   8. Step-by-step derivation

      1. unpow2 84.4%

         \[\leadsto 2 \cdot \color{blue}{{\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}} \]

   9. Simplified 84.4%

      \[\leadsto 2 \cdot \color{blue}{{\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-162}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell \cdot \sqrt{\frac{1}{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+219}:\\ \;\;\;\;\left(2 \cdot \left(\cos k \cdot \left(\frac{\ell}{k} \cdot {\left(t \cdot \sin k\right)}^{-2}\right)\right)\right) \cdot \frac{\ell}{\frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.5 \cdot 10^{-162}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell \cdot \sqrt{\frac{1}{t\_m}}}{{k}^{2}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 1.6 \cdot 10^{+219}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t\_m \cdot \sin k\right)}^{-2}\right) \cdot \left(t\_m \cdot \frac{\ell}{k}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t\_m}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.5e-162)
    (* 2.0 (pow (/ (* l (sqrt (/ 1.0 t_m))) (pow k 2.0)) 2.0))
    (if (<= t_m 1.6e+219)
      (*
       (/ l k)
       (* 2.0 (* (* (cos k) (pow (* t_m (sin k)) -2.0)) (* t_m (/ l k)))))
      (* 2.0 (pow (/ l (* (pow k 2.0) (sqrt t_m))) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.5e-162) {
		tmp = 2.0 * pow(((l * sqrt((1.0 / t_m))) / pow(k, 2.0)), 2.0);
	} else if (t_m <= 1.6e+219) {
		tmp = (l / k) * (2.0 * ((cos(k) * pow((t_m * sin(k)), -2.0)) * (t_m * (l / k))));
	} else {
		tmp = 2.0 * pow((l / (pow(k, 2.0) * sqrt(t_m))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.5d-162) then
        tmp = 2.0d0 * (((l * sqrt((1.0d0 / t_m))) / (k ** 2.0d0)) ** 2.0d0)
    else if (t_m <= 1.6d+219) then
        tmp = (l / k) * (2.0d0 * ((cos(k) * ((t_m * sin(k)) ** (-2.0d0))) * (t_m * (l / k))))
    else
        tmp = 2.0d0 * ((l / ((k ** 2.0d0) * sqrt(t_m))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (t_m <= 1.5e-162) {
		tmp = 2.0 * Math.pow(((l * Math.sqrt((1.0 / t_m))) / Math.pow(k, 2.0)), 2.0);
	} else if (t_m <= 1.6e+219) {
		tmp = (l / k) * (2.0 * ((Math.cos(k) * Math.pow((t_m * Math.sin(k)), -2.0)) * (t_m * (l / k))));
	} else {
		tmp = 2.0 * Math.pow((l / (Math.pow(k, 2.0) * Math.sqrt(t_m))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.5e-162:
		tmp = 2.0 * math.pow(((l * math.sqrt((1.0 / t_m))) / math.pow(k, 2.0)), 2.0)
	elif t_m <= 1.6e+219:
		tmp = (l / k) * (2.0 * ((math.cos(k) * math.pow((t_m * math.sin(k)), -2.0)) * (t_m * (l / k))))
	else:
		tmp = 2.0 * math.pow((l / (math.pow(k, 2.0) * math.sqrt(t_m))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.5e-162)
		tmp = Float64(2.0 * (Float64(Float64(l * sqrt(Float64(1.0 / t_m))) / (k ^ 2.0)) ^ 2.0));
	elseif (t_m <= 1.6e+219)
		tmp = Float64(Float64(l / k) * Float64(2.0 * Float64(Float64(cos(k) * (Float64(t_m * sin(k)) ^ -2.0)) * Float64(t_m * Float64(l / k)))));
	else
		tmp = Float64(2.0 * (Float64(l / Float64((k ^ 2.0) * sqrt(t_m))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.5e-162)
		tmp = 2.0 * (((l * sqrt((1.0 / t_m))) / (k ^ 2.0)) ^ 2.0);
	elseif (t_m <= 1.6e+219)
		tmp = (l / k) * (2.0 * ((cos(k) * ((t_m * sin(k)) ^ -2.0)) * (t_m * (l / k))));
	else
		tmp = 2.0 * ((l / ((k ^ 2.0) * sqrt(t_m))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.5e-162], N[(2.0 * N[Power[N[(N[(l * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.6e+219], N[(N[(l / k), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.49999999999999999e-162

   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-/r* 35.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-/l/ 35.2%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 35.8%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/r/ 35.1%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 35.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 35.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 35.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg2 35.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(-\frac{k}{t}\right) \cdot \color{blue}{\frac{k}{-t}} + 1\right) - 1} \]

      9. distribute-frac-neg2 35.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{-t}} \cdot \frac{k}{-t} + 1\right) - 1} \]

      10. unpow2 35.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) - 1} \]

      11. +-rgt-identity 35.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 35.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + \color{blue}{\left(1 - 1\right)}\right) + 1\right) - 1} \]

      13. associate--l+ 35.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + 1\right) - 1\right)} + 1\right) - 1} \]

      14. +-commutative 35.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} - 1\right) + 1\right) - 1} \]

      15. associate--l+ 35.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) - 1\right) + \left(1 - 1\right)}} \]

   3. Simplified 41.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 54.9%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 32.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right)} \]

      2. sqrt-div 7.4%

         \[\leadsto 2 \cdot \left(\color{blue}{\frac{\sqrt{{\ell}^{2}}}{\sqrt{{k}^{4} \cdot t}}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      3. sqrt-pow1 4.8%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}}{\sqrt{{k}^{4} \cdot t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      4. metadata-eval 4.8%

         \[\leadsto 2 \cdot \left(\frac{{\ell}^{\color{blue}{1}}}{\sqrt{{k}^{4} \cdot t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      5. pow1 4.8%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell}}{\sqrt{{k}^{4} \cdot t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      6. sqrt-prod 4.8%

         \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\sqrt{{k}^{4}} \cdot \sqrt{t}}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      7. sqrt-pow1 4.8%

         \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{{k}^{\left(\frac{4}{2}\right)}} \cdot \sqrt{t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      8. metadata-eval 4.8%

         \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{\color{blue}{2}} \cdot \sqrt{t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      9. sqrt-div 4.8%

         \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \color{blue}{\frac{\sqrt{{\ell}^{2}}}{\sqrt{{k}^{4} \cdot t}}}\right) \]

      10. sqrt-pow1 9.4%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}}{\sqrt{{k}^{4} \cdot t}}\right) \]

      11. metadata-eval 9.4%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{{\ell}^{\color{blue}{1}}}{\sqrt{{k}^{4} \cdot t}}\right) \]

      12. pow1 9.4%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\color{blue}{\ell}}{\sqrt{{k}^{4} \cdot t}}\right) \]

      13. sqrt-prod 9.4%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\ell}{\color{blue}{\sqrt{{k}^{4}} \cdot \sqrt{t}}}\right) \]

      14. sqrt-pow1 10.0%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\ell}{\color{blue}{{k}^{\left(\frac{4}{2}\right)}} \cdot \sqrt{t}}\right) \]

      15. metadata-eval 10.0%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\ell}{{k}^{\color{blue}{2}} \cdot \sqrt{t}}\right) \]

   7. Applied egg-rr 10.0%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)} \]
   8. Step-by-step derivation

      1. unpow2 10.0%

         \[\leadsto 2 \cdot \color{blue}{{\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}} \]

   9. Simplified 10.0%

      \[\leadsto 2 \cdot \color{blue}{{\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}} \]

   10. Taylor expanded in l around 0 10.0%

       \[\leadsto 2 \cdot {\color{blue}{\left(\frac{\ell}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]
   11. Step-by-step derivation

       1. associate-*l/ 10.0%

          \[\leadsto 2 \cdot {\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{t}}}{{k}^{2}}\right)}}^{2} \]

   12. Simplified 10.0%

       \[\leadsto 2 \cdot {\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{t}}}{{k}^{2}}\right)}}^{2} \]

   if 1.49999999999999999e-162 < t < 1.60000000000000013e219

   1. Initial program 50.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 50.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-/l/ 50.8%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 53.3%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/r/ 53.3%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 53.3%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 53.3%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 53.3%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg2 53.3%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(-\frac{k}{t}\right) \cdot \color{blue}{\frac{k}{-t}} + 1\right) - 1} \]

      9. distribute-frac-neg2 53.3%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{-t}} \cdot \frac{k}{-t} + 1\right) - 1} \]

      10. unpow2 53.3%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) - 1} \]

      11. +-rgt-identity 53.3%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 53.3%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + \color{blue}{\left(1 - 1\right)}\right) + 1\right) - 1} \]

      13. associate--l+ 53.3%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + 1\right) - 1\right)} + 1\right) - 1} \]

      14. +-commutative 53.3%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} - 1\right) + 1\right) - 1} \]

      15. associate--l+ 53.3%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) - 1\right) + \left(1 - 1\right)}} \]

   3. Simplified 63.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 71.8%

         \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. unpow2 71.8%

         \[\leadsto \frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      3. times-frac 76.7%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

      4. div-inv 76.7%

         \[\leadsto \frac{\color{blue}{\left(\frac{2}{\tan k} \cdot \frac{1}{{t}^{3} \cdot \sin k}\right)} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

      5. frac-times 76.8%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

      6. metadata-eval 76.8%

         \[\leadsto \frac{\frac{\color{blue}{2}}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

   6. Applied egg-rr 76.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

   7. Taylor expanded in k around inf 90.0%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{\ell \cdot \cos k}{k \cdot \left({t}^{2} \cdot {\sin k}^{2}\right)}\right)} \cdot \frac{\ell}{\frac{k}{t}} \]
   8. Step-by-step derivation

      1. times-frac 91.1%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\cos k}{{t}^{2} \cdot {\sin k}^{2}}\right)}\right) \cdot \frac{\ell}{\frac{k}{t}} \]

      2. *-commutative 91.1%

         \[\leadsto \left(2 \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{\color{blue}{{\sin k}^{2} \cdot {t}^{2}}}\right)\right) \cdot \frac{\ell}{\frac{k}{t}} \]

   9. Simplified 91.1%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot {t}^{2}}\right)\right)} \cdot \frac{\ell}{\frac{k}{t}} \]
   10. Step-by-step derivation

       1. associate-*r/ 91.0%

          \[\leadsto \color{blue}{\frac{\left(2 \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{{\sin k}^{2} \cdot {t}^{2}}\right)\right) \cdot \ell}{\frac{k}{t}}} \]

       2. associate-*r* 91.0%

          \[\leadsto \frac{\color{blue}{\left(\left(2 \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot {t}^{2}}\right)} \cdot \ell}{\frac{k}{t}} \]

       3. div-inv 91.0%

          \[\leadsto \frac{\left(\left(2 \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\cos k \cdot \frac{1}{{\sin k}^{2} \cdot {t}^{2}}\right)}\right) \cdot \ell}{\frac{k}{t}} \]

       4. pow-prod-down 92.1%

          \[\leadsto \frac{\left(\left(2 \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \frac{1}{\color{blue}{{\left(\sin k \cdot t\right)}^{2}}}\right)\right) \cdot \ell}{\frac{k}{t}} \]

       5. pow-flip 92.8%

          \[\leadsto \frac{\left(\left(2 \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \color{blue}{{\left(\sin k \cdot t\right)}^{\left(-2\right)}}\right)\right) \cdot \ell}{\frac{k}{t}} \]

       6. metadata-eval 92.8%

          \[\leadsto \frac{\left(\left(2 \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot {\left(\sin k \cdot t\right)}^{\color{blue}{-2}}\right)\right) \cdot \ell}{\frac{k}{t}} \]

   11. Applied egg-rr 92.8%

       \[\leadsto \color{blue}{\frac{\left(\left(2 \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot {\left(\sin k \cdot t\right)}^{-2}\right)\right) \cdot \ell}{\frac{k}{t}}} \]
   12. Step-by-step derivation

       1. associate-/l* 93.0%

          \[\leadsto \color{blue}{\left(\left(2 \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot {\left(\sin k \cdot t\right)}^{-2}\right)\right) \cdot \frac{\ell}{\frac{k}{t}}} \]

       2. associate-/r/ 94.2%

          \[\leadsto \left(\left(2 \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot {\left(\sin k \cdot t\right)}^{-2}\right)\right) \cdot \color{blue}{\left(\frac{\ell}{k} \cdot t\right)} \]

       3. *-commutative 94.2%

          \[\leadsto \left(\left(2 \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot {\left(\sin k \cdot t\right)}^{-2}\right)\right) \cdot \color{blue}{\left(t \cdot \frac{\ell}{k}\right)} \]

       4. associate-*l* 95.3%

          \[\leadsto \color{blue}{\left(2 \cdot \frac{\ell}{k}\right) \cdot \left(\left(\cos k \cdot {\left(\sin k \cdot t\right)}^{-2}\right) \cdot \left(t \cdot \frac{\ell}{k}\right)\right)} \]

       5. *-commutative 95.3%

          \[\leadsto \color{blue}{\left(\frac{\ell}{k} \cdot 2\right)} \cdot \left(\left(\cos k \cdot {\left(\sin k \cdot t\right)}^{-2}\right) \cdot \left(t \cdot \frac{\ell}{k}\right)\right) \]

       6. associate-*l* 95.3%

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(\sin k \cdot t\right)}^{-2}\right) \cdot \left(t \cdot \frac{\ell}{k}\right)\right)\right)} \]

       7. *-commutative 95.3%

          \[\leadsto \frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\color{blue}{\left(t \cdot \sin k\right)}}^{-2}\right) \cdot \left(t \cdot \frac{\ell}{k}\right)\right)\right) \]

   13. Simplified 95.3%

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t \cdot \sin k\right)}^{-2}\right) \cdot \left(t \cdot \frac{\ell}{k}\right)\right)\right)} \]

   if 1.60000000000000013e219 < t

   1. Initial program 0.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 0.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-/l/ 0.0%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 0.0%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/r/ 0.0%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 0.0%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 0.0%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 0.0%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg2 0.0%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(-\frac{k}{t}\right) \cdot \color{blue}{\frac{k}{-t}} + 1\right) - 1} \]

      9. distribute-frac-neg2 0.0%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{-t}} \cdot \frac{k}{-t} + 1\right) - 1} \]

      10. unpow2 0.0%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) - 1} \]

      11. +-rgt-identity 0.0%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 0.0%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + \color{blue}{\left(1 - 1\right)}\right) + 1\right) - 1} \]

      13. associate--l+ 0.0%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + 1\right) - 1\right)} + 1\right) - 1} \]

      14. +-commutative 0.0%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} - 1\right) + 1\right) - 1} \]

      15. associate--l+ 0.0%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) - 1\right) + \left(1 - 1\right)}} \]

   3. Simplified 11.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 73.3%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 73.3%

         \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right)} \]

      2. sqrt-div 73.3%

         \[\leadsto 2 \cdot \left(\color{blue}{\frac{\sqrt{{\ell}^{2}}}{\sqrt{{k}^{4} \cdot t}}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      3. sqrt-pow1 62.3%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}}{\sqrt{{k}^{4} \cdot t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      4. metadata-eval 62.3%

         \[\leadsto 2 \cdot \left(\frac{{\ell}^{\color{blue}{1}}}{\sqrt{{k}^{4} \cdot t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      5. pow1 62.3%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell}}{\sqrt{{k}^{4} \cdot t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      6. sqrt-prod 62.3%

         \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\sqrt{{k}^{4}} \cdot \sqrt{t}}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      7. sqrt-pow1 62.3%

         \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{{k}^{\left(\frac{4}{2}\right)}} \cdot \sqrt{t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      8. metadata-eval 62.3%

         \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{\color{blue}{2}} \cdot \sqrt{t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      9. sqrt-div 62.3%

         \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \color{blue}{\frac{\sqrt{{\ell}^{2}}}{\sqrt{{k}^{4} \cdot t}}}\right) \]

      10. sqrt-pow1 79.0%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}}{\sqrt{{k}^{4} \cdot t}}\right) \]

      11. metadata-eval 79.0%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{{\ell}^{\color{blue}{1}}}{\sqrt{{k}^{4} \cdot t}}\right) \]

      12. pow1 79.0%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\color{blue}{\ell}}{\sqrt{{k}^{4} \cdot t}}\right) \]

      13. sqrt-prod 79.0%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\ell}{\color{blue}{\sqrt{{k}^{4}} \cdot \sqrt{t}}}\right) \]

      14. sqrt-pow1 84.4%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\ell}{\color{blue}{{k}^{\left(\frac{4}{2}\right)}} \cdot \sqrt{t}}\right) \]

      15. metadata-eval 84.4%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\ell}{{k}^{\color{blue}{2}} \cdot \sqrt{t}}\right) \]

   7. Applied egg-rr 84.4%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)} \]
   8. Step-by-step derivation

      1. unpow2 84.4%

         \[\leadsto 2 \cdot \color{blue}{{\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}} \]

   9. Simplified 84.4%

      \[\leadsto 2 \cdot \color{blue}{{\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-162}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell \cdot \sqrt{\frac{1}{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+219}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(2 \cdot \left(\left(\cos k \cdot {\left(t \cdot \sin k\right)}^{-2}\right) \cdot \left(t \cdot \frac{\ell}{k}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-23}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{\ell}{k}, 2 \cdot \frac{\ell}{{k}^{3}}\right)}{k \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6.5e-23)
    (* 2.0 (pow (/ l (* (pow k 2.0) (sqrt t_m))) 2.0))
    (*
     l
     (/
      (fma -0.3333333333333333 (/ l k) (* 2.0 (/ l (pow k 3.0))))
      (* k t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.5e-23) {
		tmp = 2.0 * pow((l / (pow(k, 2.0) * sqrt(t_m))), 2.0);
	} else {
		tmp = l * (fma(-0.3333333333333333, (l / k), (2.0 * (l / pow(k, 3.0)))) / (k * t_m));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6.5e-23)
		tmp = Float64(2.0 * (Float64(l / Float64((k ^ 2.0) * sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(l * Float64(fma(-0.3333333333333333, Float64(l / k), Float64(2.0 * Float64(l / (k ^ 3.0)))) / Float64(k * t_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 6.5e-23], N[(2.0 * N[Power[N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(-0.3333333333333333 * N[(l / k), $MachinePrecision] + N[(2.0 * N[(l / N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.5e-23

   1. Initial program 40.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 40.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-/l/ 40.8%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 42.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/r/ 42.5%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 42.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 42.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 42.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg2 42.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(-\frac{k}{t}\right) \cdot \color{blue}{\frac{k}{-t}} + 1\right) - 1} \]

      9. distribute-frac-neg2 42.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{-t}} \cdot \frac{k}{-t} + 1\right) - 1} \]

      10. unpow2 42.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) - 1} \]

      11. +-rgt-identity 42.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 42.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + \color{blue}{\left(1 - 1\right)}\right) + 1\right) - 1} \]

      13. associate--l+ 42.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + 1\right) - 1\right)} + 1\right) - 1} \]

      14. +-commutative 42.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} - 1\right) + 1\right) - 1} \]

      15. associate--l+ 42.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) - 1\right) + \left(1 - 1\right)}} \]

   3. Simplified 51.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 67.3%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 48.9%

         \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right)} \]

      2. sqrt-div 39.0%

         \[\leadsto 2 \cdot \left(\color{blue}{\frac{\sqrt{{\ell}^{2}}}{\sqrt{{k}^{4} \cdot t}}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      3. sqrt-pow1 25.2%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}}{\sqrt{{k}^{4} \cdot t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      4. metadata-eval 25.2%

         \[\leadsto 2 \cdot \left(\frac{{\ell}^{\color{blue}{1}}}{\sqrt{{k}^{4} \cdot t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      5. pow1 25.2%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell}}{\sqrt{{k}^{4} \cdot t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      6. sqrt-prod 25.2%

         \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\sqrt{{k}^{4}} \cdot \sqrt{t}}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      7. sqrt-pow1 25.2%

         \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{{k}^{\left(\frac{4}{2}\right)}} \cdot \sqrt{t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      8. metadata-eval 25.2%

         \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{\color{blue}{2}} \cdot \sqrt{t}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}\right) \]

      9. sqrt-div 25.2%

         \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \color{blue}{\frac{\sqrt{{\ell}^{2}}}{\sqrt{{k}^{4} \cdot t}}}\right) \]

      10. sqrt-pow1 45.3%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\color{blue}{{\ell}^{\left(\frac{2}{2}\right)}}}{\sqrt{{k}^{4} \cdot t}}\right) \]

      11. metadata-eval 45.3%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{{\ell}^{\color{blue}{1}}}{\sqrt{{k}^{4} \cdot t}}\right) \]

      12. pow1 45.3%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\color{blue}{\ell}}{\sqrt{{k}^{4} \cdot t}}\right) \]

      13. sqrt-prod 45.2%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\ell}{\color{blue}{\sqrt{{k}^{4}} \cdot \sqrt{t}}}\right) \]

      14. sqrt-pow1 46.3%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\ell}{\color{blue}{{k}^{\left(\frac{4}{2}\right)}} \cdot \sqrt{t}}\right) \]

      15. metadata-eval 46.3%

          \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\ell}{{k}^{\color{blue}{2}} \cdot \sqrt{t}}\right) \]

   7. Applied egg-rr 46.3%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}} \cdot \frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)} \]
   8. Step-by-step derivation

      1. unpow2 46.3%

         \[\leadsto 2 \cdot \color{blue}{{\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}} \]

   9. Simplified 46.3%

      \[\leadsto 2 \cdot \color{blue}{{\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}} \]

   if 6.5e-23 < k

   1. Initial program 31.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 31.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-/l/ 31.3%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 31.3%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/r/ 30.1%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 30.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 30.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 30.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg2 30.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(-\frac{k}{t}\right) \cdot \color{blue}{\frac{k}{-t}} + 1\right) - 1} \]

      9. distribute-frac-neg2 30.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{-t}} \cdot \frac{k}{-t} + 1\right) - 1} \]

      10. unpow2 30.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) - 1} \]

      11. +-rgt-identity 30.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 30.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + \color{blue}{\left(1 - 1\right)}\right) + 1\right) - 1} \]

      13. associate--l+ 30.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + 1\right) - 1\right)} + 1\right) - 1} \]

      14. +-commutative 30.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} - 1\right) + 1\right) - 1} \]

      15. associate--l+ 30.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) - 1\right) + \left(1 - 1\right)}} \]

   3. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 37.9%

         \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. unpow2 37.9%

         \[\leadsto \frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      3. times-frac 51.8%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

      4. div-inv 51.8%

         \[\leadsto \frac{\color{blue}{\left(\frac{2}{\tan k} \cdot \frac{1}{{t}^{3} \cdot \sin k}\right)} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

      5. frac-times 51.8%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

      6. metadata-eval 51.8%

         \[\leadsto \frac{\frac{\color{blue}{2}}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

   6. Applied egg-rr 51.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

   7. Taylor expanded in k around 0 44.5%

      \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot \frac{\ell}{k \cdot {t}^{2}} + 2 \cdot \frac{\ell}{{k}^{3} \cdot {t}^{2}}\right)} \cdot \frac{\ell}{\frac{k}{t}} \]

   8. Taylor expanded in t around 0 54.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(-0.3333333333333333 \cdot \frac{\ell}{k} + 2 \cdot \frac{\ell}{{k}^{3}}\right)}{k \cdot t}} \]
   9. Step-by-step derivation

      1. associate-/l* 55.0%

         \[\leadsto \color{blue}{\ell \cdot \frac{-0.3333333333333333 \cdot \frac{\ell}{k} + 2 \cdot \frac{\ell}{{k}^{3}}}{k \cdot t}} \]

      2. fma-define 55.0%

         \[\leadsto \ell \cdot \frac{\color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{\ell}{k}, 2 \cdot \frac{\ell}{{k}^{3}}\right)}}{k \cdot t} \]

   10. Simplified 55.0%

       \[\leadsto \color{blue}{\ell \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{\ell}{k}, 2 \cdot \frac{\ell}{{k}^{3}}\right)}{k \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-23}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{{k}^{2} \cdot \sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{\ell}{k}, 2 \cdot \frac{\ell}{{k}^{3}}\right)}{k \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 75.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{\ell}{k}, 2 \cdot \frac{\ell}{{k}^{3}}\right)}{k \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 3.3e-19)
    (/ 2.0 (pow (* (/ (pow k 2.0) l) (sqrt t_m)) 2.0))
    (*
     l
     (/
      (fma -0.3333333333333333 (/ l k) (* 2.0 (/ l (pow k 3.0))))
      (* k t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.3e-19) {
		tmp = 2.0 / pow(((pow(k, 2.0) / l) * sqrt(t_m)), 2.0);
	} else {
		tmp = l * (fma(-0.3333333333333333, (l / k), (2.0 * (l / pow(k, 3.0)))) / (k * t_m));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.3e-19)
		tmp = Float64(2.0 / (Float64(Float64((k ^ 2.0) / l) * sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(l * Float64(fma(-0.3333333333333333, Float64(l / k), Float64(2.0 * Float64(l / (k ^ 3.0)))) / Float64(k * t_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 3.3e-19], N[(2.0 / N[Power[N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(-0.3333333333333333 * N[(l / k), $MachinePrecision] + N[(2.0 * N[(l / N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.2999999999999998e-19

   1. Initial program 40.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0 76.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   4. Step-by-step derivation

      1. associate-/l* 77.7%

         \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]

   5. Simplified 77.7%

      \[\leadsto \frac{2}{\color{blue}{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 40.3%

         \[\leadsto \frac{2}{\color{blue}{\sqrt{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \cdot \sqrt{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}}} \]

      2. pow2 40.3%

         \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\right)}^{2}}} \]

   7. Applied egg-rr 42.8%

      \[\leadsto \frac{2}{\color{blue}{{\left(k \cdot \frac{\sin k \cdot \sqrt{t}}{\ell \cdot \sqrt{\cos k}}\right)}^{2}}} \]
   8. Step-by-step derivation

      1. associate-/r* 42.8%

         \[\leadsto \frac{2}{{\left(k \cdot \color{blue}{\frac{\frac{\sin k \cdot \sqrt{t}}{\ell}}{\sqrt{\cos k}}}\right)}^{2}} \]

   9. Simplified 42.8%

      \[\leadsto \frac{2}{\color{blue}{{\left(k \cdot \frac{\frac{\sin k \cdot \sqrt{t}}{\ell}}{\sqrt{\cos k}}\right)}^{2}}} \]

   10. Taylor expanded in k around 0 46.6%

       \[\leadsto \frac{2}{{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}}^{2}} \]

   if 3.2999999999999998e-19 < k

   1. Initial program 31.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 31.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-/l/ 31.7%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 31.7%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/r/ 30.5%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 30.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 30.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 30.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg2 30.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(-\frac{k}{t}\right) \cdot \color{blue}{\frac{k}{-t}} + 1\right) - 1} \]

      9. distribute-frac-neg2 30.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{-t}} \cdot \frac{k}{-t} + 1\right) - 1} \]

      10. unpow2 30.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) - 1} \]

      11. +-rgt-identity 30.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 30.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + \color{blue}{\left(1 - 1\right)}\right) + 1\right) - 1} \]

      13. associate--l+ 30.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + 1\right) - 1\right)} + 1\right) - 1} \]

      14. +-commutative 30.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} - 1\right) + 1\right) - 1} \]

      15. associate--l+ 30.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) - 1\right) + \left(1 - 1\right)}} \]

   3. Simplified 36.9%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 38.4%

         \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. unpow2 38.4%

         \[\leadsto \frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      3. times-frac 52.4%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

      4. div-inv 52.4%

         \[\leadsto \frac{\color{blue}{\left(\frac{2}{\tan k} \cdot \frac{1}{{t}^{3} \cdot \sin k}\right)} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

      5. frac-times 52.4%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

      6. metadata-eval 52.4%

         \[\leadsto \frac{\frac{\color{blue}{2}}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

   6. Applied egg-rr 52.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

   7. Taylor expanded in k around 0 43.8%

      \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot \frac{\ell}{k \cdot {t}^{2}} + 2 \cdot \frac{\ell}{{k}^{3} \cdot {t}^{2}}\right)} \cdot \frac{\ell}{\frac{k}{t}} \]

   8. Taylor expanded in t around 0 54.4%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(-0.3333333333333333 \cdot \frac{\ell}{k} + 2 \cdot \frac{\ell}{{k}^{3}}\right)}{k \cdot t}} \]
   9. Step-by-step derivation

      1. associate-/l* 54.4%

         \[\leadsto \color{blue}{\ell \cdot \frac{-0.3333333333333333 \cdot \frac{\ell}{k} + 2 \cdot \frac{\ell}{{k}^{3}}}{k \cdot t}} \]

      2. fma-define 54.4%

         \[\leadsto \ell \cdot \frac{\color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{\ell}{k}, 2 \cdot \frac{\ell}{{k}^{3}}\right)}}{k \cdot t} \]

   10. Simplified 54.4%

       \[\leadsto \color{blue}{\ell \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{\ell}{k}, 2 \cdot \frac{\ell}{{k}^{3}}\right)}{k \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{{\left(\frac{{k}^{2}}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{\ell}{k}, 2 \cdot \frac{\ell}{{k}^{3}}\right)}{k \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 70.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\ell}{t\_m \cdot {k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{\ell}{k}, 2 \cdot \frac{\ell}{{k}^{3}}\right)}{k \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 5e-24)
    (* 2.0 (* l (/ l (* t_m (pow k 4.0)))))
    (*
     l
     (/
      (fma -0.3333333333333333 (/ l k) (* 2.0 (/ l (pow k 3.0))))
      (* k t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5e-24) {
		tmp = 2.0 * (l * (l / (t_m * pow(k, 4.0))));
	} else {
		tmp = l * (fma(-0.3333333333333333, (l / k), (2.0 * (l / pow(k, 3.0)))) / (k * t_m));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 5e-24)
		tmp = Float64(2.0 * Float64(l * Float64(l / Float64(t_m * (k ^ 4.0)))));
	else
		tmp = Float64(l * Float64(fma(-0.3333333333333333, Float64(l / k), Float64(2.0 * Float64(l / (k ^ 3.0)))) / Float64(k * t_m)));
	end
	return Float64(t_s * tmp)
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 5e-24], N[(2.0 * N[(l * N[(l / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(-0.3333333333333333 * N[(l / k), $MachinePrecision] + N[(2.0 * N[(l / N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.9999999999999998e-24

   1. Initial program 40.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 40.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-/l/ 40.8%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 42.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/r/ 42.5%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 42.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 42.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 42.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg2 42.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(-\frac{k}{t}\right) \cdot \color{blue}{\frac{k}{-t}} + 1\right) - 1} \]

      9. distribute-frac-neg2 42.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{-t}} \cdot \frac{k}{-t} + 1\right) - 1} \]

      10. unpow2 42.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) - 1} \]

      11. +-rgt-identity 42.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 42.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + \color{blue}{\left(1 - 1\right)}\right) + 1\right) - 1} \]

      13. associate--l+ 42.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + 1\right) - 1\right)} + 1\right) - 1} \]

      14. +-commutative 42.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} - 1\right) + 1\right) - 1} \]

      15. associate--l+ 42.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) - 1\right) + \left(1 - 1\right)}} \]

   3. Simplified 51.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 67.3%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   6. Step-by-step derivation

      1. pow2 67.3%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-un-lft-identity 67.3%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot \left({k}^{4} \cdot t\right)}} \]

      3. times-frac 77.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{{k}^{4} \cdot t}\right)} \]

      4. *-commutative 77.7%

         \[\leadsto 2 \cdot \left(\frac{\ell}{1} \cdot \frac{\ell}{\color{blue}{t \cdot {k}^{4}}}\right) \]

   7. Applied egg-rr 77.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{t \cdot {k}^{4}}\right)} \]

   if 4.9999999999999998e-24 < k

   1. Initial program 31.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 31.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-/l/ 31.3%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 31.3%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/r/ 30.1%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 30.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 30.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 30.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg2 30.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(-\frac{k}{t}\right) \cdot \color{blue}{\frac{k}{-t}} + 1\right) - 1} \]

      9. distribute-frac-neg2 30.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{-t}} \cdot \frac{k}{-t} + 1\right) - 1} \]

      10. unpow2 30.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) - 1} \]

      11. +-rgt-identity 30.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 30.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + \color{blue}{\left(1 - 1\right)}\right) + 1\right) - 1} \]

      13. associate--l+ 30.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + 1\right) - 1\right)} + 1\right) - 1} \]

      14. +-commutative 30.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} - 1\right) + 1\right) - 1} \]

      15. associate--l+ 30.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) - 1\right) + \left(1 - 1\right)}} \]

   3. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 37.9%

         \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. unpow2 37.9%

         \[\leadsto \frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      3. times-frac 51.8%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

      4. div-inv 51.8%

         \[\leadsto \frac{\color{blue}{\left(\frac{2}{\tan k} \cdot \frac{1}{{t}^{3} \cdot \sin k}\right)} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

      5. frac-times 51.8%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

      6. metadata-eval 51.8%

         \[\leadsto \frac{\frac{\color{blue}{2}}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

   6. Applied egg-rr 51.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

   7. Taylor expanded in k around 0 44.5%

      \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot \frac{\ell}{k \cdot {t}^{2}} + 2 \cdot \frac{\ell}{{k}^{3} \cdot {t}^{2}}\right)} \cdot \frac{\ell}{\frac{k}{t}} \]

   8. Taylor expanded in t around 0 54.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(-0.3333333333333333 \cdot \frac{\ell}{k} + 2 \cdot \frac{\ell}{{k}^{3}}\right)}{k \cdot t}} \]
   9. Step-by-step derivation

      1. associate-/l* 55.0%

         \[\leadsto \color{blue}{\ell \cdot \frac{-0.3333333333333333 \cdot \frac{\ell}{k} + 2 \cdot \frac{\ell}{{k}^{3}}}{k \cdot t}} \]

      2. fma-define 55.0%

         \[\leadsto \ell \cdot \frac{\color{blue}{\mathsf{fma}\left(-0.3333333333333333, \frac{\ell}{k}, 2 \cdot \frac{\ell}{{k}^{3}}\right)}}{k \cdot t} \]

   10. Simplified 55.0%

       \[\leadsto \color{blue}{\ell \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{\ell}{k}, 2 \cdot \frac{\ell}{{k}^{3}}\right)}{k \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{\ell}{k}, 2 \cdot \frac{\ell}{{k}^{3}}\right)}{k \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.55 \cdot 10^{-20}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\ell}{t\_m \cdot {k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{\ell}{{k}^{3}} + \frac{\ell}{k} \cdot -0.3333333333333333\right)}{k \cdot t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.55e-20)
    (* 2.0 (* l (/ l (* t_m (pow k 4.0)))))
    (/
     (* l (+ (* 2.0 (/ l (pow k 3.0))) (* (/ l k) -0.3333333333333333)))
     (* k t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.55e-20) {
		tmp = 2.0 * (l * (l / (t_m * pow(k, 4.0))));
	} else {
		tmp = (l * ((2.0 * (l / pow(k, 3.0))) + ((l / k) * -0.3333333333333333))) / (k * t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.55d-20) then
        tmp = 2.0d0 * (l * (l / (t_m * (k ** 4.0d0))))
    else
        tmp = (l * ((2.0d0 * (l / (k ** 3.0d0))) + ((l / k) * (-0.3333333333333333d0)))) / (k * t_m)
    end if
    code = t_s * tmp
end function

Java

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) {
	double tmp;
	if (k <= 2.55e-20) {
		tmp = 2.0 * (l * (l / (t_m * Math.pow(k, 4.0))));
	} else {
		tmp = (l * ((2.0 * (l / Math.pow(k, 3.0))) + ((l / k) * -0.3333333333333333))) / (k * t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 2.55e-20:
		tmp = 2.0 * (l * (l / (t_m * math.pow(k, 4.0))))
	else:
		tmp = (l * ((2.0 * (l / math.pow(k, 3.0))) + ((l / k) * -0.3333333333333333))) / (k * t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.55e-20)
		tmp = Float64(2.0 * Float64(l * Float64(l / Float64(t_m * (k ^ 4.0)))));
	else
		tmp = Float64(Float64(l * Float64(Float64(2.0 * Float64(l / (k ^ 3.0))) + Float64(Float64(l / k) * -0.3333333333333333))) / Float64(k * t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2.55e-20)
		tmp = 2.0 * (l * (l / (t_m * (k ^ 4.0))));
	else
		tmp = (l * ((2.0 * (l / (k ^ 3.0))) + ((l / k) * -0.3333333333333333))) / (k * t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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_] := N[(t$95$s * If[LessEqual[k, 2.55e-20], N[(2.0 * N[(l * N[(l / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(N[(2.0 * N[(l / N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l / k), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.55000000000000009e-20

   1. Initial program 40.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 40.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-/l/ 40.8%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 42.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/r/ 42.5%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 42.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 42.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 42.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg2 42.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(-\frac{k}{t}\right) \cdot \color{blue}{\frac{k}{-t}} + 1\right) - 1} \]

      9. distribute-frac-neg2 42.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{-t}} \cdot \frac{k}{-t} + 1\right) - 1} \]

      10. unpow2 42.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) - 1} \]

      11. +-rgt-identity 42.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 42.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + \color{blue}{\left(1 - 1\right)}\right) + 1\right) - 1} \]

      13. associate--l+ 42.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + 1\right) - 1\right)} + 1\right) - 1} \]

      14. +-commutative 42.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} - 1\right) + 1\right) - 1} \]

      15. associate--l+ 42.5%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) - 1\right) + \left(1 - 1\right)}} \]

   3. Simplified 51.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 67.3%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   6. Step-by-step derivation

      1. pow2 67.3%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-un-lft-identity 67.3%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot \left({k}^{4} \cdot t\right)}} \]

      3. times-frac 77.7%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{{k}^{4} \cdot t}\right)} \]

      4. *-commutative 77.7%

         \[\leadsto 2 \cdot \left(\frac{\ell}{1} \cdot \frac{\ell}{\color{blue}{t \cdot {k}^{4}}}\right) \]

   7. Applied egg-rr 77.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{t \cdot {k}^{4}}\right)} \]

   if 2.55000000000000009e-20 < k

   1. Initial program 31.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 31.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

      2. associate-/l/ 31.3%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      3. associate-*l/ 31.3%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      4. associate-/r/ 30.1%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

      5. +-commutative 30.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

      6. unpow2 30.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

      7. sqr-neg 30.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

      8. distribute-frac-neg2 30.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(-\frac{k}{t}\right) \cdot \color{blue}{\frac{k}{-t}} + 1\right) - 1} \]

      9. distribute-frac-neg2 30.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{-t}} \cdot \frac{k}{-t} + 1\right) - 1} \]

      10. unpow2 30.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) - 1} \]

      11. +-rgt-identity 30.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

      12. metadata-eval 30.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + \color{blue}{\left(1 - 1\right)}\right) + 1\right) - 1} \]

      13. associate--l+ 30.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + 1\right) - 1\right)} + 1\right) - 1} \]

      14. +-commutative 30.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} - 1\right) + 1\right) - 1} \]

      15. associate--l+ 30.1%

          \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) - 1\right) + \left(1 - 1\right)}} \]

   3. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 37.9%

         \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}}{{\left(\frac{k}{t}\right)}^{2}} \]

      2. unpow2 37.9%

         \[\leadsto \frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}{\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}} \]

      3. times-frac 51.8%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

      4. div-inv 51.8%

         \[\leadsto \frac{\color{blue}{\left(\frac{2}{\tan k} \cdot \frac{1}{{t}^{3} \cdot \sin k}\right)} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

      5. frac-times 51.8%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

      6. metadata-eval 51.8%

         \[\leadsto \frac{\frac{\color{blue}{2}}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}} \]

   6. Applied egg-rr 51.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \ell}{\frac{k}{t}} \cdot \frac{\ell}{\frac{k}{t}}} \]

   7. Taylor expanded in k around 0 44.5%

      \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot \frac{\ell}{k \cdot {t}^{2}} + 2 \cdot \frac{\ell}{{k}^{3} \cdot {t}^{2}}\right)} \cdot \frac{\ell}{\frac{k}{t}} \]

   8. Taylor expanded in t around 0 54.9%

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(-0.3333333333333333 \cdot \frac{\ell}{k} + 2 \cdot \frac{\ell}{{k}^{3}}\right)}{k \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.55 \cdot 10^{-20}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{\ell}{{k}^{3}} + \frac{\ell}{k} \cdot -0.3333333333333333\right)}{k \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 68.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \left(\left(\ell \cdot {k}^{-4}\right) \cdot \frac{\ell}{t\_m}\right)\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (* (* l (pow k -4.0)) (/ l t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((l * pow(k, -4.0)) * (l / t_m)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * ((l * (k ** (-4.0d0))) * (l / t_m)))
end function

Java

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) {
	return t_s * (2.0 * ((l * Math.pow(k, -4.0)) * (l / t_m)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * ((l * math.pow(k, -4.0)) * (l / t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64(Float64(l * (k ^ -4.0)) * Float64(l / t_m))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * ((l * (k ^ -4.0)) * (l / t_m)));
end

Wolfram

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_] := N[(t$95$s * N[(2.0 * N[(N[(l * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 37.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

   2. associate-/l/ 37.8%

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   3. associate-*l/ 39.0%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   4. associate-/r/ 38.6%

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   5. +-commutative 38.6%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

   6. unpow2 38.6%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

   7. sqr-neg 38.6%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

   8. distribute-frac-neg2 38.6%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(-\frac{k}{t}\right) \cdot \color{blue}{\frac{k}{-t}} + 1\right) - 1} \]

   9. distribute-frac-neg2 38.6%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{-t}} \cdot \frac{k}{-t} + 1\right) - 1} \]

   10. unpow2 38.6%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) - 1} \]

   11. +-rgt-identity 38.6%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

   12. metadata-eval 38.6%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + \color{blue}{\left(1 - 1\right)}\right) + 1\right) - 1} \]

   13. associate--l+ 38.6%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + 1\right) - 1\right)} + 1\right) - 1} \]

   14. +-commutative 38.6%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} - 1\right) + 1\right) - 1} \]

   15. associate--l+ 38.6%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) - 1\right) + \left(1 - 1\right)}} \]

3. Simplified 46.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2}}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 62.7%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
6. Step-by-step derivation

   1. pow2 62.7%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. times-frac 69.5%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)} \]

7. Applied egg-rr 69.5%

   \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)} \]
8. Step-by-step derivation

   1. associate-*r/ 70.0%

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{{k}^{4}} \cdot \ell}{t}} \]

   2. div-inv 69.6%

      \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \frac{1}{{k}^{4}}\right)} \cdot \ell}{t} \]

   3. pow-flip 69.6%

      \[\leadsto 2 \cdot \frac{\left(\ell \cdot \color{blue}{{k}^{\left(-4\right)}}\right) \cdot \ell}{t} \]

   4. metadata-eval 69.6%

      \[\leadsto 2 \cdot \frac{\left(\ell \cdot {k}^{\color{blue}{-4}}\right) \cdot \ell}{t} \]

9. Applied egg-rr 69.6%

   \[\leadsto 2 \cdot \color{blue}{\frac{\left(\ell \cdot {k}^{-4}\right) \cdot \ell}{t}} \]
10. Step-by-step derivation

    1. associate-/l* 69.1%

       \[\leadsto 2 \cdot \color{blue}{\left(\left(\ell \cdot {k}^{-4}\right) \cdot \frac{\ell}{t}\right)} \]

11. Simplified 69.1%

    \[\leadsto 2 \cdot \color{blue}{\left(\left(\ell \cdot {k}^{-4}\right) \cdot \frac{\ell}{t}\right)} \]

12. Final simplification 69.1%

    \[\leadsto 2 \cdot \left(\left(\ell \cdot {k}^{-4}\right) \cdot \frac{\ell}{t}\right) \]
13. Add Preprocessing

Alternative 18: 68.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t\_m}\right)\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (* (/ l (pow k 4.0)) (/ l t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((l / pow(k, 4.0)) * (l / t_m)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * ((l / (k ** 4.0d0)) * (l / t_m)))
end function

Java

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) {
	return t_s * (2.0 * ((l / Math.pow(k, 4.0)) * (l / t_m)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * ((l / math.pow(k, 4.0)) * (l / t_m)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64(Float64(l / (k ^ 4.0)) * Float64(l / t_m))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * ((l / (k ^ 4.0)) * (l / t_m)));
end

Wolfram

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_] := N[(t$95$s * N[(2.0 * N[(N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 37.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

   2. associate-/l/ 37.8%

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   3. associate-*l/ 39.0%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   4. associate-/r/ 38.6%

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   5. +-commutative 38.6%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

   6. unpow2 38.6%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

   7. sqr-neg 38.6%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

   8. distribute-frac-neg2 38.6%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(-\frac{k}{t}\right) \cdot \color{blue}{\frac{k}{-t}} + 1\right) - 1} \]

   9. distribute-frac-neg2 38.6%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{-t}} \cdot \frac{k}{-t} + 1\right) - 1} \]

   10. unpow2 38.6%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) - 1} \]

   11. +-rgt-identity 38.6%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

   12. metadata-eval 38.6%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + \color{blue}{\left(1 - 1\right)}\right) + 1\right) - 1} \]

   13. associate--l+ 38.6%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + 1\right) - 1\right)} + 1\right) - 1} \]

   14. +-commutative 38.6%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} - 1\right) + 1\right) - 1} \]

   15. associate--l+ 38.6%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) - 1\right) + \left(1 - 1\right)}} \]

3. Simplified 46.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2}}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 62.7%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
6. Step-by-step derivation

   1. pow2 62.7%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. times-frac 69.5%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)} \]

7. Applied egg-rr 69.5%

   \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)} \]

8. Final simplification 69.5%

   \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right) \]
9. Add Preprocessing

Alternative 19: 68.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \frac{\ell \cdot \left(\ell \cdot {k}^{-4}\right)}{t\_m}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (/ (* l (* l (pow k -4.0))) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((l * (l * pow(k, -4.0))) / t_m));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * ((l * (l * (k ** (-4.0d0)))) / t_m))
end function

Java

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) {
	return t_s * (2.0 * ((l * (l * Math.pow(k, -4.0))) / t_m));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * ((l * (l * math.pow(k, -4.0))) / t_m))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64(Float64(l * Float64(l * (k ^ -4.0))) / t_m)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * ((l * (l * (k ^ -4.0))) / t_m));
end

Wolfram

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_] := N[(t$95$s * N[(2.0 * N[(N[(l * N[(l * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.9%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 37.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]

   2. associate-/l/ 37.8%

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   3. associate-*l/ 39.0%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   4. associate-/r/ 38.6%

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   5. +-commutative 38.6%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]

   6. unpow2 38.6%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) - 1} \]

   7. sqr-neg 38.6%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) - 1} \]

   8. distribute-frac-neg2 38.6%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(-\frac{k}{t}\right) \cdot \color{blue}{\frac{k}{-t}} + 1\right) - 1} \]

   9. distribute-frac-neg2 38.6%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\frac{k}{-t}} \cdot \frac{k}{-t} + 1\right) - 1} \]

   10. unpow2 38.6%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) - 1} \]

   11. +-rgt-identity 38.6%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left({\left(\frac{k}{-t}\right)}^{2} + 0\right)} + 1\right) - 1} \]

   12. metadata-eval 38.6%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + \color{blue}{\left(1 - 1\right)}\right) + 1\right) - 1} \]

   13. associate--l+ 38.6%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\color{blue}{\left(\left({\left(\frac{k}{-t}\right)}^{2} + 1\right) - 1\right)} + 1\right) - 1} \]

   14. +-commutative 38.6%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\left(\left(\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} - 1\right) + 1\right) - 1} \]

   15. associate--l+ 38.6%

       \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) - 1\right) + \left(1 - 1\right)}} \]

3. Simplified 46.7%

   \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{k}{t}\right)}^{2}}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 62.7%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
6. Step-by-step derivation

   1. pow2 62.7%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. times-frac 69.5%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)} \]

7. Applied egg-rr 69.5%

   \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)} \]
8. Step-by-step derivation

   1. associate-*r/ 70.0%

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{{k}^{4}} \cdot \ell}{t}} \]

   2. div-inv 69.6%

      \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \frac{1}{{k}^{4}}\right)} \cdot \ell}{t} \]

   3. pow-flip 69.6%

      \[\leadsto 2 \cdot \frac{\left(\ell \cdot \color{blue}{{k}^{\left(-4\right)}}\right) \cdot \ell}{t} \]

   4. metadata-eval 69.6%

      \[\leadsto 2 \cdot \frac{\left(\ell \cdot {k}^{\color{blue}{-4}}\right) \cdot \ell}{t} \]

9. Applied egg-rr 69.6%

   \[\leadsto 2 \cdot \color{blue}{\frac{\left(\ell \cdot {k}^{-4}\right) \cdot \ell}{t}} \]

10. Final simplification 69.6%

    \[\leadsto 2 \cdot \frac{\ell \cdot \left(\ell \cdot {k}^{-4}\right)}{t} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(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))))