Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.4% → 94.7%

Time: 24.1s

Alternatives: 9

Speedup: 2.0×

• 35.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 34.4% 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: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 5.2 \cdot 10^{-38}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k_m} \cdot \frac{\sqrt{\frac{\cos k_m}{t_m}}}{\sin k_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k_m \cdot \left(\frac{\ell}{k_m} \cdot \frac{\ell}{k_m}\right)}{t_m \cdot {\sin k_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 5.2e-38)
    (* 2.0 (pow (* (/ l k_m) (/ (sqrt (/ (cos k_m) t_m)) (sin k_m))) 2.0))
    (*
     2.0
     (/ (* (cos k_m) (* (/ l k_m) (/ l k_m))) (* t_m (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5.2e-38) {
		tmp = 2.0 * pow(((l / k_m) * (sqrt((cos(k_m) / t_m)) / sin(k_m))), 2.0);
	} else {
		tmp = 2.0 * ((cos(k_m) * ((l / k_m) * (l / k_m))) / (t_m * pow(sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5.2d-38) then
        tmp = 2.0d0 * (((l / k_m) * (sqrt((cos(k_m) / t_m)) / sin(k_m))) ** 2.0d0)
    else
        tmp = 2.0d0 * ((cos(k_m) * ((l / k_m) * (l / k_m))) / (t_m * (sin(k_m) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5.2e-38) {
		tmp = 2.0 * Math.pow(((l / k_m) * (Math.sqrt((Math.cos(k_m) / t_m)) / Math.sin(k_m))), 2.0);
	} else {
		tmp = 2.0 * ((Math.cos(k_m) * ((l / k_m) * (l / k_m))) / (t_m * Math.pow(Math.sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 5.2e-38:
		tmp = 2.0 * math.pow(((l / k_m) * (math.sqrt((math.cos(k_m) / t_m)) / math.sin(k_m))), 2.0)
	else:
		tmp = 2.0 * ((math.cos(k_m) * ((l / k_m) * (l / k_m))) / (t_m * math.pow(math.sin(k_m), 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 5.2e-38)
		tmp = Float64(2.0 * (Float64(Float64(l / k_m) * Float64(sqrt(Float64(cos(k_m) / t_m)) / sin(k_m))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) * Float64(Float64(l / k_m) * Float64(l / k_m))) / Float64(t_m * (sin(k_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 5.2e-38)
		tmp = 2.0 * (((l / k_m) * (sqrt((cos(k_m) / t_m)) / sin(k_m))) ^ 2.0);
	else
		tmp = 2.0 * ((cos(k_m) * ((l / k_m) * (l / k_m))) / (t_m * (sin(k_m) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 5.2e-38], N[(2.0 * N[Power[N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.20000000000000022e-38

   1. Initial program 39.4%

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

      1. associate-*l* 39.4%

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

      2. associate-/r* 39.4%

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

      3. sub-neg 39.4%

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

      4. distribute-rgt-in 33.6%

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

      5. unpow2 33.6%

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

      6. times-frac 23.2%

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

      7. sqr-neg 23.2%

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

      8. times-frac 33.6%

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

      9. unpow2 33.6%

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

      10. distribute-rgt-in 39.4%

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

      11. +-commutative 39.4%

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

      12. associate-+l+ 44.5%

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

   3. Simplified 44.5%

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

   5. Taylor expanded in t around 0 76.9%

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

      1. times-frac 78.9%

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

   7. Simplified 78.9%

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

      1. associate-*l/ 77.3%

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

   9. Applied egg-rr 77.3%

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

       1. expm1-log1p-u 44.1%

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

       2. expm1-udef 39.2%

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

   11. Applied egg-rr 37.0%

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

       1. expm1-def 42.4%

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

       2. expm1-log1p 43.1%

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

       3. associate-/l* 40.5%

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

       4. associate-/r/ 43.2%

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

   13. Simplified 43.2%

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

   if 5.20000000000000022e-38 < k

   1. Initial program 29.2%

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

      1. associate-*l* 29.2%

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

      2. associate-/r* 29.2%

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

      3. sub-neg 29.2%

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

      4. distribute-rgt-in 29.2%

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

      5. unpow2 29.2%

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

      6. times-frac 25.5%

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

      7. sqr-neg 25.5%

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

      8. times-frac 29.2%

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

      9. unpow2 29.2%

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

      10. distribute-rgt-in 29.2%

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

      11. +-commutative 29.2%

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

      12. associate-+l+ 46.0%

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

   3. Simplified 46.0%

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

   5. Taylor expanded in t around 0 73.0%

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

      1. times-frac 73.8%

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

   7. Simplified 73.8%

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

      1. associate-*r/ 73.7%

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

      2. div-inv 73.7%

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

      3. pow-flip 73.8%

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

      4. metadata-eval 73.8%

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

   9. Applied egg-rr 73.8%

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

   10. Taylor expanded in l around 0 73.7%

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

       1. unpow2 73.7%

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

       2. unpow2 73.7%

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

       3. times-frac 95.1%

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

       4. *-rgt-identity 95.1%

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

       5. associate-*r/ 95.1%

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

       6. *-rgt-identity 95.1%

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

       7. associate-*r/ 95.0%

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

       8. unpow2 95.0%

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

       9. associate-*r/ 95.1%

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

       10. *-rgt-identity 95.1%

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

   12. Simplified 95.1%

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

       1. unpow2 95.1%

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

   14. Applied egg-rr 95.1%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-38}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k} \cdot \frac{\sqrt{\frac{\cos k}{t}}}{\sin k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t \cdot {\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-315}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell \cdot {k_m}^{-2}}{\sqrt{t_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k_m \cdot \left(\frac{\ell}{k_m} \cdot \frac{\ell}{k_m}\right)}{t_m \cdot {\sin k_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 5e-315)
    (* 2.0 (pow (/ (* l (pow k_m -2.0)) (sqrt t_m)) 2.0))
    (*
     2.0
     (/ (* (cos k_m) (* (/ l k_m) (/ l k_m))) (* t_m (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 5e-315) {
		tmp = 2.0 * pow(((l * pow(k_m, -2.0)) / sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 * ((cos(k_m) * ((l / k_m) * (l / k_m))) / (t_m * pow(sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 5d-315) then
        tmp = 2.0d0 * (((l * (k_m ** (-2.0d0))) / sqrt(t_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * ((cos(k_m) * ((l / k_m) * (l / k_m))) / (t_m * (sin(k_m) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 5e-315) {
		tmp = 2.0 * Math.pow(((l * Math.pow(k_m, -2.0)) / Math.sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 * ((Math.cos(k_m) * ((l / k_m) * (l / k_m))) / (t_m * Math.pow(Math.sin(k_m), 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 5e-315:
		tmp = 2.0 * math.pow(((l * math.pow(k_m, -2.0)) / math.sqrt(t_m)), 2.0)
	else:
		tmp = 2.0 * ((math.cos(k_m) * ((l / k_m) * (l / k_m))) / (t_m * math.pow(math.sin(k_m), 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 5e-315)
		tmp = Float64(2.0 * (Float64(Float64(l * (k_m ^ -2.0)) / sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k_m) * Float64(Float64(l / k_m) * Float64(l / k_m))) / Float64(t_m * (sin(k_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 5e-315)
		tmp = 2.0 * (((l * (k_m ^ -2.0)) / sqrt(t_m)) ^ 2.0);
	else
		tmp = 2.0 * ((cos(k_m) * ((l / k_m) * (l / k_m))) / (t_m * (sin(k_m) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 5e-315], N[(2.0 * N[Power[N[(N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.0000000023e-315

   1. Initial program 23.1%

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

      1. associate-*l* 23.1%

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

      2. associate-/r* 23.1%

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

      3. sub-neg 23.1%

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

      4. distribute-rgt-in 23.1%

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

      5. unpow2 23.1%

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

      6. times-frac 21.5%

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

      7. sqr-neg 21.5%

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

      8. times-frac 23.1%

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

      9. unpow2 23.1%

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

      10. distribute-rgt-in 23.1%

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

      11. +-commutative 23.1%

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

      12. associate-+l+ 43.7%

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

   3. Simplified 43.7%

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

   5. Taylor expanded in k around 0 59.4%

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

      1. *-commutative 59.4%

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

      2. associate-/r* 60.5%

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

   7. Simplified 60.5%

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

      1. expm1-log1p-u 60.5%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\right)\right)} \]

      2. expm1-udef 60.4%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\right)} - 1\right)} \]

      3. div-inv 60.4%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{4}}}\right)} - 1\right) \]

      4. pow-flip 60.4%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1\right) \]

      5. metadata-eval 60.4%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{\color{blue}{-4}}\right)} - 1\right) \]

   9. Applied egg-rr 60.4%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 60.4%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)\right)} \]

       2. expm1-log1p 60.4%

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

   11. Simplified 60.4%

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

       1. rem-cbrt-cube 59.4%

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

       2. unpow1/3 59.4%

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

       3. sqr-pow 59.4%

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

       4. pow2 59.4%

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

       5. sqrt-pow1 59.4%

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

       6. pow-pow 60.4%

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

       7. metadata-eval 60.4%

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

       8. pow1 60.4%

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

       9. *-commutative 60.4%

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

       10. sqrt-prod 58.8%

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

       11. sqrt-pow1 59.9%

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

       12. metadata-eval 59.9%

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

       13. sqrt-div 33.4%

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

       14. pow2 33.4%

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

       15. sqrt-prod 14.9%

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

       16. add-sqr-sqrt 44.5%

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

   13. Applied egg-rr 44.5%

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

       1. associate-*r/ 44.5%

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

   15. Simplified 44.5%

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

   if 5.0000000023e-315 < (*.f64 l l)

   1. Initial program 40.2%

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

      1. associate-*l* 40.2%

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

      2. associate-/r* 40.2%

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

      3. sub-neg 40.2%

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

      4. distribute-rgt-in 35.0%

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

      5. unpow2 35.0%

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

      6. times-frac 24.7%

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

      7. sqr-neg 24.7%

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

      8. times-frac 35.0%

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

      9. unpow2 35.0%

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

      10. distribute-rgt-in 40.2%

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

      11. +-commutative 40.2%

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

      12. associate-+l+ 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in t around 0 80.5%

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

      1. times-frac 82.5%

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

   7. Simplified 82.5%

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

      1. associate-*r/ 82.5%

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

      2. div-inv 82.5%

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

      3. pow-flip 82.5%

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

      4. metadata-eval 82.5%

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

   9. Applied egg-rr 82.5%

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

   10. Taylor expanded in l around 0 82.5%

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

       1. unpow2 82.5%

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

       2. unpow2 82.5%

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

       3. times-frac 98.1%

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

       4. *-rgt-identity 98.1%

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

       5. associate-*r/ 98.2%

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

       6. *-rgt-identity 98.2%

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

       7. associate-*r/ 98.1%

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

       8. unpow2 98.1%

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

       9. associate-*r/ 98.1%

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

       10. *-rgt-identity 98.1%

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

   12. Simplified 98.1%

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

       1. unpow2 98.1%

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

   14. Applied egg-rr 98.1%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-315}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell \cdot {k}^{-2}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t \cdot {\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.1 \cdot 10^{+74}:\\ \;\;\;\;2 \cdot {\left({k_m}^{-2} \cdot \frac{\ell}{\sqrt{t_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \frac{{\left(\frac{\ell}{k_m}\right)}^{2}}{t_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.1e+74)
    (* 2.0 (pow (* (pow k_m -2.0) (/ l (sqrt t_m))) 2.0))
    (* 2.0 (* -0.16666666666666666 (/ (pow (/ l k_m) 2.0) t_m))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.1e+74) {
		tmp = 2.0 * pow((pow(k_m, -2.0) * (l / sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * (-0.16666666666666666 * (pow((l / k_m), 2.0) / t_m));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.1d+74) then
        tmp = 2.0d0 * (((k_m ** (-2.0d0)) * (l / sqrt(t_m))) ** 2.0d0)
    else
        tmp = 2.0d0 * ((-0.16666666666666666d0) * (((l / k_m) ** 2.0d0) / t_m))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.1e+74) {
		tmp = 2.0 * Math.pow((Math.pow(k_m, -2.0) * (l / Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 * (-0.16666666666666666 * (Math.pow((l / k_m), 2.0) / t_m));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.1e+74:
		tmp = 2.0 * math.pow((math.pow(k_m, -2.0) * (l / math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 * (-0.16666666666666666 * (math.pow((l / k_m), 2.0) / t_m))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.1e+74)
		tmp = Float64(2.0 * (Float64((k_m ^ -2.0) * Float64(l / sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(-0.16666666666666666 * Float64((Float64(l / k_m) ^ 2.0) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.1e+74)
		tmp = 2.0 * (((k_m ^ -2.0) * (l / sqrt(t_m))) ^ 2.0);
	else
		tmp = 2.0 * (-0.16666666666666666 * (((l / k_m) ^ 2.0) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.1e+74], N[(2.0 * N[Power[N[(N[Power[k$95$m, -2.0], $MachinePrecision] * N[(l / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(-0.16666666666666666 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.1000000000000001e74

   1. Initial program 37.4%

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

      1. associate-*l* 37.4%

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

      2. associate-/r* 37.4%

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

      3. sub-neg 37.4%

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

      4. distribute-rgt-in 32.4%

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

      5. unpow2 32.4%

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

      6. times-frac 23.5%

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

      7. sqr-neg 23.5%

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

      8. times-frac 32.4%

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

      9. unpow2 32.4%

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

      10. distribute-rgt-in 37.4%

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

      11. +-commutative 37.4%

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

      12. associate-+l+ 44.8%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in k around 0 67.4%

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

      1. *-commutative 67.4%

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

      2. associate-/r* 67.5%

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

   7. Simplified 67.5%

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

      1. expm1-log1p-u 40.1%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\right)\right)} \]

      2. expm1-udef 38.6%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\right)} - 1\right)} \]

      3. div-inv 38.5%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{4}}}\right)} - 1\right) \]

      4. pow-flip 38.5%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1\right) \]

      5. metadata-eval 38.5%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{\color{blue}{-4}}\right)} - 1\right) \]

   9. Applied egg-rr 38.5%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 40.0%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)\right)} \]

       2. expm1-log1p 67.5%

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

   11. Simplified 67.5%

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

       1. rem-cbrt-cube 64.8%

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

       2. unpow1/3 39.1%

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

       3. sqr-pow 39.1%

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

       4. pow2 39.1%

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

       5. sqrt-pow1 39.1%

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

       6. pow-pow 38.8%

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

       7. metadata-eval 38.8%

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

       8. pow1 38.8%

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

       9. *-commutative 38.8%

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

       10. sqrt-prod 34.7%

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

       11. sqrt-pow1 35.6%

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

       12. metadata-eval 35.6%

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

       13. sqrt-div 29.5%

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

       14. pow2 29.5%

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

       15. sqrt-prod 15.0%

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

       16. add-sqr-sqrt 32.9%

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

   13. Applied egg-rr 32.9%

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

   if 1.1000000000000001e74 < k

   1. Initial program 31.3%

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

      1. associate-*l* 31.3%

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

      2. associate-/r* 31.3%

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

      3. sub-neg 31.3%

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

      4. distribute-rgt-in 31.3%

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

      5. unpow2 31.3%

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

      6. times-frac 25.7%

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

      7. sqr-neg 25.7%

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

      8. times-frac 31.3%

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

      9. unpow2 31.3%

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

      10. distribute-rgt-in 31.3%

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

      11. +-commutative 31.3%

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

      12. associate-+l+ 45.8%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in t around 0 64.0%

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

      1. times-frac 62.4%

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

   7. Simplified 62.4%

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

      1. associate-*l/ 62.3%

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

   9. Applied egg-rr 62.3%

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

   10. Taylor expanded in k around 0 55.6%

       \[\leadsto 2 \cdot \frac{\color{blue}{\left(-0.5 \cdot \frac{{\ell}^{2}}{t} + \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}}{{k}^{2}} \]
   11. Step-by-step derivation

       1. +-commutative 55.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + -0.5 \cdot \frac{{\ell}^{2}}{t}\right)} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} \]

       2. associate--l+ 55.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}}{{k}^{2}} \]

       3. associate-/r* 55.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       4. unpow2 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       5. unpow2 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       6. times-frac 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       7. *-rgt-identity 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \frac{\color{blue}{\ell \cdot 1}}{k}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       8. associate-*r/ 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \color{blue}{\left(\ell \cdot \frac{1}{k}\right)}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       9. *-rgt-identity 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot 1}}{k} \cdot \left(\ell \cdot \frac{1}{k}\right)}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       10. associate-*r/ 55.6%

           \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\left(\ell \cdot \frac{1}{k}\right)} \cdot \left(\ell \cdot \frac{1}{k}\right)}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       11. unpow2 55.6%

           \[\leadsto 2 \cdot \frac{\frac{\color{blue}{{\left(\ell \cdot \frac{1}{k}\right)}^{2}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       12. associate-*r/ 55.6%

           \[\leadsto 2 \cdot \frac{\frac{{\color{blue}{\left(\frac{\ell \cdot 1}{k}\right)}}^{2}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       13. *-rgt-identity 55.6%

           \[\leadsto 2 \cdot \frac{\frac{{\left(\frac{\color{blue}{\ell}}{k}\right)}^{2}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       14. distribute-rgt-out-- 56.0%

           \[\leadsto 2 \cdot \frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} + \color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}} \]

       15. metadata-eval 56.0%

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

       16. associate-*l/ 56.0%

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

   12. Simplified 56.0%

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

   13. Taylor expanded in k around inf 56.3%

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

       1. associate-/r* 56.5%

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

       2. unpow2 56.5%

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

       3. unpow2 56.5%

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

       4. times-frac 66.1%

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

       5. unpow2 66.1%

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

   15. Simplified 66.1%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+74}:\\ \;\;\;\;2 \cdot {\left({k}^{-2} \cdot \frac{\ell}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.1 \cdot 10^{+74}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell \cdot {k_m}^{-2}}{\sqrt{t_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \frac{{\left(\frac{\ell}{k_m}\right)}^{2}}{t_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.1e+74)
    (* 2.0 (pow (/ (* l (pow k_m -2.0)) (sqrt t_m)) 2.0))
    (* 2.0 (* -0.16666666666666666 (/ (pow (/ l k_m) 2.0) t_m))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.1e+74) {
		tmp = 2.0 * pow(((l * pow(k_m, -2.0)) / sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 * (-0.16666666666666666 * (pow((l / k_m), 2.0) / t_m));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.1d+74) then
        tmp = 2.0d0 * (((l * (k_m ** (-2.0d0))) / sqrt(t_m)) ** 2.0d0)
    else
        tmp = 2.0d0 * ((-0.16666666666666666d0) * (((l / k_m) ** 2.0d0) / t_m))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.1e+74) {
		tmp = 2.0 * Math.pow(((l * Math.pow(k_m, -2.0)) / Math.sqrt(t_m)), 2.0);
	} else {
		tmp = 2.0 * (-0.16666666666666666 * (Math.pow((l / k_m), 2.0) / t_m));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.1e+74:
		tmp = 2.0 * math.pow(((l * math.pow(k_m, -2.0)) / math.sqrt(t_m)), 2.0)
	else:
		tmp = 2.0 * (-0.16666666666666666 * (math.pow((l / k_m), 2.0) / t_m))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.1e+74)
		tmp = Float64(2.0 * (Float64(Float64(l * (k_m ^ -2.0)) / sqrt(t_m)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(-0.16666666666666666 * Float64((Float64(l / k_m) ^ 2.0) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.1e+74)
		tmp = 2.0 * (((l * (k_m ^ -2.0)) / sqrt(t_m)) ^ 2.0);
	else
		tmp = 2.0 * (-0.16666666666666666 * (((l / k_m) ^ 2.0) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.1e+74], N[(2.0 * N[Power[N[(N[(l * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(-0.16666666666666666 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.1000000000000001e74

   1. Initial program 37.4%

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

      1. associate-*l* 37.4%

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

      2. associate-/r* 37.4%

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

      3. sub-neg 37.4%

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

      4. distribute-rgt-in 32.4%

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

      5. unpow2 32.4%

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

      6. times-frac 23.5%

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

      7. sqr-neg 23.5%

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

      8. times-frac 32.4%

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

      9. unpow2 32.4%

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

      10. distribute-rgt-in 37.4%

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

      11. +-commutative 37.4%

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

      12. associate-+l+ 44.8%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in k around 0 67.4%

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

      1. *-commutative 67.4%

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

      2. associate-/r* 67.5%

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

   7. Simplified 67.5%

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

      1. expm1-log1p-u 40.1%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\right)\right)} \]

      2. expm1-udef 38.6%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\right)} - 1\right)} \]

      3. div-inv 38.5%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{4}}}\right)} - 1\right) \]

      4. pow-flip 38.5%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1\right) \]

      5. metadata-eval 38.5%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{\color{blue}{-4}}\right)} - 1\right) \]

   9. Applied egg-rr 38.5%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 40.0%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)\right)} \]

       2. expm1-log1p 67.5%

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

   11. Simplified 67.5%

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

       1. rem-cbrt-cube 64.8%

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

       2. unpow1/3 39.1%

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

       3. sqr-pow 39.1%

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

       4. pow2 39.1%

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

       5. sqrt-pow1 39.1%

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

       6. pow-pow 38.8%

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

       7. metadata-eval 38.8%

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

       8. pow1 38.8%

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

       9. *-commutative 38.8%

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

       10. sqrt-prod 34.7%

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

       11. sqrt-pow1 35.6%

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

       12. metadata-eval 35.6%

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

       13. sqrt-div 29.5%

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

       14. pow2 29.5%

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

       15. sqrt-prod 15.0%

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

       16. add-sqr-sqrt 32.9%

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

   13. Applied egg-rr 32.9%

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

       1. associate-*r/ 32.9%

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

   15. Simplified 32.9%

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

   if 1.1000000000000001e74 < k

   1. Initial program 31.3%

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

      1. associate-*l* 31.3%

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

      2. associate-/r* 31.3%

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

      3. sub-neg 31.3%

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

      4. distribute-rgt-in 31.3%

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

      5. unpow2 31.3%

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

      6. times-frac 25.7%

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

      7. sqr-neg 25.7%

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

      8. times-frac 31.3%

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

      9. unpow2 31.3%

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

      10. distribute-rgt-in 31.3%

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

      11. +-commutative 31.3%

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

      12. associate-+l+ 45.8%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in t around 0 64.0%

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

      1. times-frac 62.4%

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

   7. Simplified 62.4%

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

      1. associate-*l/ 62.3%

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

   9. Applied egg-rr 62.3%

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

   10. Taylor expanded in k around 0 55.6%

       \[\leadsto 2 \cdot \frac{\color{blue}{\left(-0.5 \cdot \frac{{\ell}^{2}}{t} + \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}}{{k}^{2}} \]
   11. Step-by-step derivation

       1. +-commutative 55.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + -0.5 \cdot \frac{{\ell}^{2}}{t}\right)} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} \]

       2. associate--l+ 55.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}}{{k}^{2}} \]

       3. associate-/r* 55.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       4. unpow2 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       5. unpow2 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       6. times-frac 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       7. *-rgt-identity 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \frac{\color{blue}{\ell \cdot 1}}{k}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       8. associate-*r/ 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \color{blue}{\left(\ell \cdot \frac{1}{k}\right)}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       9. *-rgt-identity 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot 1}}{k} \cdot \left(\ell \cdot \frac{1}{k}\right)}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       10. associate-*r/ 55.6%

           \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\left(\ell \cdot \frac{1}{k}\right)} \cdot \left(\ell \cdot \frac{1}{k}\right)}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       11. unpow2 55.6%

           \[\leadsto 2 \cdot \frac{\frac{\color{blue}{{\left(\ell \cdot \frac{1}{k}\right)}^{2}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       12. associate-*r/ 55.6%

           \[\leadsto 2 \cdot \frac{\frac{{\color{blue}{\left(\frac{\ell \cdot 1}{k}\right)}}^{2}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       13. *-rgt-identity 55.6%

           \[\leadsto 2 \cdot \frac{\frac{{\left(\frac{\color{blue}{\ell}}{k}\right)}^{2}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       14. distribute-rgt-out-- 56.0%

           \[\leadsto 2 \cdot \frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} + \color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}} \]

       15. metadata-eval 56.0%

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

       16. associate-*l/ 56.0%

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

   12. Simplified 56.0%

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

   13. Taylor expanded in k around inf 56.3%

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

       1. associate-/r* 56.5%

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

       2. unpow2 56.5%

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

       3. unpow2 56.5%

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

       4. times-frac 66.1%

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

       5. unpow2 66.1%

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

   15. Simplified 66.1%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+74}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell \cdot {k}^{-2}}{\sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.1 \cdot 10^{+74}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \frac{1}{t_m}}{{k_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \frac{{\left(\frac{\ell}{k_m}\right)}^{2}}{t_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.1e+74)
    (* 2.0 (/ (* (pow l 2.0) (/ 1.0 t_m)) (pow k_m 4.0)))
    (* 2.0 (* -0.16666666666666666 (/ (pow (/ l k_m) 2.0) t_m))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.1e+74) {
		tmp = 2.0 * ((pow(l, 2.0) * (1.0 / t_m)) / pow(k_m, 4.0));
	} else {
		tmp = 2.0 * (-0.16666666666666666 * (pow((l / k_m), 2.0) / t_m));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.1d+74) then
        tmp = 2.0d0 * (((l ** 2.0d0) * (1.0d0 / t_m)) / (k_m ** 4.0d0))
    else
        tmp = 2.0d0 * ((-0.16666666666666666d0) * (((l / k_m) ** 2.0d0) / t_m))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.1e+74) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * (1.0 / t_m)) / Math.pow(k_m, 4.0));
	} else {
		tmp = 2.0 * (-0.16666666666666666 * (Math.pow((l / k_m), 2.0) / t_m));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.1e+74:
		tmp = 2.0 * ((math.pow(l, 2.0) * (1.0 / t_m)) / math.pow(k_m, 4.0))
	else:
		tmp = 2.0 * (-0.16666666666666666 * (math.pow((l / k_m), 2.0) / t_m))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.1e+74)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * Float64(1.0 / t_m)) / (k_m ^ 4.0)));
	else
		tmp = Float64(2.0 * Float64(-0.16666666666666666 * Float64((Float64(l / k_m) ^ 2.0) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.1e+74)
		tmp = 2.0 * (((l ^ 2.0) * (1.0 / t_m)) / (k_m ^ 4.0));
	else
		tmp = 2.0 * (-0.16666666666666666 * (((l / k_m) ^ 2.0) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.1e+74], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(-0.16666666666666666 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.1000000000000001e74

   1. Initial program 37.4%

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

      1. associate-*l* 37.4%

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

      2. associate-/r* 37.4%

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

      3. sub-neg 37.4%

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

      4. distribute-rgt-in 32.4%

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

      5. unpow2 32.4%

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

      6. times-frac 23.5%

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

      7. sqr-neg 23.5%

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

      8. times-frac 32.4%

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

      9. unpow2 32.4%

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

      10. distribute-rgt-in 37.4%

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

      11. +-commutative 37.4%

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

      12. associate-+l+ 44.8%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in k around 0 67.4%

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

      1. *-commutative 67.4%

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

      2. associate-/r* 67.5%

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

   7. Simplified 67.5%

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

      1. div-inv 67.5%

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

   9. Applied egg-rr 67.5%

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

   if 1.1000000000000001e74 < k

   1. Initial program 31.3%

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

      1. associate-*l* 31.3%

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

      2. associate-/r* 31.3%

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

      3. sub-neg 31.3%

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

      4. distribute-rgt-in 31.3%

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

      5. unpow2 31.3%

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

      6. times-frac 25.7%

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

      7. sqr-neg 25.7%

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

      8. times-frac 31.3%

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

      9. unpow2 31.3%

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

      10. distribute-rgt-in 31.3%

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

      11. +-commutative 31.3%

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

      12. associate-+l+ 45.8%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in t around 0 64.0%

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

      1. times-frac 62.4%

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

   7. Simplified 62.4%

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

      1. associate-*l/ 62.3%

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

   9. Applied egg-rr 62.3%

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

   10. Taylor expanded in k around 0 55.6%

       \[\leadsto 2 \cdot \frac{\color{blue}{\left(-0.5 \cdot \frac{{\ell}^{2}}{t} + \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}}{{k}^{2}} \]
   11. Step-by-step derivation

       1. +-commutative 55.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + -0.5 \cdot \frac{{\ell}^{2}}{t}\right)} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} \]

       2. associate--l+ 55.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}}{{k}^{2}} \]

       3. associate-/r* 55.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       4. unpow2 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       5. unpow2 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       6. times-frac 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       7. *-rgt-identity 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \frac{\color{blue}{\ell \cdot 1}}{k}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       8. associate-*r/ 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \color{blue}{\left(\ell \cdot \frac{1}{k}\right)}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       9. *-rgt-identity 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot 1}}{k} \cdot \left(\ell \cdot \frac{1}{k}\right)}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       10. associate-*r/ 55.6%

           \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\left(\ell \cdot \frac{1}{k}\right)} \cdot \left(\ell \cdot \frac{1}{k}\right)}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       11. unpow2 55.6%

           \[\leadsto 2 \cdot \frac{\frac{\color{blue}{{\left(\ell \cdot \frac{1}{k}\right)}^{2}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       12. associate-*r/ 55.6%

           \[\leadsto 2 \cdot \frac{\frac{{\color{blue}{\left(\frac{\ell \cdot 1}{k}\right)}}^{2}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       13. *-rgt-identity 55.6%

           \[\leadsto 2 \cdot \frac{\frac{{\left(\frac{\color{blue}{\ell}}{k}\right)}^{2}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       14. distribute-rgt-out-- 56.0%

           \[\leadsto 2 \cdot \frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} + \color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}} \]

       15. metadata-eval 56.0%

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

       16. associate-*l/ 56.0%

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

   12. Simplified 56.0%

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

   13. Taylor expanded in k around inf 56.3%

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

       1. associate-/r* 56.5%

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

       2. unpow2 56.5%

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

       3. unpow2 56.5%

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

       4. times-frac 66.1%

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

       5. unpow2 66.1%

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

   15. Simplified 66.1%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+74}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \frac{1}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{\left(\frac{\ell}{k_m}\right)}^{2}}{t_m}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.1 \cdot 10^{+74}:\\ \;\;\;\;2 \cdot \frac{t_2}{{k_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot t_2\right)\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (pow (/ l k_m) 2.0) t_m)))
   (*
    t_s
    (if (<= k_m 1.1e+74)
      (* 2.0 (/ t_2 (pow k_m 2.0)))
      (* 2.0 (* -0.16666666666666666 t_2))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = pow((l / k_m), 2.0) / t_m;
	double tmp;
	if (k_m <= 1.1e+74) {
		tmp = 2.0 * (t_2 / pow(k_m, 2.0));
	} else {
		tmp = 2.0 * (-0.16666666666666666 * t_2);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = ((l / k_m) ** 2.0d0) / t_m
    if (k_m <= 1.1d+74) then
        tmp = 2.0d0 * (t_2 / (k_m ** 2.0d0))
    else
        tmp = 2.0d0 * ((-0.16666666666666666d0) * t_2)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.pow((l / k_m), 2.0) / t_m;
	double tmp;
	if (k_m <= 1.1e+74) {
		tmp = 2.0 * (t_2 / Math.pow(k_m, 2.0));
	} else {
		tmp = 2.0 * (-0.16666666666666666 * t_2);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = math.pow((l / k_m), 2.0) / t_m
	tmp = 0
	if k_m <= 1.1e+74:
		tmp = 2.0 * (t_2 / math.pow(k_m, 2.0))
	else:
		tmp = 2.0 * (-0.16666666666666666 * t_2)
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64((Float64(l / k_m) ^ 2.0) / t_m)
	tmp = 0.0
	if (k_m <= 1.1e+74)
		tmp = Float64(2.0 * Float64(t_2 / (k_m ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(-0.16666666666666666 * t_2));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = ((l / k_m) ^ 2.0) / t_m;
	tmp = 0.0;
	if (k_m <= 1.1e+74)
		tmp = 2.0 * (t_2 / (k_m ^ 2.0));
	else
		tmp = 2.0 * (-0.16666666666666666 * t_2);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.1e+74], N[(2.0 * N[(t$95$2 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(-0.16666666666666666 * t$95$2), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.1000000000000001e74

   1. Initial program 37.4%

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

      1. associate-*l* 37.4%

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

      2. associate-/r* 37.4%

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

      3. sub-neg 37.4%

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

      4. distribute-rgt-in 32.4%

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

      5. unpow2 32.4%

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

      6. times-frac 23.5%

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

      7. sqr-neg 23.5%

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

      8. times-frac 32.4%

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

      9. unpow2 32.4%

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

      10. distribute-rgt-in 37.4%

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

      11. +-commutative 37.4%

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

      12. associate-+l+ 44.8%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in t around 0 78.9%

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

      1. times-frac 81.3%

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

   7. Simplified 81.3%

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

      1. associate-*l/ 80.1%

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

   9. Applied egg-rr 80.1%

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

   10. Taylor expanded in k around 0 69.0%

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

       1. associate-/r* 69.4%

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

       2. unpow2 69.4%

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

       3. unpow2 69.4%

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

       4. times-frac 76.4%

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

       5. *-rgt-identity 76.4%

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

       6. associate-*r/ 76.4%

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

       7. *-rgt-identity 76.4%

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

       8. associate-*r/ 76.4%

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

       9. unpow2 76.4%

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

       10. associate-*r/ 76.4%

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

       11. *-rgt-identity 76.4%

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

   12. Simplified 76.4%

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

   if 1.1000000000000001e74 < k

   1. Initial program 31.3%

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

      1. associate-*l* 31.3%

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

      2. associate-/r* 31.3%

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

      3. sub-neg 31.3%

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

      4. distribute-rgt-in 31.3%

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

      5. unpow2 31.3%

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

      6. times-frac 25.7%

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

      7. sqr-neg 25.7%

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

      8. times-frac 31.3%

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

      9. unpow2 31.3%

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

      10. distribute-rgt-in 31.3%

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

      11. +-commutative 31.3%

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

      12. associate-+l+ 45.8%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in t around 0 64.0%

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

      1. times-frac 62.4%

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

   7. Simplified 62.4%

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

      1. associate-*l/ 62.3%

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

   9. Applied egg-rr 62.3%

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

   10. Taylor expanded in k around 0 55.6%

       \[\leadsto 2 \cdot \frac{\color{blue}{\left(-0.5 \cdot \frac{{\ell}^{2}}{t} + \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}}{{k}^{2}} \]
   11. Step-by-step derivation

       1. +-commutative 55.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + -0.5 \cdot \frac{{\ell}^{2}}{t}\right)} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} \]

       2. associate--l+ 55.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}}{{k}^{2}} \]

       3. associate-/r* 55.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       4. unpow2 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       5. unpow2 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       6. times-frac 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       7. *-rgt-identity 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \frac{\color{blue}{\ell \cdot 1}}{k}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       8. associate-*r/ 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \color{blue}{\left(\ell \cdot \frac{1}{k}\right)}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       9. *-rgt-identity 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot 1}}{k} \cdot \left(\ell \cdot \frac{1}{k}\right)}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       10. associate-*r/ 55.6%

           \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\left(\ell \cdot \frac{1}{k}\right)} \cdot \left(\ell \cdot \frac{1}{k}\right)}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       11. unpow2 55.6%

           \[\leadsto 2 \cdot \frac{\frac{\color{blue}{{\left(\ell \cdot \frac{1}{k}\right)}^{2}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       12. associate-*r/ 55.6%

           \[\leadsto 2 \cdot \frac{\frac{{\color{blue}{\left(\frac{\ell \cdot 1}{k}\right)}}^{2}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       13. *-rgt-identity 55.6%

           \[\leadsto 2 \cdot \frac{\frac{{\left(\frac{\color{blue}{\ell}}{k}\right)}^{2}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       14. distribute-rgt-out-- 56.0%

           \[\leadsto 2 \cdot \frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} + \color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}} \]

       15. metadata-eval 56.0%

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

       16. associate-*l/ 56.0%

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

   12. Simplified 56.0%

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

   13. Taylor expanded in k around inf 56.3%

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

       1. associate-/r* 56.5%

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

       2. unpow2 56.5%

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

       3. unpow2 56.5%

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

       4. times-frac 66.1%

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

       5. unpow2 66.1%

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

   15. Simplified 66.1%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+74}:\\ \;\;\;\;2 \cdot \frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}}{{k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.1 \cdot 10^{+74}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t_m} \cdot {k_m}^{-4}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \frac{{\left(\frac{\ell}{k_m}\right)}^{2}}{t_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.1e+74)
    (* 2.0 (* (/ (pow l 2.0) t_m) (pow k_m -4.0)))
    (* 2.0 (* -0.16666666666666666 (/ (pow (/ l k_m) 2.0) t_m))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.1e+74) {
		tmp = 2.0 * ((pow(l, 2.0) / t_m) * pow(k_m, -4.0));
	} else {
		tmp = 2.0 * (-0.16666666666666666 * (pow((l / k_m), 2.0) / t_m));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.1d+74) then
        tmp = 2.0d0 * (((l ** 2.0d0) / t_m) * (k_m ** (-4.0d0)))
    else
        tmp = 2.0d0 * ((-0.16666666666666666d0) * (((l / k_m) ** 2.0d0) / t_m))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.1e+74) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / t_m) * Math.pow(k_m, -4.0));
	} else {
		tmp = 2.0 * (-0.16666666666666666 * (Math.pow((l / k_m), 2.0) / t_m));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.1e+74:
		tmp = 2.0 * ((math.pow(l, 2.0) / t_m) * math.pow(k_m, -4.0))
	else:
		tmp = 2.0 * (-0.16666666666666666 * (math.pow((l / k_m), 2.0) / t_m))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.1e+74)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / t_m) * (k_m ^ -4.0)));
	else
		tmp = Float64(2.0 * Float64(-0.16666666666666666 * Float64((Float64(l / k_m) ^ 2.0) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.1e+74)
		tmp = 2.0 * (((l ^ 2.0) / t_m) * (k_m ^ -4.0));
	else
		tmp = 2.0 * (-0.16666666666666666 * (((l / k_m) ^ 2.0) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.1e+74], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(-0.16666666666666666 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.1000000000000001e74

   1. Initial program 37.4%

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

      1. associate-*l* 37.4%

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

      2. associate-/r* 37.4%

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

      3. sub-neg 37.4%

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

      4. distribute-rgt-in 32.4%

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

      5. unpow2 32.4%

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

      6. times-frac 23.5%

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

      7. sqr-neg 23.5%

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

      8. times-frac 32.4%

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

      9. unpow2 32.4%

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

      10. distribute-rgt-in 37.4%

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

      11. +-commutative 37.4%

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

      12. associate-+l+ 44.8%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in k around 0 67.4%

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

      1. *-commutative 67.4%

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

      2. associate-/r* 67.5%

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

   7. Simplified 67.5%

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

      1. expm1-log1p-u 40.1%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\right)\right)} \]

      2. expm1-udef 38.6%

         \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\right)} - 1\right)} \]

      3. div-inv 38.5%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{{\ell}^{2}}{t} \cdot \frac{1}{{k}^{4}}}\right)} - 1\right) \]

      4. pow-flip 38.5%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot \color{blue}{{k}^{\left(-4\right)}}\right)} - 1\right) \]

      5. metadata-eval 38.5%

         \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{\color{blue}{-4}}\right)} - 1\right) \]

   9. Applied egg-rr 38.5%

      \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 40.0%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)\right)} \]

       2. expm1-log1p 67.5%

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

   11. Simplified 67.5%

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

   if 1.1000000000000001e74 < k

   1. Initial program 31.3%

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

      1. associate-*l* 31.3%

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

      2. associate-/r* 31.3%

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

      3. sub-neg 31.3%

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

      4. distribute-rgt-in 31.3%

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

      5. unpow2 31.3%

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

      6. times-frac 25.7%

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

      7. sqr-neg 25.7%

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

      8. times-frac 31.3%

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

      9. unpow2 31.3%

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

      10. distribute-rgt-in 31.3%

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

      11. +-commutative 31.3%

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

      12. associate-+l+ 45.8%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in t around 0 64.0%

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

      1. times-frac 62.4%

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

   7. Simplified 62.4%

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

      1. associate-*l/ 62.3%

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

   9. Applied egg-rr 62.3%

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

   10. Taylor expanded in k around 0 55.6%

       \[\leadsto 2 \cdot \frac{\color{blue}{\left(-0.5 \cdot \frac{{\ell}^{2}}{t} + \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}}{{k}^{2}} \]
   11. Step-by-step derivation

       1. +-commutative 55.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + -0.5 \cdot \frac{{\ell}^{2}}{t}\right)} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} \]

       2. associate--l+ 55.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}}{{k}^{2}} \]

       3. associate-/r* 55.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       4. unpow2 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       5. unpow2 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       6. times-frac 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       7. *-rgt-identity 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \frac{\color{blue}{\ell \cdot 1}}{k}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       8. associate-*r/ 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \color{blue}{\left(\ell \cdot \frac{1}{k}\right)}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       9. *-rgt-identity 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot 1}}{k} \cdot \left(\ell \cdot \frac{1}{k}\right)}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       10. associate-*r/ 55.6%

           \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\left(\ell \cdot \frac{1}{k}\right)} \cdot \left(\ell \cdot \frac{1}{k}\right)}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       11. unpow2 55.6%

           \[\leadsto 2 \cdot \frac{\frac{\color{blue}{{\left(\ell \cdot \frac{1}{k}\right)}^{2}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       12. associate-*r/ 55.6%

           \[\leadsto 2 \cdot \frac{\frac{{\color{blue}{\left(\frac{\ell \cdot 1}{k}\right)}}^{2}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       13. *-rgt-identity 55.6%

           \[\leadsto 2 \cdot \frac{\frac{{\left(\frac{\color{blue}{\ell}}{k}\right)}^{2}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       14. distribute-rgt-out-- 56.0%

           \[\leadsto 2 \cdot \frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} + \color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}} \]

       15. metadata-eval 56.0%

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

       16. associate-*l/ 56.0%

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

   12. Simplified 56.0%

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

   13. Taylor expanded in k around inf 56.3%

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

       1. associate-/r* 56.5%

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

       2. unpow2 56.5%

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

       3. unpow2 56.5%

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

       4. times-frac 66.1%

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

       5. unpow2 66.1%

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

   15. Simplified 66.1%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+74}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{t} \cdot {k}^{-4}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.1 \cdot 10^{+74}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t_m}}{{k_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \frac{{\left(\frac{\ell}{k_m}\right)}^{2}}{t_m}\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.1e+74)
    (* 2.0 (/ (/ (pow l 2.0) t_m) (pow k_m 4.0)))
    (* 2.0 (* -0.16666666666666666 (/ (pow (/ l k_m) 2.0) t_m))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.1e+74) {
		tmp = 2.0 * ((pow(l, 2.0) / t_m) / pow(k_m, 4.0));
	} else {
		tmp = 2.0 * (-0.16666666666666666 * (pow((l / k_m), 2.0) / t_m));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.1d+74) then
        tmp = 2.0d0 * (((l ** 2.0d0) / t_m) / (k_m ** 4.0d0))
    else
        tmp = 2.0d0 * ((-0.16666666666666666d0) * (((l / k_m) ** 2.0d0) / t_m))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.1e+74) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / t_m) / Math.pow(k_m, 4.0));
	} else {
		tmp = 2.0 * (-0.16666666666666666 * (Math.pow((l / k_m), 2.0) / t_m));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.1e+74:
		tmp = 2.0 * ((math.pow(l, 2.0) / t_m) / math.pow(k_m, 4.0))
	else:
		tmp = 2.0 * (-0.16666666666666666 * (math.pow((l / k_m), 2.0) / t_m))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.1e+74)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / t_m) / (k_m ^ 4.0)));
	else
		tmp = Float64(2.0 * Float64(-0.16666666666666666 * Float64((Float64(l / k_m) ^ 2.0) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.1e+74)
		tmp = 2.0 * (((l ^ 2.0) / t_m) / (k_m ^ 4.0));
	else
		tmp = 2.0 * (-0.16666666666666666 * (((l / k_m) ^ 2.0) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.1e+74], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(-0.16666666666666666 * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.1000000000000001e74

   1. Initial program 37.4%

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

      1. associate-*l* 37.4%

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

      2. associate-/r* 37.4%

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

      3. sub-neg 37.4%

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

      4. distribute-rgt-in 32.4%

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

      5. unpow2 32.4%

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

      6. times-frac 23.5%

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

      7. sqr-neg 23.5%

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

      8. times-frac 32.4%

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

      9. unpow2 32.4%

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

      10. distribute-rgt-in 37.4%

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

      11. +-commutative 37.4%

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

      12. associate-+l+ 44.8%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in k around 0 67.4%

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

      1. *-commutative 67.4%

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

      2. associate-/r* 67.5%

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

   7. Simplified 67.5%

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

   if 1.1000000000000001e74 < k

   1. Initial program 31.3%

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

      1. associate-*l* 31.3%

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

      2. associate-/r* 31.3%

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

      3. sub-neg 31.3%

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

      4. distribute-rgt-in 31.3%

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

      5. unpow2 31.3%

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

      6. times-frac 25.7%

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

      7. sqr-neg 25.7%

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

      8. times-frac 31.3%

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

      9. unpow2 31.3%

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

      10. distribute-rgt-in 31.3%

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

      11. +-commutative 31.3%

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

      12. associate-+l+ 45.8%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in t around 0 64.0%

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

      1. times-frac 62.4%

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

   7. Simplified 62.4%

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

      1. associate-*l/ 62.3%

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

   9. Applied egg-rr 62.3%

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

   10. Taylor expanded in k around 0 55.6%

       \[\leadsto 2 \cdot \frac{\color{blue}{\left(-0.5 \cdot \frac{{\ell}^{2}}{t} + \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}}{{k}^{2}} \]
   11. Step-by-step derivation

       1. +-commutative 55.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + -0.5 \cdot \frac{{\ell}^{2}}{t}\right)} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} \]

       2. associate--l+ 55.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}}{{k}^{2}} \]

       3. associate-/r* 55.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       4. unpow2 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       5. unpow2 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       6. times-frac 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       7. *-rgt-identity 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \frac{\color{blue}{\ell \cdot 1}}{k}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       8. associate-*r/ 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \color{blue}{\left(\ell \cdot \frac{1}{k}\right)}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       9. *-rgt-identity 55.6%

          \[\leadsto 2 \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot 1}}{k} \cdot \left(\ell \cdot \frac{1}{k}\right)}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       10. associate-*r/ 55.6%

           \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\left(\ell \cdot \frac{1}{k}\right)} \cdot \left(\ell \cdot \frac{1}{k}\right)}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       11. unpow2 55.6%

           \[\leadsto 2 \cdot \frac{\frac{\color{blue}{{\left(\ell \cdot \frac{1}{k}\right)}^{2}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       12. associate-*r/ 55.6%

           \[\leadsto 2 \cdot \frac{\frac{{\color{blue}{\left(\frac{\ell \cdot 1}{k}\right)}}^{2}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       13. *-rgt-identity 55.6%

           \[\leadsto 2 \cdot \frac{\frac{{\left(\frac{\color{blue}{\ell}}{k}\right)}^{2}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

       14. distribute-rgt-out-- 56.0%

           \[\leadsto 2 \cdot \frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} + \color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}} \]

       15. metadata-eval 56.0%

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

       16. associate-*l/ 56.0%

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

   12. Simplified 56.0%

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

   13. Taylor expanded in k around inf 56.3%

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

       1. associate-/r* 56.5%

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

       2. unpow2 56.5%

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

       3. unpow2 56.5%

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

       4. times-frac 66.1%

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

       5. unpow2 66.1%

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

   15. Simplified 66.1%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+74}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.5% accurate, 3.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (* -0.16666666666666666 (/ (pow (/ l k_m) 2.0) t_m)))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * (-0.16666666666666666 * (pow((l / k_m), 2.0) / t_m)));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.1%

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

   1. associate-*l* 36.1%

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

   2. associate-/r* 36.1%

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

   3. sub-neg 36.1%

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

   4. distribute-rgt-in 32.2%

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

   5. unpow2 32.2%

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

   6. times-frac 24.0%

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

   7. sqr-neg 24.0%

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

   8. times-frac 32.2%

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

   9. unpow2 32.2%

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

   10. distribute-rgt-in 36.1%

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

   11. +-commutative 36.1%

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

   12. associate-+l+ 45.0%

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

3. Simplified 45.0%

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

5. Taylor expanded in t around 0 75.7%

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

   1. times-frac 77.2%

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

7. Simplified 77.2%

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

   1. associate-*l/ 76.3%

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

9. Applied egg-rr 76.3%

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

10. Taylor expanded in k around 0 47.7%

    \[\leadsto 2 \cdot \frac{\color{blue}{\left(-0.5 \cdot \frac{{\ell}^{2}}{t} + \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}}{{k}^{2}} \]
11. Step-by-step derivation

    1. +-commutative 47.7%

       \[\leadsto 2 \cdot \frac{\color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2} \cdot t} + -0.5 \cdot \frac{{\ell}^{2}}{t}\right)} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}} \]

    2. associate--l+ 47.7%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}}{{k}^{2}} \]

    3. associate-/r* 48.0%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

    4. unpow2 48.0%

       \[\leadsto 2 \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

    5. unpow2 48.0%

       \[\leadsto 2 \cdot \frac{\frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

    6. times-frac 53.4%

       \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

    7. *-rgt-identity 53.4%

       \[\leadsto 2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \frac{\color{blue}{\ell \cdot 1}}{k}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

    8. associate-*r/ 53.4%

       \[\leadsto 2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \color{blue}{\left(\ell \cdot \frac{1}{k}\right)}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

    9. *-rgt-identity 53.4%

       \[\leadsto 2 \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot 1}}{k} \cdot \left(\ell \cdot \frac{1}{k}\right)}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

    10. associate-*r/ 53.4%

        \[\leadsto 2 \cdot \frac{\frac{\color{blue}{\left(\ell \cdot \frac{1}{k}\right)} \cdot \left(\ell \cdot \frac{1}{k}\right)}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

    11. unpow2 53.4%

        \[\leadsto 2 \cdot \frac{\frac{\color{blue}{{\left(\ell \cdot \frac{1}{k}\right)}^{2}}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

    12. associate-*r/ 53.4%

        \[\leadsto 2 \cdot \frac{\frac{{\color{blue}{\left(\frac{\ell \cdot 1}{k}\right)}}^{2}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

    13. *-rgt-identity 53.4%

        \[\leadsto 2 \cdot \frac{\frac{{\left(\frac{\color{blue}{\ell}}{k}\right)}^{2}}{t} + \left(-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{2}} \]

    14. distribute-rgt-out-- 53.5%

        \[\leadsto 2 \cdot \frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} + \color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}} \]

    15. metadata-eval 53.5%

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

    16. associate-*l/ 53.5%

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

12. Simplified 53.5%

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

13. Taylor expanded in k around inf 31.2%

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

    1. associate-/r* 31.3%

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

    2. unpow2 31.3%

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

    3. unpow2 31.3%

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

    4. times-frac 34.0%

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

    5. unpow2 34.0%

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

15. Simplified 34.0%

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

16. Final simplification 34.0%

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

Reproduce

herbie shell --seed 2024019 
(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))))