Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.4% → 99.3%

Time: 27.7s

Alternatives: 17

Speedup: 3.8×

• 39.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.4% 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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m}{\ell} \cdot \left(k\_m \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt{\frac{2}{t\_m}} \cdot \frac{\ell}{k\_m}\right)}^{2}}{\tan k\_m \cdot \sin k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.5e-43)
    (/ 2.0 (pow (* (/ k_m l) (* k_m (sqrt t_m))) 2.0))
    (/ (pow (* (sqrt (/ 2.0 t_m)) (/ l k_m)) 2.0) (* (tan k_m) (sin k_m))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.5e-43) {
		tmp = 2.0 / pow(((k_m / l) * (k_m * sqrt(t_m))), 2.0);
	} else {
		tmp = pow((sqrt((2.0 / t_m)) * (l / k_m)), 2.0) / (tan(k_m) * sin(k_m));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.5d-43) then
        tmp = 2.0d0 / (((k_m / l) * (k_m * sqrt(t_m))) ** 2.0d0)
    else
        tmp = ((sqrt((2.0d0 / t_m)) * (l / k_m)) ** 2.0d0) / (tan(k_m) * sin(k_m))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.5e-43) {
		tmp = 2.0 / Math.pow(((k_m / l) * (k_m * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = Math.pow((Math.sqrt((2.0 / t_m)) * (l / k_m)), 2.0) / (Math.tan(k_m) * Math.sin(k_m));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.5e-43:
		tmp = 2.0 / math.pow(((k_m / l) * (k_m * math.sqrt(t_m))), 2.0)
	else:
		tmp = math.pow((math.sqrt((2.0 / t_m)) * (l / k_m)), 2.0) / (math.tan(k_m) * math.sin(k_m))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.5e-43)
		tmp = Float64(2.0 / (Float64(Float64(k_m / l) * Float64(k_m * sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64((Float64(sqrt(Float64(2.0 / t_m)) * Float64(l / k_m)) ^ 2.0) / Float64(tan(k_m) * sin(k_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.5e-43)
		tmp = 2.0 / (((k_m / l) * (k_m * sqrt(t_m))) ^ 2.0);
	else
		tmp = ((sqrt((2.0 / t_m)) * (l / k_m)) ^ 2.0) / (tan(k_m) * sin(k_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.5e-43], N[(2.0 / N[Power[N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Sqrt[N[(2.0 / t$95$m), $MachinePrecision]], $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.50000000000000009e-43

   1. Initial program 38.4%

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

   3. Simplified 44.6%

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

   5. Taylor expanded in t around 0 73.2%

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

      1. times-frac 72.4%

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

   7. Simplified 72.4%

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

      1. associate-*l/ 73.2%

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

      2. associate-/l* 73.2%

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

      3. unpow2 73.2%

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

      4. *-un-lft-identity 73.2%

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

      5. times-frac 73.2%

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

      6. tan-quot 73.2%

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

   9. Applied egg-rr 73.2%

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

       1. *-commutative 73.2%

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

       2. associate-/l* 72.4%

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

       3. unpow2 72.4%

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

       4. associate-/r* 84.1%

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

       5. unpow2 84.1%

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

       6. associate-*r/ 87.3%

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

       7. associate-*l/ 89.3%

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

       8. unpow2 89.3%

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

       9. *-commutative 89.3%

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

       10. /-rgt-identity 89.3%

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

   11. Simplified 89.3%

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

   12. Taylor expanded in k around 0 76.8%

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

       1. add-sqr-sqrt 45.6%

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

       2. pow2 45.6%

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

       3. sqrt-prod 45.6%

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

       4. unpow2 45.6%

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

       5. sqrt-prod 26.2%

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

       6. add-sqr-sqrt 45.6%

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

       7. *-commutative 45.6%

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

       8. sqrt-prod 45.7%

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

       9. unpow2 45.7%

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

       10. sqrt-prod 18.0%

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

       11. add-sqr-sqrt 47.7%

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

   14. Applied egg-rr 47.7%

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

   if 2.50000000000000009e-43 < k

   1. Initial program 30.8%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in t around 0 72.3%

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

      1. times-frac 72.4%

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

   7. Simplified 72.4%

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

      1. div-inv 72.4%

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

      2. add-sqr-sqrt 72.3%

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

      3. pow2 72.3%

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

      4. sqrt-div 72.4%

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

      5. unpow2 72.4%

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

      6. sqrt-prod 76.4%

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

      7. add-sqr-sqrt 76.6%

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

      8. unpow2 76.6%

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

      9. sqrt-prod 51.9%

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

      10. add-sqr-sqrt 93.6%

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

      11. associate-/l* 93.6%

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

      12. unpow2 93.6%

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

      13. *-un-lft-identity 93.6%

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

      14. times-frac 93.6%

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

      15. tan-quot 93.7%

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

   9. Applied egg-rr 93.7%

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

       1. associate-*r/ 93.7%

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

       2. metadata-eval 93.7%

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

       3. associate-/r* 93.6%

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

       4. /-rgt-identity 93.6%

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

       5. associate-/r* 93.6%

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

   11. Simplified 93.6%

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

       1. add-sqr-sqrt 73.2%

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

       2. sqrt-div 44.4%

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

       3. sqrt-div 44.4%

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

       4. unpow2 44.4%

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

       5. sqrt-prod 28.7%

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

       6. add-sqr-sqrt 40.4%

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

       7. sqrt-div 40.4%

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

       8. sqrt-div 40.4%

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

       9. unpow2 40.4%

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

       10. sqrt-prod 31.3%

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

       11. add-sqr-sqrt 49.1%

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

   13. Applied egg-rr 49.1%

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

       1. unpow2 49.1%

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

       2. associate-/l/ 48.9%

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

   15. Simplified 48.9%

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

       1. *-un-lft-identity 48.9%

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

       2. associate-/r* 49.0%

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

       3. sqrt-undiv 49.0%

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

   17. Applied egg-rr 49.0%

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

       1. associate-*r/ 49.0%

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

       2. associate-*l/ 49.0%

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

       3. *-commutative 49.0%

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

       4. associate-/r/ 49.0%

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

       5. associate-*l/ 48.9%

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

       6. *-lft-identity 48.9%

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

   19. Simplified 48.9%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\ell} \cdot \left(k \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt{\frac{2}{t}} \cdot \frac{\ell}{k}\right)}^{2}}{\tan k \cdot \sin k}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt{\frac{2}{t\_m}}}{\frac{k\_m}{\ell}}\\ t\_s \cdot \left(\frac{t\_2}{\tan k\_m} \cdot \frac{t\_2}{\sin k\_m}\right) \end{array} \end{array} \]

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
 (let* ((t_2 (/ (sqrt (/ 2.0 t_m)) (/ k_m l))))
   (* t_s (* (/ t_2 (tan k_m)) (/ t_2 (sin k_m))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = sqrt((2.0 / t_m)) / (k_m / l);
	return t_s * ((t_2 / tan(k_m)) * (t_2 / sin(k_m)));
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    t_2 = sqrt((2.0d0 / t_m)) / (k_m / l)
    code = t_s * ((t_2 / tan(k_m)) * (t_2 / sin(k_m)))
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.sqrt((2.0 / t_m)) / (k_m / l);
	return t_s * ((t_2 / Math.tan(k_m)) * (t_2 / Math.sin(k_m)));
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = math.sqrt((2.0 / t_m)) / (k_m / l)
	return t_s * ((t_2 / math.tan(k_m)) * (t_2 / math.sin(k_m)))

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(sqrt(Float64(2.0 / t_m)) / Float64(k_m / l))
	return Float64(t_s * Float64(Float64(t_2 / tan(k_m)) * Float64(t_2 / sin(k_m))))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	t_2 = sqrt((2.0 / t_m)) / (k_m / l);
	tmp = t_s * ((t_2 / tan(k_m)) * (t_2 / sin(k_m)));
end

Wolfram

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

Derivation

1. Initial program 36.2%

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

3. Simplified 45.3%

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

5. Taylor expanded in t around 0 73.0%

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

   1. times-frac 72.4%

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

7. Simplified 72.4%

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

   1. div-inv 72.4%

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

   2. add-sqr-sqrt 72.3%

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

   3. pow2 72.3%

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

   4. sqrt-div 72.4%

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

   5. unpow2 72.4%

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

   6. sqrt-prod 41.7%

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

   7. add-sqr-sqrt 75.2%

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

   8. unpow2 75.2%

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

   9. sqrt-prod 50.8%

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

   10. add-sqr-sqrt 90.6%

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

   11. associate-/l* 90.6%

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

   12. unpow2 90.6%

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

   13. *-un-lft-identity 90.6%

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

   14. times-frac 90.5%

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

   15. tan-quot 90.6%

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

9. Applied egg-rr 90.6%

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

    1. associate-*r/ 90.6%

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

    2. metadata-eval 90.6%

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

    3. associate-/r* 90.7%

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

    4. /-rgt-identity 90.7%

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

    5. associate-/r* 90.8%

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

11. Simplified 90.8%

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

    1. add-sqr-sqrt 63.4%

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

    2. sqrt-div 51.2%

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

    3. sqrt-div 51.2%

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

    4. unpow2 51.2%

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

    5. sqrt-prod 28.7%

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

    6. add-sqr-sqrt 31.4%

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

    7. sqrt-div 31.4%

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

    8. sqrt-div 31.4%

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

    9. unpow2 31.4%

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

    10. sqrt-prod 26.6%

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

    11. add-sqr-sqrt 54.3%

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

13. Applied egg-rr 54.3%

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

    1. unpow2 54.3%

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

    2. associate-/l/ 54.3%

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

15. Simplified 54.3%

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

    1. unpow2 54.3%

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

    2. *-commutative 54.3%

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

    3. times-frac 55.6%

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

    4. associate-/r* 55.6%

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

    5. sqrt-undiv 55.6%

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

    6. associate-/r* 55.6%

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

    7. sqrt-undiv 55.6%

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

17. Applied egg-rr 55.6%

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

18. Final simplification 55.6%

    \[\leadsto \frac{\frac{\sqrt{\frac{2}{t}}}{\frac{k}{\ell}}}{\tan k} \cdot \frac{\frac{\sqrt{\frac{2}{t}}}{\frac{k}{\ell}}}{\sin k} \]
19. Add Preprocessing

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 6.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m}{\ell} \cdot \left(k\_m \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k\_m \cdot \sin k\_m\right) \cdot {\left(\frac{k\_m}{\ell} \cdot \sqrt{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

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 6.8e-48)
    (/ 2.0 (pow (* (/ k_m l) (* k_m (sqrt t_m))) 2.0))
    (/ 2.0 (* (* (tan k_m) (sin k_m)) (pow (* (/ k_m l) (sqrt t_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 <= 6.8e-48) {
		tmp = 2.0 / pow(((k_m / l) * (k_m * sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 / ((tan(k_m) * sin(k_m)) * pow(((k_m / l) * sqrt(t_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 <= 6.8d-48) then
        tmp = 2.0d0 / (((k_m / l) * (k_m * sqrt(t_m))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((tan(k_m) * sin(k_m)) * (((k_m / l) * sqrt(t_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 <= 6.8e-48) {
		tmp = 2.0 / Math.pow(((k_m / l) * (k_m * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 / ((Math.tan(k_m) * Math.sin(k_m)) * Math.pow(((k_m / l) * Math.sqrt(t_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 <= 6.8e-48:
		tmp = 2.0 / math.pow(((k_m / l) * (k_m * math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 / ((math.tan(k_m) * math.sin(k_m)) * math.pow(((k_m / l) * math.sqrt(t_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 <= 6.8e-48)
		tmp = Float64(2.0 / (Float64(Float64(k_m / l) * Float64(k_m * sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * sin(k_m)) * (Float64(Float64(k_m / l) * sqrt(t_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 <= 6.8e-48)
		tmp = 2.0 / (((k_m / l) * (k_m * sqrt(t_m))) ^ 2.0);
	else
		tmp = 2.0 / ((tan(k_m) * sin(k_m)) * (((k_m / l) * sqrt(t_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, 6.8e-48], N[(2.0 / N[Power[N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(k$95$m / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.80000000000000056e-48

   1. Initial program 38.4%

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

   3. Simplified 44.6%

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

   5. Taylor expanded in t around 0 73.2%

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

      1. times-frac 72.4%

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

   7. Simplified 72.4%

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

      1. associate-*l/ 73.2%

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

      2. associate-/l* 73.2%

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

      3. unpow2 73.2%

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

      4. *-un-lft-identity 73.2%

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

      5. times-frac 73.2%

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

      6. tan-quot 73.2%

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

   9. Applied egg-rr 73.2%

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

       1. *-commutative 73.2%

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

       2. associate-/l* 72.4%

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

       3. unpow2 72.4%

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

       4. associate-/r* 84.1%

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

       5. unpow2 84.1%

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

       6. associate-*r/ 87.3%

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

       7. associate-*l/ 89.3%

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

       8. unpow2 89.3%

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

       9. *-commutative 89.3%

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

       10. /-rgt-identity 89.3%

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

   11. Simplified 89.3%

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

   12. Taylor expanded in k around 0 76.8%

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

       1. add-sqr-sqrt 45.6%

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

       2. pow2 45.6%

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

       3. sqrt-prod 45.6%

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

       4. unpow2 45.6%

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

       5. sqrt-prod 26.2%

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

       6. add-sqr-sqrt 45.6%

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

       7. *-commutative 45.6%

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

       8. sqrt-prod 45.7%

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

       9. unpow2 45.7%

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

       10. sqrt-prod 18.0%

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

       11. add-sqr-sqrt 47.7%

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

   14. Applied egg-rr 47.7%

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

   if 6.80000000000000056e-48 < k

   1. Initial program 30.8%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in t around 0 72.3%

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

      1. times-frac 72.4%

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

   7. Simplified 72.4%

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

      1. div-inv 72.4%

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

      2. add-sqr-sqrt 72.3%

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

      3. pow2 72.3%

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

      4. sqrt-div 72.4%

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

      5. unpow2 72.4%

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

      6. sqrt-prod 76.4%

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

      7. add-sqr-sqrt 76.6%

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

      8. unpow2 76.6%

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

      9. sqrt-prod 51.9%

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

      10. add-sqr-sqrt 93.6%

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

      11. associate-/l* 93.6%

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

      12. unpow2 93.6%

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

      13. *-un-lft-identity 93.6%

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

      14. times-frac 93.6%

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

      15. tan-quot 93.7%

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

   9. Applied egg-rr 93.7%

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

       1. associate-*r/ 93.7%

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

       2. metadata-eval 93.7%

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

       3. associate-/r* 93.6%

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

       4. /-rgt-identity 93.6%

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

       5. associate-/r* 93.6%

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

   11. Simplified 93.6%

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

       1. add-sqr-sqrt 73.2%

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

       2. sqrt-div 44.4%

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

       3. sqrt-div 44.4%

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

       4. unpow2 44.4%

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

       5. sqrt-prod 28.7%

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

       6. add-sqr-sqrt 40.4%

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

       7. sqrt-div 40.4%

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

       8. sqrt-div 40.4%

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

       9. unpow2 40.4%

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

       10. sqrt-prod 31.3%

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

       11. add-sqr-sqrt 49.1%

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

   13. Applied egg-rr 49.1%

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

       1. unpow2 49.1%

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

       2. associate-/l/ 48.9%

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

   15. Simplified 48.9%

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

       1. div-inv 48.9%

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

       2. unpow2 48.9%

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

       3. frac-times 48.1%

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

       4. rem-square-sqrt 48.2%

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

       5. pow2 48.2%

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

   17. Applied egg-rr 48.2%

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

       1. associate-*r/ 48.2%

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

       2. *-rgt-identity 48.2%

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

       3. associate-/l/ 48.2%

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

       4. *-commutative 48.2%

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

   19. Simplified 48.2%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\ell} \cdot \left(k \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \sin k\right) \cdot {\left(\frac{k}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.5% accurate, 1.0× speedup

Math

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

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
  (*
   (/ 1.0 (tan k_m))
   (/ (/ 2.0 (pow (* (/ k_m l) (sqrt t_m)) 2.0)) (sin k_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.2%

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

3. Simplified 45.3%

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

5. Taylor expanded in t around 0 73.0%

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

   1. times-frac 72.4%

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

7. Simplified 72.4%

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

   1. div-inv 72.4%

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

   2. add-sqr-sqrt 72.3%

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

   3. pow2 72.3%

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

   4. sqrt-div 72.4%

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

   5. unpow2 72.4%

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

   6. sqrt-prod 41.7%

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

   7. add-sqr-sqrt 75.2%

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

   8. unpow2 75.2%

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

   9. sqrt-prod 50.8%

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

   10. add-sqr-sqrt 90.6%

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

   11. associate-/l* 90.6%

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

   12. unpow2 90.6%

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

   13. *-un-lft-identity 90.6%

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

   14. times-frac 90.5%

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

   15. tan-quot 90.6%

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

9. Applied egg-rr 90.6%

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

    1. associate-*r/ 90.6%

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

    2. metadata-eval 90.6%

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

    3. associate-/r* 90.7%

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

    4. /-rgt-identity 90.7%

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

    5. associate-/r* 90.8%

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

11. Simplified 90.8%

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

    1. add-sqr-sqrt 63.4%

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

    2. sqrt-div 51.2%

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

    3. sqrt-div 51.2%

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

    4. unpow2 51.2%

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

    5. sqrt-prod 28.7%

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

    6. add-sqr-sqrt 31.4%

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

    7. sqrt-div 31.4%

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

    8. sqrt-div 31.4%

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

    9. unpow2 31.4%

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

    10. sqrt-prod 26.6%

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

    11. add-sqr-sqrt 54.3%

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

13. Applied egg-rr 54.3%

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

    1. unpow2 54.3%

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

    2. associate-/l/ 54.3%

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

15. Simplified 54.3%

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

    1. *-un-lft-identity 54.3%

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

    2. *-commutative 54.3%

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

    3. times-frac 55.6%

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

    4. unpow2 55.6%

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

    5. frac-times 55.1%

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

    6. rem-square-sqrt 55.2%

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

    7. pow2 55.2%

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

17. Applied egg-rr 55.2%

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

18. Final simplification 55.2%

    \[\leadsto \frac{1}{\tan k} \cdot \frac{\frac{2}{{\left(\frac{k}{\ell} \cdot \sqrt{t}\right)}^{2}}}{\sin k} \]
19. Add Preprocessing

Alternative 5: 97.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m}{\ell} \cdot \left(k\_m \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 1.5 \cdot 10^{+83}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{t\_m}{\ell} \cdot \frac{{\left(k\_m \cdot \sin k\_m\right)}^{2}}{\cos k\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k\_m}{\ell}\right)}^{2}}}{\tan k\_m \cdot \left(t\_m \cdot \sin k\_m\right)}\\ \end{array} \end{array} \]

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 2.5e-43)
    (/ 2.0 (pow (* (/ k_m l) (* k_m (sqrt t_m))) 2.0))
    (if (<= k_m 1.5e+83)
      (* l (/ 2.0 (* (/ t_m l) (/ (pow (* k_m (sin k_m)) 2.0) (cos k_m)))))
      (/ (/ 2.0 (pow (/ k_m l) 2.0)) (* (tan k_m) (* t_m (sin k_m))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.5e-43) {
		tmp = 2.0 / pow(((k_m / l) * (k_m * sqrt(t_m))), 2.0);
	} else if (k_m <= 1.5e+83) {
		tmp = l * (2.0 / ((t_m / l) * (pow((k_m * sin(k_m)), 2.0) / cos(k_m))));
	} else {
		tmp = (2.0 / pow((k_m / l), 2.0)) / (tan(k_m) * (t_m * sin(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.5d-43) then
        tmp = 2.0d0 / (((k_m / l) * (k_m * sqrt(t_m))) ** 2.0d0)
    else if (k_m <= 1.5d+83) then
        tmp = l * (2.0d0 / ((t_m / l) * (((k_m * sin(k_m)) ** 2.0d0) / cos(k_m))))
    else
        tmp = (2.0d0 / ((k_m / l) ** 2.0d0)) / (tan(k_m) * (t_m * sin(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.5e-43) {
		tmp = 2.0 / Math.pow(((k_m / l) * (k_m * Math.sqrt(t_m))), 2.0);
	} else if (k_m <= 1.5e+83) {
		tmp = l * (2.0 / ((t_m / l) * (Math.pow((k_m * Math.sin(k_m)), 2.0) / Math.cos(k_m))));
	} else {
		tmp = (2.0 / Math.pow((k_m / l), 2.0)) / (Math.tan(k_m) * (t_m * Math.sin(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.5e-43:
		tmp = 2.0 / math.pow(((k_m / l) * (k_m * math.sqrt(t_m))), 2.0)
	elif k_m <= 1.5e+83:
		tmp = l * (2.0 / ((t_m / l) * (math.pow((k_m * math.sin(k_m)), 2.0) / math.cos(k_m))))
	else:
		tmp = (2.0 / math.pow((k_m / l), 2.0)) / (math.tan(k_m) * (t_m * math.sin(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.5e-43)
		tmp = Float64(2.0 / (Float64(Float64(k_m / l) * Float64(k_m * sqrt(t_m))) ^ 2.0));
	elseif (k_m <= 1.5e+83)
		tmp = Float64(l * Float64(2.0 / Float64(Float64(t_m / l) * Float64((Float64(k_m * sin(k_m)) ^ 2.0) / cos(k_m)))));
	else
		tmp = Float64(Float64(2.0 / (Float64(k_m / l) ^ 2.0)) / Float64(tan(k_m) * Float64(t_m * sin(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.5e-43)
		tmp = 2.0 / (((k_m / l) * (k_m * sqrt(t_m))) ^ 2.0);
	elseif (k_m <= 1.5e+83)
		tmp = l * (2.0 / ((t_m / l) * (((k_m * sin(k_m)) ^ 2.0) / cos(k_m))));
	else
		tmp = (2.0 / ((k_m / l) ^ 2.0)) / (tan(k_m) * (t_m * sin(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.5e-43], N[(2.0 / N[Power[N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.5e+83], N[(l * N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Power[N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k$95$m], $MachinePrecision] * N[(t$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.50000000000000009e-43

   1. Initial program 38.4%

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

   3. Simplified 44.6%

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

   5. Taylor expanded in t around 0 73.2%

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

      1. times-frac 72.4%

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

   7. Simplified 72.4%

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

      1. associate-*l/ 73.2%

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

      2. associate-/l* 73.2%

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

      3. unpow2 73.2%

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

      4. *-un-lft-identity 73.2%

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

      5. times-frac 73.2%

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

      6. tan-quot 73.2%

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

   9. Applied egg-rr 73.2%

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

       1. *-commutative 73.2%

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

       2. associate-/l* 72.4%

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

       3. unpow2 72.4%

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

       4. associate-/r* 84.1%

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

       5. unpow2 84.1%

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

       6. associate-*r/ 87.3%

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

       7. associate-*l/ 89.3%

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

       8. unpow2 89.3%

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

       9. *-commutative 89.3%

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

       10. /-rgt-identity 89.3%

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

   11. Simplified 89.3%

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

   12. Taylor expanded in k around 0 76.8%

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

       1. add-sqr-sqrt 45.6%

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

       2. pow2 45.6%

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

       3. sqrt-prod 45.6%

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

       4. unpow2 45.6%

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

       5. sqrt-prod 26.2%

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

       6. add-sqr-sqrt 45.6%

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

       7. *-commutative 45.6%

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

       8. sqrt-prod 45.7%

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

       9. unpow2 45.7%

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

       10. sqrt-prod 18.0%

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

       11. add-sqr-sqrt 47.7%

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

   14. Applied egg-rr 47.7%

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

   if 2.50000000000000009e-43 < k < 1.5e83

   1. Initial program 30.2%

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

   3. Simplified 51.0%

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

      1. associate-/r* 54.8%

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

      2. associate-*l/ 54.8%

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

      3. +-rgt-identity 54.8%

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

      4. associate-*l* 54.8%

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

   6. Applied egg-rr 54.8%

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

   7. Taylor expanded in t around 0 91.8%

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

      1. associate-*r* 91.7%

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

      2. *-commutative 91.7%

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

      3. *-commutative 91.7%

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

   9. Simplified 91.7%

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

       1. associate-/r/ 91.8%

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

       2. associate-*l* 91.9%

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

       3. pow-prod-down 91.7%

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

       4. *-commutative 91.7%

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

   11. Applied egg-rr 91.7%

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

       1. *-commutative 91.7%

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

       2. times-frac 99.6%

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

   13. Simplified 99.6%

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

   if 1.5e83 < k

   1. Initial program 31.1%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in t around 0 64.8%

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

      1. times-frac 66.7%

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

   7. Simplified 66.7%

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

      1. div-inv 66.7%

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

      2. add-sqr-sqrt 66.7%

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

      3. pow2 66.7%

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

      4. sqrt-div 66.7%

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

      5. unpow2 66.7%

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

      6. sqrt-prod 72.8%

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

      7. add-sqr-sqrt 72.9%

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

      8. unpow2 72.9%

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

      9. sqrt-prod 56.8%

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

      10. add-sqr-sqrt 96.4%

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

      11. associate-/l* 96.4%

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

      12. unpow2 96.4%

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

      13. *-un-lft-identity 96.4%

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

      14. times-frac 96.3%

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

      15. tan-quot 96.4%

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

   9. Applied egg-rr 96.4%

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

       1. associate-*r/ 96.4%

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

       2. metadata-eval 96.4%

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

       3. associate-/r* 96.5%

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

       4. /-rgt-identity 96.5%

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

       5. associate-/r* 96.5%

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

   11. Simplified 96.5%

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

       1. associate-/l/ 96.5%

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

       2. *-commutative 96.5%

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

       3. associate-/r* 96.4%

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

       4. div-inv 96.4%

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

   13. Applied egg-rr 96.4%

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

       1. associate-*r/ 96.4%

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

       2. metadata-eval 96.4%

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

       3. associate-/r* 96.5%

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

       4. associate-*r* 96.6%

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

   15. Simplified 96.6%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\ell} \cdot \left(k \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+83}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{t}{\ell} \cdot \frac{{\left(k \cdot \sin k\right)}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{\ell}\right)}^{2}}}{\tan k \cdot \left(t \cdot \sin k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m}{\ell} \cdot \left(k\_m \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k\_m}{\ell}\right)}^{2}}}{\tan k\_m \cdot \left(t\_m \cdot \sin k\_m\right)}\\ \end{array} \end{array} \]

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 2.4e-35)
    (/ 2.0 (pow (* (/ k_m l) (* k_m (sqrt t_m))) 2.0))
    (/ (/ 2.0 (pow (/ k_m l) 2.0)) (* (tan k_m) (* t_m (sin k_m)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-35) {
		tmp = 2.0 / pow(((k_m / l) * (k_m * sqrt(t_m))), 2.0);
	} else {
		tmp = (2.0 / pow((k_m / l), 2.0)) / (tan(k_m) * (t_m * sin(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.4d-35) then
        tmp = 2.0d0 / (((k_m / l) * (k_m * sqrt(t_m))) ** 2.0d0)
    else
        tmp = (2.0d0 / ((k_m / l) ** 2.0d0)) / (tan(k_m) * (t_m * sin(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-35) {
		tmp = 2.0 / Math.pow(((k_m / l) * (k_m * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = (2.0 / Math.pow((k_m / l), 2.0)) / (Math.tan(k_m) * (t_m * Math.sin(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.4e-35:
		tmp = 2.0 / math.pow(((k_m / l) * (k_m * math.sqrt(t_m))), 2.0)
	else:
		tmp = (2.0 / math.pow((k_m / l), 2.0)) / (math.tan(k_m) * (t_m * math.sin(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.4e-35)
		tmp = Float64(2.0 / (Float64(Float64(k_m / l) * Float64(k_m * sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(Float64(2.0 / (Float64(k_m / l) ^ 2.0)) / Float64(tan(k_m) * Float64(t_m * sin(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.4e-35)
		tmp = 2.0 / (((k_m / l) * (k_m * sqrt(t_m))) ^ 2.0);
	else
		tmp = (2.0 / ((k_m / l) ^ 2.0)) / (tan(k_m) * (t_m * sin(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.4e-35], N[(2.0 / N[Power[N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[N[(k$95$m / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k$95$m], $MachinePrecision] * N[(t$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.4000000000000001e-35

   1. Initial program 38.2%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in t around 0 72.9%

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

      1. times-frac 72.5%

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

   7. Simplified 72.5%

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

      1. associate-*l/ 72.8%

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

      2. associate-/l* 72.9%

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

      3. unpow2 72.9%

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

      4. *-un-lft-identity 72.9%

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

      5. times-frac 72.9%

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

      6. tan-quot 72.8%

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

   9. Applied egg-rr 72.8%

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

       1. *-commutative 72.8%

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

       2. associate-/l* 72.5%

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

       3. unpow2 72.5%

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

       4. associate-/r* 84.2%

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

       5. unpow2 84.2%

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

       6. associate-*r/ 87.4%

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

       7. associate-*l/ 89.4%

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

       8. unpow2 89.4%

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

       9. *-commutative 89.4%

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

       10. /-rgt-identity 89.4%

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

   11. Simplified 89.4%

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

   12. Taylor expanded in k around 0 76.9%

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

       1. add-sqr-sqrt 45.4%

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

       2. pow2 45.4%

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

       3. sqrt-prod 45.4%

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

       4. unpow2 45.4%

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

       5. sqrt-prod 26.1%

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

       6. add-sqr-sqrt 45.4%

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

       7. *-commutative 45.4%

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

       8. sqrt-prod 45.5%

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

       9. unpow2 45.5%

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

       10. sqrt-prod 17.9%

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

       11. add-sqr-sqrt 47.4%

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

   14. Applied egg-rr 47.4%

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

   if 2.4000000000000001e-35 < k

   1. Initial program 31.1%

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

   3. Simplified 47.8%

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

   5. Taylor expanded in t around 0 73.2%

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

      1. times-frac 72.0%

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

   7. Simplified 72.0%

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

      1. div-inv 72.0%

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

      2. add-sqr-sqrt 72.0%

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

      3. pow2 72.0%

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

      4. sqrt-div 72.0%

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

      5. unpow2 72.0%

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

      6. sqrt-prod 76.2%

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

      7. add-sqr-sqrt 76.3%

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

      8. unpow2 76.3%

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

      9. sqrt-prod 51.2%

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

      10. add-sqr-sqrt 93.5%

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

      11. associate-/l* 93.6%

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

      12. unpow2 93.6%

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

      13. *-un-lft-identity 93.6%

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

      14. times-frac 93.5%

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

      15. tan-quot 93.6%

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

   9. Applied egg-rr 93.6%

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

       1. associate-*r/ 93.6%

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

       2. metadata-eval 93.6%

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

       3. associate-/r* 93.6%

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

       4. /-rgt-identity 93.6%

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

       5. associate-/r* 93.6%

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

   11. Simplified 93.6%

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

       1. associate-/l/ 93.6%

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

       2. *-commutative 93.6%

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

       3. associate-/r* 93.6%

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

       4. div-inv 93.6%

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

   13. Applied egg-rr 93.6%

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

       1. associate-*r/ 93.6%

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

       2. metadata-eval 93.6%

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

       3. associate-/r* 93.6%

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

       4. associate-*r* 93.7%

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

   15. Simplified 93.7%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\ell} \cdot \left(k \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k}{\ell}\right)}^{2}}}{\tan k \cdot \left(t \cdot \sin k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m}{\ell} \cdot \left(k\_m \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot {\left(\frac{k\_m}{\ell}\right)}^{-2}}{t\_m}}{\tan k\_m \cdot \sin k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.4e-35)
    (/ 2.0 (pow (* (/ k_m l) (* k_m (sqrt t_m))) 2.0))
    (/ (/ (* 2.0 (pow (/ k_m l) -2.0)) t_m) (* (tan k_m) (sin k_m))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-35) {
		tmp = 2.0 / pow(((k_m / l) * (k_m * sqrt(t_m))), 2.0);
	} else {
		tmp = ((2.0 * pow((k_m / l), -2.0)) / t_m) / (tan(k_m) * sin(k_m));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.4d-35) then
        tmp = 2.0d0 / (((k_m / l) * (k_m * sqrt(t_m))) ** 2.0d0)
    else
        tmp = ((2.0d0 * ((k_m / l) ** (-2.0d0))) / t_m) / (tan(k_m) * sin(k_m))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-35) {
		tmp = 2.0 / Math.pow(((k_m / l) * (k_m * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = ((2.0 * Math.pow((k_m / l), -2.0)) / t_m) / (Math.tan(k_m) * Math.sin(k_m));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.4e-35:
		tmp = 2.0 / math.pow(((k_m / l) * (k_m * math.sqrt(t_m))), 2.0)
	else:
		tmp = ((2.0 * math.pow((k_m / l), -2.0)) / t_m) / (math.tan(k_m) * math.sin(k_m))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.4e-35)
		tmp = Float64(2.0 / (Float64(Float64(k_m / l) * Float64(k_m * sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(Float64(Float64(2.0 * (Float64(k_m / l) ^ -2.0)) / t_m) / Float64(tan(k_m) * sin(k_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.4e-35)
		tmp = 2.0 / (((k_m / l) * (k_m * sqrt(t_m))) ^ 2.0);
	else
		tmp = ((2.0 * ((k_m / l) ^ -2.0)) / t_m) / (tan(k_m) * sin(k_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.4e-35], N[(2.0 / N[Power[N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Power[N[(k$95$m / l), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.4000000000000001e-35

   1. Initial program 38.2%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in t around 0 72.9%

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

      1. times-frac 72.5%

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

   7. Simplified 72.5%

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

      1. associate-*l/ 72.8%

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

      2. associate-/l* 72.9%

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

      3. unpow2 72.9%

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

      4. *-un-lft-identity 72.9%

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

      5. times-frac 72.9%

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

      6. tan-quot 72.8%

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

   9. Applied egg-rr 72.8%

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

       1. *-commutative 72.8%

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

       2. associate-/l* 72.5%

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

       3. unpow2 72.5%

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

       4. associate-/r* 84.2%

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

       5. unpow2 84.2%

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

       6. associate-*r/ 87.4%

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

       7. associate-*l/ 89.4%

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

       8. unpow2 89.4%

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

       9. *-commutative 89.4%

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

       10. /-rgt-identity 89.4%

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

   11. Simplified 89.4%

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

   12. Taylor expanded in k around 0 76.9%

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

       1. add-sqr-sqrt 45.4%

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

       2. pow2 45.4%

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

       3. sqrt-prod 45.4%

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

       4. unpow2 45.4%

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

       5. sqrt-prod 26.1%

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

       6. add-sqr-sqrt 45.4%

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

       7. *-commutative 45.4%

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

       8. sqrt-prod 45.5%

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

       9. unpow2 45.5%

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

       10. sqrt-prod 17.9%

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

       11. add-sqr-sqrt 47.4%

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

   14. Applied egg-rr 47.4%

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

   if 2.4000000000000001e-35 < k

   1. Initial program 31.1%

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

   3. Simplified 47.8%

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

   5. Taylor expanded in t around 0 73.2%

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

      1. times-frac 72.0%

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

   7. Simplified 72.0%

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

      1. div-inv 72.0%

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

      2. add-sqr-sqrt 72.0%

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

      3. pow2 72.0%

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

      4. sqrt-div 72.0%

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

      5. unpow2 72.0%

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

      6. sqrt-prod 76.2%

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

      7. add-sqr-sqrt 76.3%

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

      8. unpow2 76.3%

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

      9. sqrt-prod 51.2%

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

      10. add-sqr-sqrt 93.5%

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

      11. associate-/l* 93.6%

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

      12. unpow2 93.6%

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

      13. *-un-lft-identity 93.6%

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

      14. times-frac 93.5%

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

      15. tan-quot 93.6%

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

   9. Applied egg-rr 93.6%

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

       1. associate-*r/ 93.6%

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

       2. metadata-eval 93.6%

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

       3. associate-/r* 93.6%

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

       4. /-rgt-identity 93.6%

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

       5. associate-/r* 93.6%

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

   11. Simplified 93.6%

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

       1. div-inv 93.6%

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

       2. div-inv 93.6%

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

       3. pow-flip 93.5%

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

       4. metadata-eval 93.5%

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

   13. Applied egg-rr 93.5%

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

       1. associate-*r/ 93.6%

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

       2. *-rgt-identity 93.6%

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

   15. Simplified 93.6%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\ell} \cdot \left(k \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot {\left(\frac{k}{\ell}\right)}^{-2}}{t}}{\tan k \cdot \sin k}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.8 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m}{\ell} \cdot \left(k\_m \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{1}{\frac{\ell}{k\_m} \cdot \frac{\ell}{k\_m}}}}{t\_m}}{\tan k\_m \cdot \sin k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.8e-43)
    (/ 2.0 (pow (* (/ k_m l) (* k_m (sqrt t_m))) 2.0))
    (/
     (/ (/ 2.0 (/ 1.0 (* (/ l k_m) (/ l k_m)))) t_m)
     (* (tan k_m) (sin k_m))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.8e-43) {
		tmp = 2.0 / pow(((k_m / l) * (k_m * sqrt(t_m))), 2.0);
	} else {
		tmp = ((2.0 / (1.0 / ((l / k_m) * (l / k_m)))) / t_m) / (tan(k_m) * sin(k_m));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.8d-43) then
        tmp = 2.0d0 / (((k_m / l) * (k_m * sqrt(t_m))) ** 2.0d0)
    else
        tmp = ((2.0d0 / (1.0d0 / ((l / k_m) * (l / k_m)))) / t_m) / (tan(k_m) * sin(k_m))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.8e-43) {
		tmp = 2.0 / Math.pow(((k_m / l) * (k_m * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = ((2.0 / (1.0 / ((l / k_m) * (l / k_m)))) / t_m) / (Math.tan(k_m) * Math.sin(k_m));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.8e-43:
		tmp = 2.0 / math.pow(((k_m / l) * (k_m * math.sqrt(t_m))), 2.0)
	else:
		tmp = ((2.0 / (1.0 / ((l / k_m) * (l / k_m)))) / t_m) / (math.tan(k_m) * math.sin(k_m))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.8e-43)
		tmp = Float64(2.0 / (Float64(Float64(k_m / l) * Float64(k_m * sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(1.0 / Float64(Float64(l / k_m) * Float64(l / k_m)))) / t_m) / Float64(tan(k_m) * sin(k_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.8e-43)
		tmp = 2.0 / (((k_m / l) * (k_m * sqrt(t_m))) ^ 2.0);
	else
		tmp = ((2.0 / (1.0 / ((l / k_m) * (l / k_m)))) / t_m) / (tan(k_m) * sin(k_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.8e-43], N[(2.0 / N[Power[N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(1.0 / N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.7999999999999998e-43

   1. Initial program 38.4%

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

   3. Simplified 44.6%

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

   5. Taylor expanded in t around 0 73.2%

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

      1. times-frac 72.4%

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

   7. Simplified 72.4%

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

      1. associate-*l/ 73.2%

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

      2. associate-/l* 73.2%

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

      3. unpow2 73.2%

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

      4. *-un-lft-identity 73.2%

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

      5. times-frac 73.2%

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

      6. tan-quot 73.2%

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

   9. Applied egg-rr 73.2%

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

       1. *-commutative 73.2%

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

       2. associate-/l* 72.4%

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

       3. unpow2 72.4%

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

       4. associate-/r* 84.1%

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

       5. unpow2 84.1%

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

       6. associate-*r/ 87.3%

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

       7. associate-*l/ 89.3%

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

       8. unpow2 89.3%

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

       9. *-commutative 89.3%

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

       10. /-rgt-identity 89.3%

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

   11. Simplified 89.3%

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

   12. Taylor expanded in k around 0 76.8%

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

       1. add-sqr-sqrt 45.6%

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

       2. pow2 45.6%

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

       3. sqrt-prod 45.6%

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

       4. unpow2 45.6%

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

       5. sqrt-prod 26.2%

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

       6. add-sqr-sqrt 45.6%

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

       7. *-commutative 45.6%

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

       8. sqrt-prod 45.7%

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

       9. unpow2 45.7%

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

       10. sqrt-prod 18.0%

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

       11. add-sqr-sqrt 47.7%

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

   14. Applied egg-rr 47.7%

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

   if 2.7999999999999998e-43 < k

   1. Initial program 30.8%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in t around 0 72.3%

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

      1. times-frac 72.4%

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

   7. Simplified 72.4%

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

      1. div-inv 72.4%

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

      2. add-sqr-sqrt 72.3%

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

      3. pow2 72.3%

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

      4. sqrt-div 72.4%

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

      5. unpow2 72.4%

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

      6. sqrt-prod 76.4%

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

      7. add-sqr-sqrt 76.6%

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

      8. unpow2 76.6%

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

      9. sqrt-prod 51.9%

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

      10. add-sqr-sqrt 93.6%

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

      11. associate-/l* 93.6%

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

      12. unpow2 93.6%

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

      13. *-un-lft-identity 93.6%

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

      14. times-frac 93.6%

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

      15. tan-quot 93.7%

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

   9. Applied egg-rr 93.7%

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

       1. associate-*r/ 93.7%

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

       2. metadata-eval 93.7%

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

       3. associate-/r* 93.6%

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

       4. /-rgt-identity 93.6%

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

       5. associate-/r* 93.6%

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

   11. Simplified 93.6%

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

       1. unpow2 93.6%

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

       2. clear-num 93.7%

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

       3. clear-num 93.6%

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

       4. frac-times 93.7%

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

       5. metadata-eval 93.7%

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

   13. Applied egg-rr 93.7%

       \[\leadsto \frac{\frac{\frac{2}{\color{blue}{\frac{1}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}}}{t}}{\sin k \cdot \tan k} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-43}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\ell} \cdot \left(k \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{1}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}}{t}}{\tan k \cdot \sin k}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m}{\ell} \cdot \left(k\_m \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m}{\ell} \cdot \frac{k\_m}{\ell}\right) \cdot \left(t\_m \cdot \left(\tan k\_m \cdot \sin k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.4e-35)
    (/ 2.0 (pow (* (/ k_m l) (* k_m (sqrt t_m))) 2.0))
    (/ 2.0 (* (* (/ k_m l) (/ k_m l)) (* t_m (* (tan k_m) (sin k_m))))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-35) {
		tmp = 2.0 / pow(((k_m / l) * (k_m * sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 / (((k_m / l) * (k_m / l)) * (t_m * (tan(k_m) * sin(k_m))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.4d-35) then
        tmp = 2.0d0 / (((k_m / l) * (k_m * sqrt(t_m))) ** 2.0d0)
    else
        tmp = 2.0d0 / (((k_m / l) * (k_m / l)) * (t_m * (tan(k_m) * sin(k_m))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-35) {
		tmp = 2.0 / Math.pow(((k_m / l) * (k_m * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 / (((k_m / l) * (k_m / l)) * (t_m * (Math.tan(k_m) * Math.sin(k_m))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.4e-35:
		tmp = 2.0 / math.pow(((k_m / l) * (k_m * math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 / (((k_m / l) * (k_m / l)) * (t_m * (math.tan(k_m) * math.sin(k_m))))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.4e-35)
		tmp = Float64(2.0 / (Float64(Float64(k_m / l) * Float64(k_m * sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / l) * Float64(k_m / l)) * Float64(t_m * Float64(tan(k_m) * sin(k_m)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.4e-35)
		tmp = 2.0 / (((k_m / l) * (k_m * sqrt(t_m))) ^ 2.0);
	else
		tmp = 2.0 / (((k_m / l) * (k_m / l)) * (t_m * (tan(k_m) * sin(k_m))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.4e-35], N[(2.0 / N[Power[N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.4000000000000001e-35

   1. Initial program 38.2%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in t around 0 72.9%

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

      1. times-frac 72.5%

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

   7. Simplified 72.5%

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

      1. associate-*l/ 72.8%

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

      2. associate-/l* 72.9%

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

      3. unpow2 72.9%

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

      4. *-un-lft-identity 72.9%

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

      5. times-frac 72.9%

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

      6. tan-quot 72.8%

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

   9. Applied egg-rr 72.8%

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

       1. *-commutative 72.8%

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

       2. associate-/l* 72.5%

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

       3. unpow2 72.5%

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

       4. associate-/r* 84.2%

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

       5. unpow2 84.2%

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

       6. associate-*r/ 87.4%

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

       7. associate-*l/ 89.4%

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

       8. unpow2 89.4%

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

       9. *-commutative 89.4%

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

       10. /-rgt-identity 89.4%

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

   11. Simplified 89.4%

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

   12. Taylor expanded in k around 0 76.9%

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

       1. add-sqr-sqrt 45.4%

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

       2. pow2 45.4%

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

       3. sqrt-prod 45.4%

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

       4. unpow2 45.4%

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

       5. sqrt-prod 26.1%

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

       6. add-sqr-sqrt 45.4%

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

       7. *-commutative 45.4%

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

       8. sqrt-prod 45.5%

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

       9. unpow2 45.5%

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

       10. sqrt-prod 17.9%

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

       11. add-sqr-sqrt 47.4%

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

   14. Applied egg-rr 47.4%

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

   if 2.4000000000000001e-35 < k

   1. Initial program 31.1%

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

   3. Simplified 47.8%

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

   5. Taylor expanded in t around 0 73.2%

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

      1. times-frac 72.0%

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

   7. Simplified 72.0%

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

      1. associate-*l/ 73.2%

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

      2. associate-/l* 73.2%

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

      3. unpow2 73.2%

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

      4. *-un-lft-identity 73.2%

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

      5. times-frac 73.1%

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

      6. tan-quot 73.2%

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

   9. Applied egg-rr 73.2%

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

       1. *-commutative 73.2%

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

       2. associate-/l* 72.1%

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

       3. unpow2 72.1%

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

       4. associate-/r* 79.1%

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

       5. unpow2 79.1%

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

       6. associate-*r/ 91.1%

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

       7. associate-*l/ 93.6%

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

       8. unpow2 93.6%

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

       9. *-commutative 93.6%

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

       10. /-rgt-identity 93.6%

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

   11. Simplified 93.6%

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

       1. unpow2 93.6%

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

   13. Applied egg-rr 93.6%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\ell} \cdot \left(k \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \left(\tan k \cdot \sin k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m}{\ell} \cdot \left(k\_m \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{k\_m}{\ell} \cdot \frac{k\_m}{\ell}}}{t\_m}}{\tan k\_m \cdot \sin k\_m}\\ \end{array} \end{array} \]

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 1.8e-35)
    (/ 2.0 (pow (* (/ k_m l) (* k_m (sqrt t_m))) 2.0))
    (/ (/ (/ 2.0 (* (/ k_m l) (/ k_m l))) t_m) (* (tan k_m) (sin k_m))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.8e-35) {
		tmp = 2.0 / pow(((k_m / l) * (k_m * sqrt(t_m))), 2.0);
	} else {
		tmp = ((2.0 / ((k_m / l) * (k_m / l))) / t_m) / (tan(k_m) * sin(k_m));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.8d-35) then
        tmp = 2.0d0 / (((k_m / l) * (k_m * sqrt(t_m))) ** 2.0d0)
    else
        tmp = ((2.0d0 / ((k_m / l) * (k_m / l))) / t_m) / (tan(k_m) * sin(k_m))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.8e-35) {
		tmp = 2.0 / Math.pow(((k_m / l) * (k_m * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = ((2.0 / ((k_m / l) * (k_m / l))) / t_m) / (Math.tan(k_m) * Math.sin(k_m));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.8e-35:
		tmp = 2.0 / math.pow(((k_m / l) * (k_m * math.sqrt(t_m))), 2.0)
	else:
		tmp = ((2.0 / ((k_m / l) * (k_m / l))) / t_m) / (math.tan(k_m) * math.sin(k_m))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.8e-35)
		tmp = Float64(2.0 / (Float64(Float64(k_m / l) * Float64(k_m * sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(Float64(k_m / l) * Float64(k_m / l))) / t_m) / Float64(tan(k_m) * sin(k_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.8e-35)
		tmp = 2.0 / (((k_m / l) * (k_m * sqrt(t_m))) ^ 2.0);
	else
		tmp = ((2.0 / ((k_m / l) * (k_m / l))) / t_m) / (tan(k_m) * sin(k_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.8e-35], N[(2.0 / N[Power[N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(N[(k$95$m / l), $MachinePrecision] * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(N[Tan[k$95$m], $MachinePrecision] * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.80000000000000009e-35

   1. Initial program 38.2%

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

   3. Simplified 44.4%

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

   5. Taylor expanded in t around 0 72.9%

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

      1. times-frac 72.5%

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

   7. Simplified 72.5%

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

      1. associate-*l/ 72.8%

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

      2. associate-/l* 72.9%

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

      3. unpow2 72.9%

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

      4. *-un-lft-identity 72.9%

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

      5. times-frac 72.9%

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

      6. tan-quot 72.8%

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

   9. Applied egg-rr 72.8%

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

       1. *-commutative 72.8%

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

       2. associate-/l* 72.5%

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

       3. unpow2 72.5%

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

       4. associate-/r* 84.2%

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

       5. unpow2 84.2%

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

       6. associate-*r/ 87.4%

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

       7. associate-*l/ 89.4%

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

       8. unpow2 89.4%

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

       9. *-commutative 89.4%

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

       10. /-rgt-identity 89.4%

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

   11. Simplified 89.4%

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

   12. Taylor expanded in k around 0 76.9%

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

       1. add-sqr-sqrt 45.4%

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

       2. pow2 45.4%

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

       3. sqrt-prod 45.4%

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

       4. unpow2 45.4%

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

       5. sqrt-prod 26.1%

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

       6. add-sqr-sqrt 45.4%

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

       7. *-commutative 45.4%

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

       8. sqrt-prod 45.5%

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

       9. unpow2 45.5%

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

       10. sqrt-prod 17.9%

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

       11. add-sqr-sqrt 47.4%

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

   14. Applied egg-rr 47.4%

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

   if 1.80000000000000009e-35 < k

   1. Initial program 31.1%

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

   3. Simplified 47.8%

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

   5. Taylor expanded in t around 0 73.2%

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

      1. times-frac 72.0%

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

   7. Simplified 72.0%

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

      1. div-inv 72.0%

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

      2. add-sqr-sqrt 72.0%

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

      3. pow2 72.0%

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

      4. sqrt-div 72.0%

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

      5. unpow2 72.0%

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

      6. sqrt-prod 76.2%

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

      7. add-sqr-sqrt 76.3%

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

      8. unpow2 76.3%

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

      9. sqrt-prod 51.2%

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

      10. add-sqr-sqrt 93.5%

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

      11. associate-/l* 93.6%

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

      12. unpow2 93.6%

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

      13. *-un-lft-identity 93.6%

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

      14. times-frac 93.5%

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

      15. tan-quot 93.6%

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

   9. Applied egg-rr 93.6%

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

       1. associate-*r/ 93.6%

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

       2. metadata-eval 93.6%

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

       3. associate-/r* 93.6%

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

       4. /-rgt-identity 93.6%

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

       5. associate-/r* 93.6%

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

   11. Simplified 93.6%

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

       1. unpow2 93.6%

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

   13. Applied egg-rr 93.6%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\ell} \cdot \left(k \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{k}{\ell} \cdot \frac{k}{\ell}}}{t}}{\tan k \cdot \sin k}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.2%

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

3. Simplified 45.3%

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

5. Taylor expanded in t around 0 73.0%

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

   1. times-frac 72.4%

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

7. Simplified 72.4%

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

   1. associate-*l/ 72.9%

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

   2. associate-/l* 72.9%

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

   3. unpow2 72.9%

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

   4. *-un-lft-identity 72.9%

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

   5. times-frac 72.9%

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

   6. tan-quot 72.9%

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

9. Applied egg-rr 72.9%

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

    1. *-commutative 72.9%

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

    2. associate-/l* 72.4%

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

    3. unpow2 72.4%

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

    4. associate-/r* 82.8%

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

    5. unpow2 82.8%

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

    6. associate-*r/ 88.4%

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

    7. associate-*l/ 90.6%

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

    8. unpow2 90.6%

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

    9. *-commutative 90.6%

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

    10. /-rgt-identity 90.6%

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

11. Simplified 90.6%

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

12. Taylor expanded in k around 0 72.7%

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

    1. add-sqr-sqrt 41.2%

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

    2. pow2 41.2%

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

    3. sqrt-prod 41.2%

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

    4. unpow2 41.2%

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

    5. sqrt-prod 23.6%

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

    6. add-sqr-sqrt 41.2%

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

    7. *-commutative 41.2%

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

    8. sqrt-prod 41.3%

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

    9. unpow2 41.3%

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

    10. sqrt-prod 21.4%

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

    11. add-sqr-sqrt 42.6%

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

14. Applied egg-rr 42.6%

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

    1. associate-*l/ 41.6%

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

    2. associate-*r/ 42.6%

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

    3. associate-*r/ 42.6%

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

    4. *-commutative 42.6%

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

    5. associate-*l/ 42.6%

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

    6. associate-/l* 43.4%

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

16. Simplified 43.4%

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

17. Final simplification 43.4%

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

Alternative 12: 75.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.2%

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

3. Simplified 45.3%

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

5. Taylor expanded in t around 0 73.0%

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

   1. times-frac 72.4%

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

7. Simplified 72.4%

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

   1. associate-*l/ 72.9%

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

   2. associate-/l* 72.9%

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

   3. unpow2 72.9%

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

   4. *-un-lft-identity 72.9%

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

   5. times-frac 72.9%

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

   6. tan-quot 72.9%

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

9. Applied egg-rr 72.9%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot \left(\frac{\sin k}{1} \cdot \tan k\right)\right)}{{\ell}^{2}}}} \]
10. Step-by-step derivation

    1. *-commutative 72.9%

       \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot \left(\frac{\sin k}{1} \cdot \tan k\right)\right) \cdot {k}^{2}}}{{\ell}^{2}}} \]

    2. associate-/l* 72.4%

       \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{\sin k}{1} \cdot \tan k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}}} \]

    3. unpow2 72.4%

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{1} \cdot \tan k\right)\right) \cdot \frac{{k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]

    4. associate-/r* 82.8%

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{1} \cdot \tan k\right)\right) \cdot \color{blue}{\frac{\frac{{k}^{2}}{\ell}}{\ell}}} \]

    5. unpow2 82.8%

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{1} \cdot \tan k\right)\right) \cdot \frac{\frac{\color{blue}{k \cdot k}}{\ell}}{\ell}} \]

    6. associate-*r/ 88.4%

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{1} \cdot \tan k\right)\right) \cdot \frac{\color{blue}{k \cdot \frac{k}{\ell}}}{\ell}} \]

    7. associate-*l/ 90.6%

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{1} \cdot \tan k\right)\right) \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]

    8. unpow2 90.6%

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{1} \cdot \tan k\right)\right) \cdot \color{blue}{{\left(\frac{k}{\ell}\right)}^{2}}} \]

    9. *-commutative 90.6%

       \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left(t \cdot \left(\frac{\sin k}{1} \cdot \tan k\right)\right)}} \]

    10. /-rgt-identity 90.6%

        \[\leadsto \frac{2}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left(t \cdot \left(\color{blue}{\sin k} \cdot \tan k\right)\right)} \]

11. Simplified 90.6%

    \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

12. Taylor expanded in k around 0 72.7%

    \[\leadsto \frac{2}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]
13. Step-by-step derivation

    1. add-sqr-sqrt 41.2%

       \[\leadsto \frac{2}{\color{blue}{\sqrt{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left({k}^{2} \cdot t\right)} \cdot \sqrt{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left({k}^{2} \cdot t\right)}}} \]

    2. pow2 41.2%

       \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left({k}^{2} \cdot t\right)}\right)}^{2}}} \]

    3. sqrt-prod 41.2%

       \[\leadsto \frac{2}{{\color{blue}{\left(\sqrt{{\left(\frac{k}{\ell}\right)}^{2}} \cdot \sqrt{{k}^{2} \cdot t}\right)}}^{2}} \]

    4. unpow2 41.2%

       \[\leadsto \frac{2}{{\left(\sqrt{\color{blue}{\frac{k}{\ell} \cdot \frac{k}{\ell}}} \cdot \sqrt{{k}^{2} \cdot t}\right)}^{2}} \]

    5. sqrt-prod 23.6%

       \[\leadsto \frac{2}{{\left(\color{blue}{\left(\sqrt{\frac{k}{\ell}} \cdot \sqrt{\frac{k}{\ell}}\right)} \cdot \sqrt{{k}^{2} \cdot t}\right)}^{2}} \]

    6. add-sqr-sqrt 41.2%

       \[\leadsto \frac{2}{{\left(\color{blue}{\frac{k}{\ell}} \cdot \sqrt{{k}^{2} \cdot t}\right)}^{2}} \]

    7. *-commutative 41.2%

       \[\leadsto \frac{2}{{\left(\frac{k}{\ell} \cdot \sqrt{\color{blue}{t \cdot {k}^{2}}}\right)}^{2}} \]

    8. sqrt-prod 41.3%

       \[\leadsto \frac{2}{{\left(\frac{k}{\ell} \cdot \color{blue}{\left(\sqrt{t} \cdot \sqrt{{k}^{2}}\right)}\right)}^{2}} \]

    9. unpow2 41.3%

       \[\leadsto \frac{2}{{\left(\frac{k}{\ell} \cdot \left(\sqrt{t} \cdot \sqrt{\color{blue}{k \cdot k}}\right)\right)}^{2}} \]

    10. sqrt-prod 21.4%

        \[\leadsto \frac{2}{{\left(\frac{k}{\ell} \cdot \left(\sqrt{t} \cdot \color{blue}{\left(\sqrt{k} \cdot \sqrt{k}\right)}\right)\right)}^{2}} \]

    11. add-sqr-sqrt 42.6%

        \[\leadsto \frac{2}{{\left(\frac{k}{\ell} \cdot \left(\sqrt{t} \cdot \color{blue}{k}\right)\right)}^{2}} \]

14. Applied egg-rr 42.6%

    \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{\ell} \cdot \left(\sqrt{t} \cdot k\right)\right)}^{2}}} \]

15. Final simplification 42.6%

    \[\leadsto \frac{2}{{\left(\frac{k}{\ell} \cdot \left(k \cdot \sqrt{t}\right)\right)}^{2}} \]
16. Add Preprocessing

Alternative 13: 73.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \frac{1}{t\_m \cdot {\left(k\_m \cdot \frac{k\_m}{\ell}\right)}^{2}}\right) \end{array} \]

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 (/ 1.0 (* t_m (pow (* k_m (/ k_m l)) 2.0))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * (1.0 / (t_m * pow((k_m * (k_m / l)), 2.0))));
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 * (1.0d0 / (t_m * ((k_m * (k_m / l)) ** 2.0d0))))
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * (1.0 / (t_m * Math.pow((k_m * (k_m / l)), 2.0))));
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 * (1.0 / (t_m * math.pow((k_m * (k_m / l)), 2.0))))

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 * Float64(1.0 / Float64(t_m * (Float64(k_m * Float64(k_m / l)) ^ 2.0)))))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 * (1.0 / (t_m * ((k_m * (k_m / l)) ^ 2.0))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 * N[(1.0 / N[(t$95$m * N[Power[N[(k$95$m * N[(k$95$m / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 45.3%

   \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in t around 0 73.0%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
6. Step-by-step derivation

   1. times-frac 72.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]

7. Simplified 72.4%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]
8. Step-by-step derivation

   1. associate-*l/ 72.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}{{\ell}^{2}}}} \]

   2. associate-/l* 72.9%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \color{blue}{\left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right)}}{{\ell}^{2}}} \]

   3. unpow2 72.9%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \frac{\color{blue}{\sin k \cdot \sin k}}{\cos k}\right)}{{\ell}^{2}}} \]

   4. *-un-lft-identity 72.9%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \frac{\sin k \cdot \sin k}{\color{blue}{1 \cdot \cos k}}\right)}{{\ell}^{2}}} \]

   5. times-frac 72.9%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(\frac{\sin k}{1} \cdot \frac{\sin k}{\cos k}\right)}\right)}{{\ell}^{2}}} \]

   6. tan-quot 72.9%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(\frac{\sin k}{1} \cdot \color{blue}{\tan k}\right)\right)}{{\ell}^{2}}} \]

9. Applied egg-rr 72.9%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot \left(\frac{\sin k}{1} \cdot \tan k\right)\right)}{{\ell}^{2}}}} \]
10. Step-by-step derivation

    1. *-commutative 72.9%

       \[\leadsto \frac{2}{\frac{\color{blue}{\left(t \cdot \left(\frac{\sin k}{1} \cdot \tan k\right)\right) \cdot {k}^{2}}}{{\ell}^{2}}} \]

    2. associate-/l* 72.4%

       \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{\sin k}{1} \cdot \tan k\right)\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}}} \]

    3. unpow2 72.4%

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{1} \cdot \tan k\right)\right) \cdot \frac{{k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]

    4. associate-/r* 82.8%

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{1} \cdot \tan k\right)\right) \cdot \color{blue}{\frac{\frac{{k}^{2}}{\ell}}{\ell}}} \]

    5. unpow2 82.8%

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{1} \cdot \tan k\right)\right) \cdot \frac{\frac{\color{blue}{k \cdot k}}{\ell}}{\ell}} \]

    6. associate-*r/ 88.4%

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{1} \cdot \tan k\right)\right) \cdot \frac{\color{blue}{k \cdot \frac{k}{\ell}}}{\ell}} \]

    7. associate-*l/ 90.6%

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{1} \cdot \tan k\right)\right) \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}} \]

    8. unpow2 90.6%

       \[\leadsto \frac{2}{\left(t \cdot \left(\frac{\sin k}{1} \cdot \tan k\right)\right) \cdot \color{blue}{{\left(\frac{k}{\ell}\right)}^{2}}} \]

    9. *-commutative 90.6%

       \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left(t \cdot \left(\frac{\sin k}{1} \cdot \tan k\right)\right)}} \]

    10. /-rgt-identity 90.6%

        \[\leadsto \frac{2}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left(t \cdot \left(\color{blue}{\sin k} \cdot \tan k\right)\right)} \]

11. Simplified 90.6%

    \[\leadsto \frac{2}{\color{blue}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

12. Taylor expanded in k around 0 72.7%

    \[\leadsto \frac{2}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]
13. Step-by-step derivation

    1. div-inv 72.7%

       \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]

    2. associate-*r* 72.0%

       \[\leadsto 2 \cdot \frac{1}{\color{blue}{\left({\left(\frac{k}{\ell}\right)}^{2} \cdot {k}^{2}\right) \cdot t}} \]

    3. pow-prod-down 73.7%

       \[\leadsto 2 \cdot \frac{1}{\color{blue}{{\left(\frac{k}{\ell} \cdot k\right)}^{2}} \cdot t} \]

14. Applied egg-rr 73.7%

    \[\leadsto \color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{\ell} \cdot k\right)}^{2} \cdot t}} \]

15. Final simplification 73.7%

    \[\leadsto 2 \cdot \frac{1}{t \cdot {\left(k \cdot \frac{k}{\ell}\right)}^{2}} \]
16. Add Preprocessing

Alternative 14: 67.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \left(\ell \cdot \left(\frac{\ell}{t\_m} \cdot {k\_m}^{-4}\right)\right)\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (* l (* (/ l t_m) (pow k_m -4.0))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * (l * ((l / t_m) * pow(k_m, -4.0))));
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 * (l * ((l / t_m) * (k_m ** (-4.0d0)))))
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * (l * ((l / t_m) * Math.pow(k_m, -4.0))));
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 * (l * ((l / t_m) * math.pow(k_m, -4.0))))

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 * Float64(l * Float64(Float64(l / t_m) * (k_m ^ -4.0)))))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 * (l * ((l / t_m) * (k_m ^ -4.0))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 * N[(l * N[(N[(l / t$95$m), $MachinePrecision] * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 45.3%

   \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 64.5%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
6. Step-by-step derivation

   1. unpow2 64.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. times-frac 70.5%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)} \]

7. Applied egg-rr 70.5%

   \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)} \]
8. Step-by-step derivation

   1. associate-*r/ 70.8%

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{{k}^{4}} \cdot \ell}{t}} \]

   2. div-inv 70.8%

      \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \frac{1}{{k}^{4}}\right)} \cdot \ell}{t} \]

   3. pow-flip 70.8%

      \[\leadsto 2 \cdot \frac{\left(\ell \cdot \color{blue}{{k}^{\left(-4\right)}}\right) \cdot \ell}{t} \]

   4. metadata-eval 70.8%

      \[\leadsto 2 \cdot \frac{\left(\ell \cdot {k}^{\color{blue}{-4}}\right) \cdot \ell}{t} \]

9. Applied egg-rr 70.8%

   \[\leadsto 2 \cdot \color{blue}{\frac{\left(\ell \cdot {k}^{-4}\right) \cdot \ell}{t}} \]
10. Step-by-step derivation

    1. associate-/l* 70.5%

       \[\leadsto 2 \cdot \color{blue}{\left(\left(\ell \cdot {k}^{-4}\right) \cdot \frac{\ell}{t}\right)} \]

11. Simplified 70.5%

    \[\leadsto 2 \cdot \color{blue}{\left(\left(\ell \cdot {k}^{-4}\right) \cdot \frac{\ell}{t}\right)} \]
12. Step-by-step derivation

    1. associate-*r/ 70.8%

       \[\leadsto 2 \cdot \color{blue}{\frac{\left(\ell \cdot {k}^{-4}\right) \cdot \ell}{t}} \]

13. Applied egg-rr 70.8%

    \[\leadsto 2 \cdot \color{blue}{\frac{\left(\ell \cdot {k}^{-4}\right) \cdot \ell}{t}} \]
14. Step-by-step derivation

    1. associate-/l* 70.5%

       \[\leadsto 2 \cdot \color{blue}{\left(\left(\ell \cdot {k}^{-4}\right) \cdot \frac{\ell}{t}\right)} \]

    2. associate-*r* 70.5%

       \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \left({k}^{-4} \cdot \frac{\ell}{t}\right)\right)} \]

15. Simplified 70.5%

    \[\leadsto 2 \cdot \color{blue}{\left(\ell \cdot \left({k}^{-4} \cdot \frac{\ell}{t}\right)\right)} \]

16. Final simplification 70.5%

    \[\leadsto 2 \cdot \left(\ell \cdot \left(\frac{\ell}{t} \cdot {k}^{-4}\right)\right) \]
17. Add Preprocessing

Alternative 15: 67.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \left(\frac{\ell}{t\_m} \cdot \left(\ell \cdot {k\_m}^{-4}\right)\right)\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (* (/ l t_m) (* l (pow k_m -4.0))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * ((l / t_m) * (l * pow(k_m, -4.0))));
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 * ((l / t_m) * (l * (k_m ** (-4.0d0)))))
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * ((l / t_m) * (l * Math.pow(k_m, -4.0))));
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 * ((l / t_m) * (l * math.pow(k_m, -4.0))))

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 * Float64(Float64(l / t_m) * Float64(l * (k_m ^ -4.0)))))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 * ((l / t_m) * (l * (k_m ^ -4.0))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 * N[(N[(l / t$95$m), $MachinePrecision] * N[(l * N[Power[k$95$m, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 45.3%

   \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 64.5%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
6. Step-by-step derivation

   1. unpow2 64.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. times-frac 70.5%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)} \]

7. Applied egg-rr 70.5%

   \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)} \]
8. Step-by-step derivation

   1. associate-*r/ 70.8%

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{{k}^{4}} \cdot \ell}{t}} \]

   2. div-inv 70.8%

      \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \frac{1}{{k}^{4}}\right)} \cdot \ell}{t} \]

   3. pow-flip 70.8%

      \[\leadsto 2 \cdot \frac{\left(\ell \cdot \color{blue}{{k}^{\left(-4\right)}}\right) \cdot \ell}{t} \]

   4. metadata-eval 70.8%

      \[\leadsto 2 \cdot \frac{\left(\ell \cdot {k}^{\color{blue}{-4}}\right) \cdot \ell}{t} \]

9. Applied egg-rr 70.8%

   \[\leadsto 2 \cdot \color{blue}{\frac{\left(\ell \cdot {k}^{-4}\right) \cdot \ell}{t}} \]
10. Step-by-step derivation

    1. associate-/l* 70.5%

       \[\leadsto 2 \cdot \color{blue}{\left(\left(\ell \cdot {k}^{-4}\right) \cdot \frac{\ell}{t}\right)} \]

11. Simplified 70.5%

    \[\leadsto 2 \cdot \color{blue}{\left(\left(\ell \cdot {k}^{-4}\right) \cdot \frac{\ell}{t}\right)} \]

12. Final simplification 70.5%

    \[\leadsto 2 \cdot \left(\frac{\ell}{t} \cdot \left(\ell \cdot {k}^{-4}\right)\right) \]
13. Add Preprocessing

Alternative 16: 68.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \left(\frac{\ell}{{k\_m}^{4}} \cdot \frac{\ell}{t\_m}\right)\right) \end{array} \]

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 (* (/ l (pow k_m 4.0)) (/ l 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 * ((l / pow(k_m, 4.0)) * (l / t_m)));
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 * ((l / (k_m ** 4.0d0)) * (l / 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 * ((l / Math.pow(k_m, 4.0)) * (l / 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 * ((l / math.pow(k_m, 4.0)) * (l / t_m)))

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 * Float64(Float64(l / (k_m ^ 4.0)) * Float64(l / t_m))))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 * ((l / (k_m ^ 4.0)) * (l / t_m)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 * N[(N[(l / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 45.3%

   \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 64.5%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
6. Step-by-step derivation

   1. unpow2 64.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. times-frac 70.5%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)} \]

7. Applied egg-rr 70.5%

   \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)} \]

8. Final simplification 70.5%

   \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right) \]
9. Add Preprocessing

Alternative 17: 69.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \frac{\ell}{t\_m \cdot \frac{{k\_m}^{4}}{\ell}}\right) \end{array} \]

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 (/ l (* t_m (/ (pow k_m 4.0) l))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * (l / (t_m * (pow(k_m, 4.0) / l))));
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 * (l / (t_m * ((k_m ** 4.0d0) / l))))
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * (l / (t_m * (Math.pow(k_m, 4.0) / l))));
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 * (l / (t_m * (math.pow(k_m, 4.0) / l))))

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 * Float64(l / Float64(t_m * Float64((k_m ^ 4.0) / l)))))
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 * (l / (t_m * ((k_m ^ 4.0) / l))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 * N[(l / N[(t$95$m * N[(N[Power[k$95$m, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

3. Simplified 45.3%

   \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in k around 0 64.5%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
6. Step-by-step derivation

   1. unpow2 64.5%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. times-frac 70.5%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)} \]

7. Applied egg-rr 70.5%

   \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)} \]
8. Step-by-step derivation

   1. clear-num 70.5%

      \[\leadsto 2 \cdot \left(\color{blue}{\frac{1}{\frac{{k}^{4}}{\ell}}} \cdot \frac{\ell}{t}\right) \]

   2. frac-times 70.8%

      \[\leadsto 2 \cdot \color{blue}{\frac{1 \cdot \ell}{\frac{{k}^{4}}{\ell} \cdot t}} \]

   3. *-un-lft-identity 70.8%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell}}{\frac{{k}^{4}}{\ell} \cdot t} \]

9. Applied egg-rr 70.8%

   \[\leadsto 2 \cdot \color{blue}{\frac{\ell}{\frac{{k}^{4}}{\ell} \cdot t}} \]

10. Final simplification 70.8%

    \[\leadsto 2 \cdot \frac{\ell}{t \cdot \frac{{k}^{4}}{\ell}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(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))))