Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.2% → 89.4%

Time: 19.6s

Alternatives: 14

Speedup: 3.8×

• 26.4% 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: 55.2% 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: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t_m \cdot {\sin k_m}^{2}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 8.2 \cdot 10^{-16}:\\ \;\;\;\;{\left({t_m}^{-0.5} \cdot \frac{\ell}{k_m \cdot t_m}\right)}^{2}\\ \mathbf{elif}\;k_m \leq 2.2 \cdot 10^{+78}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k_m}^{2} \cdot t_2}{\ell \cdot \cos k_m}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t_2}{{\left(\frac{\ell}{k_m}\right)}^{2}}}{\cos k_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* t_m (pow (sin k_m) 2.0))))
   (*
    t_s
    (if (<= k_m 8.2e-16)
      (pow (* (pow t_m -0.5) (/ l (* k_m t_m))) 2.0)
      (if (<= k_m 2.2e+78)
        (/ 2.0 (/ (/ (* (pow k_m 2.0) t_2) (* l (cos k_m))) l))
        (/ 2.0 (/ (/ t_2 (pow (/ l k_m) 2.0)) (cos 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 = t_m * pow(sin(k_m), 2.0);
	double tmp;
	if (k_m <= 8.2e-16) {
		tmp = pow((pow(t_m, -0.5) * (l / (k_m * t_m))), 2.0);
	} else if (k_m <= 2.2e+78) {
		tmp = 2.0 / (((pow(k_m, 2.0) * t_2) / (l * cos(k_m))) / l);
	} else {
		tmp = 2.0 / ((t_2 / pow((l / k_m), 2.0)) / cos(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) :: t_2
    real(8) :: tmp
    t_2 = t_m * (sin(k_m) ** 2.0d0)
    if (k_m <= 8.2d-16) then
        tmp = ((t_m ** (-0.5d0)) * (l / (k_m * t_m))) ** 2.0d0
    else if (k_m <= 2.2d+78) then
        tmp = 2.0d0 / ((((k_m ** 2.0d0) * t_2) / (l * cos(k_m))) / l)
    else
        tmp = 2.0d0 / ((t_2 / ((l / k_m) ** 2.0d0)) / cos(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 t_2 = t_m * Math.pow(Math.sin(k_m), 2.0);
	double tmp;
	if (k_m <= 8.2e-16) {
		tmp = Math.pow((Math.pow(t_m, -0.5) * (l / (k_m * t_m))), 2.0);
	} else if (k_m <= 2.2e+78) {
		tmp = 2.0 / (((Math.pow(k_m, 2.0) * t_2) / (l * Math.cos(k_m))) / l);
	} else {
		tmp = 2.0 / ((t_2 / Math.pow((l / k_m), 2.0)) / Math.cos(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):
	t_2 = t_m * math.pow(math.sin(k_m), 2.0)
	tmp = 0
	if k_m <= 8.2e-16:
		tmp = math.pow((math.pow(t_m, -0.5) * (l / (k_m * t_m))), 2.0)
	elif k_m <= 2.2e+78:
		tmp = 2.0 / (((math.pow(k_m, 2.0) * t_2) / (l * math.cos(k_m))) / l)
	else:
		tmp = 2.0 / ((t_2 / math.pow((l / k_m), 2.0)) / math.cos(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)
	t_2 = Float64(t_m * (sin(k_m) ^ 2.0))
	tmp = 0.0
	if (k_m <= 8.2e-16)
		tmp = Float64((t_m ^ -0.5) * Float64(l / Float64(k_m * t_m))) ^ 2.0;
	elseif (k_m <= 2.2e+78)
		tmp = Float64(2.0 / Float64(Float64(Float64((k_m ^ 2.0) * t_2) / Float64(l * cos(k_m))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(t_2 / (Float64(l / k_m) ^ 2.0)) / cos(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)
	t_2 = t_m * (sin(k_m) ^ 2.0);
	tmp = 0.0;
	if (k_m <= 8.2e-16)
		tmp = ((t_m ^ -0.5) * (l / (k_m * t_m))) ^ 2.0;
	elseif (k_m <= 2.2e+78)
		tmp = 2.0 / ((((k_m ^ 2.0) * t_2) / (l * cos(k_m))) / l);
	else
		tmp = 2.0 / ((t_2 / ((l / k_m) ^ 2.0)) / cos(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_] := Block[{t$95$2 = N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 8.2e-16], N[Power[N[(N[Power[t$95$m, -0.5], $MachinePrecision] * N[(l / N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 2.2e+78], N[(2.0 / N[(N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * t$95$2), $MachinePrecision] / N[(l * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$2 / N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 8.20000000000000012e-16

   1. Initial program 51.5%

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

   3. Simplified 53.0%

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

   4. Taylor expanded in k around 0 48.0%

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

      1. add-sqr-sqrt 33.1%

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

      2. pow2 33.1%

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

      3. sqrt-div 22.0%

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

      4. unpow2 22.0%

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

      5. sqrt-prod 11.5%

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

      6. add-sqr-sqrt 24.5%

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

      7. sqrt-prod 24.5%

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

      8. unpow2 24.5%

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

      9. sqrt-prod 7.7%

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

      10. add-sqr-sqrt 28.9%

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

      11. sqrt-pow1 32.7%

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

      12. metadata-eval 32.7%

          \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

   6. Applied egg-rr 32.7%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 32.4%

         \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right) \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}}^{2} \]

      2. pow2 32.4%

         \[\leadsto {\left(\color{blue}{{\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      3. associate-/r* 32.5%

         \[\leadsto {\left({\left(\sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      4. cbrt-div 32.5%

         \[\leadsto {\left({\color{blue}{\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      5. metadata-eval 32.5%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{\color{blue}{\left(\frac{3}{2}\right)}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      6. sqrt-pow1 27.0%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{{t}^{3}}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      7. unpow3 27.0%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\sqrt{\color{blue}{\left(t \cdot t\right) \cdot t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      8. sqrt-prod 29.6%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{t \cdot t} \cdot \sqrt{t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      9. sqrt-unprod 32.5%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      10. add-cbrt-cube 32.5%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\color{blue}{\sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      11. associate-/r* 32.9%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \]

      12. cbrt-div 32.9%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}}\right)}^{2} \]

      13. metadata-eval 32.9%

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

      14. sqrt-pow1 27.0%

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

      15. unpow3 27.0%

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

      16. sqrt-prod 30.0%

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

      17. sqrt-unprod 32.9%

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

   8. Applied egg-rr 34.4%

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

      1. pow-plus 34.4%

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

      2. metadata-eval 34.4%

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

      3. cube-div 32.9%

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

      4. rem-cube-cbrt 33.1%

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

   10. Simplified 33.1%

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

       1. *-un-lft-identity 33.1%

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

       2. cube-mult 33.1%

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

       3. add-sqr-sqrt 33.2%

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

       4. times-frac 34.7%

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

       5. pow1/2 34.7%

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

       6. pow-flip 34.7%

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

       7. metadata-eval 34.7%

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

   12. Applied egg-rr 34.7%

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

       1. associate-/l/ 35.2%

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

   14. Simplified 35.2%

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

   if 8.20000000000000012e-16 < k < 2.20000000000000014e78

   1. Initial program 35.3%

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

      1. associate-*l* 35.3%

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

      2. *-commutative 35.3%

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

      3. *-commutative 35.3%

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

      4. associate-/r* 41.6%

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

      5. distribute-rgt-in 41.6%

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

      6. unpow2 41.6%

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

      7. times-frac 41.6%

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

      8. sqr-neg 41.6%

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

      9. times-frac 41.6%

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

      10. unpow2 41.6%

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

      11. distribute-rgt-in 41.6%

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

      12. +-commutative 41.6%

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

   3. Simplified 41.6%

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

      1. associate-*l/ 41.6%

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

      2. associate-*l/ 41.6%

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

      3. clear-num 41.6%

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

      4. associate-*l/ 41.6%

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

      5. *-un-lft-identity 41.6%

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

   5. Applied egg-rr 41.6%

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

   6. Taylor expanded in k around inf 94.0%

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

   if 2.20000000000000014e78 < k

   1. Initial program 54.9%

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

      1. associate-*l* 54.9%

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

      2. *-commutative 54.9%

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

      3. *-commutative 54.9%

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

      4. associate-/r* 59.4%

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

      5. distribute-rgt-in 59.4%

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

      6. unpow2 59.4%

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

      7. times-frac 43.6%

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

      8. sqr-neg 43.6%

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

      9. times-frac 59.4%

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

      10. unpow2 59.4%

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

      11. distribute-rgt-in 59.4%

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

      12. +-commutative 59.4%

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

   3. Simplified 59.4%

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

      1. associate-/l/ 54.9%

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

      2. unpow3 54.9%

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

      3. times-frac 72.7%

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

      4. pow2 72.7%

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

   5. Applied egg-rr 72.7%

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

   6. Taylor expanded in t around 0 73.2%

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

      1. associate-/r* 73.2%

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

      2. *-commutative 73.2%

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

      3. associate-/l* 71.4%

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

      4. unpow2 71.4%

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

      5. associate-/r* 73.9%

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

      6. unpow2 73.9%

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

      7. associate-*l/ 84.6%

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

      8. associate-*r/ 86.9%

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

      9. unpow2 86.9%

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

   8. Simplified 86.9%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{-16}:\\ \;\;\;\;{\left({t}^{-0.5} \cdot \frac{\ell}{k \cdot t}\right)}^{2}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{+78}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{{\left(\frac{\ell}{k}\right)}^{2}}}{\cos k}}\\ \end{array} \]

Alternative 2: 88.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.5 \cdot 10^{-15}:\\ \;\;\;\;{\left({t_m}^{-0.5} \cdot \frac{\ell}{k_m \cdot t_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t_m \cdot {\sin k_m}^{2}}{{\left(\frac{\ell}{k_m}\right)}^{2}}}{\cos k_m}}\\ \end{array} \end{array} \]

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 1.5e-15)
    (pow (* (pow t_m -0.5) (/ l (* k_m t_m))) 2.0)
    (/
     2.0
     (/ (/ (* t_m (pow (sin k_m) 2.0)) (pow (/ l k_m) 2.0)) (cos 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 <= 1.5e-15) {
		tmp = pow((pow(t_m, -0.5) * (l / (k_m * t_m))), 2.0);
	} else {
		tmp = 2.0 / (((t_m * pow(sin(k_m), 2.0)) / pow((l / k_m), 2.0)) / cos(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 <= 1.5d-15) then
        tmp = ((t_m ** (-0.5d0)) * (l / (k_m * t_m))) ** 2.0d0
    else
        tmp = 2.0d0 / (((t_m * (sin(k_m) ** 2.0d0)) / ((l / k_m) ** 2.0d0)) / cos(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 <= 1.5e-15) {
		tmp = Math.pow((Math.pow(t_m, -0.5) * (l / (k_m * t_m))), 2.0);
	} else {
		tmp = 2.0 / (((t_m * Math.pow(Math.sin(k_m), 2.0)) / Math.pow((l / k_m), 2.0)) / Math.cos(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 <= 1.5e-15:
		tmp = math.pow((math.pow(t_m, -0.5) * (l / (k_m * t_m))), 2.0)
	else:
		tmp = 2.0 / (((t_m * math.pow(math.sin(k_m), 2.0)) / math.pow((l / k_m), 2.0)) / math.cos(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 <= 1.5e-15)
		tmp = Float64((t_m ^ -0.5) * Float64(l / Float64(k_m * t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (sin(k_m) ^ 2.0)) / (Float64(l / k_m) ^ 2.0)) / cos(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 <= 1.5e-15)
		tmp = ((t_m ^ -0.5) * (l / (k_m * t_m))) ^ 2.0;
	else
		tmp = 2.0 / (((t_m * (sin(k_m) ^ 2.0)) / ((l / k_m) ^ 2.0)) / cos(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, 1.5e-15], N[Power[N[(N[Power[t$95$m, -0.5], $MachinePrecision] * N[(l / N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.5e-15

   1. Initial program 51.5%

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

   3. Simplified 53.0%

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

   4. Taylor expanded in k around 0 48.0%

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

      1. add-sqr-sqrt 33.1%

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

      2. pow2 33.1%

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

      3. sqrt-div 22.0%

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

      4. unpow2 22.0%

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

      5. sqrt-prod 11.5%

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

      6. add-sqr-sqrt 24.5%

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

      7. sqrt-prod 24.5%

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

      8. unpow2 24.5%

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

      9. sqrt-prod 7.7%

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

      10. add-sqr-sqrt 28.9%

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

      11. sqrt-pow1 32.7%

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

      12. metadata-eval 32.7%

          \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

   6. Applied egg-rr 32.7%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 32.4%

         \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right) \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}}^{2} \]

      2. pow2 32.4%

         \[\leadsto {\left(\color{blue}{{\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      3. associate-/r* 32.5%

         \[\leadsto {\left({\left(\sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      4. cbrt-div 32.5%

         \[\leadsto {\left({\color{blue}{\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      5. metadata-eval 32.5%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{\color{blue}{\left(\frac{3}{2}\right)}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      6. sqrt-pow1 27.0%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{{t}^{3}}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      7. unpow3 27.0%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\sqrt{\color{blue}{\left(t \cdot t\right) \cdot t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      8. sqrt-prod 29.6%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{t \cdot t} \cdot \sqrt{t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      9. sqrt-unprod 32.5%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      10. add-cbrt-cube 32.5%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\color{blue}{\sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      11. associate-/r* 32.9%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \]

      12. cbrt-div 32.9%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}}\right)}^{2} \]

      13. metadata-eval 32.9%

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

      14. sqrt-pow1 27.0%

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

      15. unpow3 27.0%

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

      16. sqrt-prod 30.0%

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

      17. sqrt-unprod 32.9%

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

   8. Applied egg-rr 34.4%

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

      1. pow-plus 34.4%

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

      2. metadata-eval 34.4%

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

      3. cube-div 32.9%

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

      4. rem-cube-cbrt 33.1%

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

   10. Simplified 33.1%

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

       1. *-un-lft-identity 33.1%

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

       2. cube-mult 33.1%

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

       3. add-sqr-sqrt 33.2%

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

       4. times-frac 34.7%

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

       5. pow1/2 34.7%

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

       6. pow-flip 34.7%

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

       7. metadata-eval 34.7%

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

   12. Applied egg-rr 34.7%

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

       1. associate-/l/ 35.2%

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

   14. Simplified 35.2%

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

   if 1.5e-15 < k

   1. Initial program 49.4%

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

      1. associate-*l* 49.4%

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

      2. *-commutative 49.4%

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

      3. *-commutative 49.4%

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

      4. associate-/r* 54.5%

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

      5. distribute-rgt-in 54.5%

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

      6. unpow2 54.5%

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

      7. times-frac 43.1%

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

      8. sqr-neg 43.1%

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

      9. times-frac 54.5%

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

      10. unpow2 54.5%

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

      11. distribute-rgt-in 54.5%

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

      12. +-commutative 54.5%

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

   3. Simplified 54.5%

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

      1. associate-/l/ 49.4%

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

      2. unpow3 49.4%

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

      3. times-frac 67.4%

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

      4. pow2 67.4%

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

   5. Applied egg-rr 67.4%

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

   6. Taylor expanded in t around 0 77.3%

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

      1. associate-/r* 77.4%

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

      2. *-commutative 77.4%

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

      3. associate-/l* 74.6%

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

      4. unpow2 74.6%

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

      5. associate-/r* 76.3%

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

      6. unpow2 76.3%

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

      7. associate-*l/ 84.1%

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

      8. associate-*r/ 85.7%

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

      9. unpow2 85.7%

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

   8. Simplified 85.7%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-15}:\\ \;\;\;\;{\left({t}^{-0.5} \cdot \frac{\ell}{k \cdot t}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{{\left(\frac{\ell}{k}\right)}^{2}}}{\cos k}}\\ \end{array} \]

Alternative 3: 79.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.6 \cdot 10^{-24}:\\ \;\;\;\;{\left({t_m}^{-0.5} \cdot \frac{\ell}{k_m \cdot t_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k_m \cdot \left(\frac{t_m}{\ell} \cdot \left(t_m \cdot \frac{t_m}{\ell}\right)\right)\right) \cdot \left(\tan k_m \cdot \left(2 + {\left(\frac{k_m}{t_m}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.6e-24)
    (pow (* (pow t_m -0.5) (/ l (* k_m t_m))) 2.0)
    (/
     2.0
     (*
      (* (sin k_m) (* (/ t_m l) (* t_m (/ t_m l))))
      (* (tan k_m) (+ 2.0 (pow (/ k_m 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 <= 1.6e-24) {
		tmp = pow((pow(t_m, -0.5) * (l / (k_m * t_m))), 2.0);
	} else {
		tmp = 2.0 / ((sin(k_m) * ((t_m / l) * (t_m * (t_m / l)))) * (tan(k_m) * (2.0 + pow((k_m / 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 <= 1.6d-24) then
        tmp = ((t_m ** (-0.5d0)) * (l / (k_m * t_m))) ** 2.0d0
    else
        tmp = 2.0d0 / ((sin(k_m) * ((t_m / l) * (t_m * (t_m / l)))) * (tan(k_m) * (2.0d0 + ((k_m / 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 <= 1.6e-24) {
		tmp = Math.pow((Math.pow(t_m, -0.5) * (l / (k_m * t_m))), 2.0);
	} else {
		tmp = 2.0 / ((Math.sin(k_m) * ((t_m / l) * (t_m * (t_m / l)))) * (Math.tan(k_m) * (2.0 + Math.pow((k_m / 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 <= 1.6e-24:
		tmp = math.pow((math.pow(t_m, -0.5) * (l / (k_m * t_m))), 2.0)
	else:
		tmp = 2.0 / ((math.sin(k_m) * ((t_m / l) * (t_m * (t_m / l)))) * (math.tan(k_m) * (2.0 + math.pow((k_m / 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 <= 1.6e-24)
		tmp = Float64((t_m ^ -0.5) * Float64(l / Float64(k_m * t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) * Float64(Float64(t_m / l) * Float64(t_m * Float64(t_m / l)))) * Float64(tan(k_m) * Float64(2.0 + (Float64(k_m / 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 <= 1.6e-24)
		tmp = ((t_m ^ -0.5) * (l / (k_m * t_m))) ^ 2.0;
	else
		tmp = 2.0 / ((sin(k_m) * ((t_m / l) * (t_m * (t_m / l)))) * (tan(k_m) * (2.0 + ((k_m / 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, 1.6e-24], N[Power[N[(N[Power[t$95$m, -0.5], $MachinePrecision] * N[(l / N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.60000000000000006e-24

   1. Initial program 51.1%

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

   3. Simplified 52.6%

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

   4. Taylor expanded in k around 0 47.4%

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

      1. add-sqr-sqrt 32.2%

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

      2. pow2 32.2%

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

      3. sqrt-div 22.0%

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

      4. unpow2 22.0%

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

      5. sqrt-prod 11.2%

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

      6. add-sqr-sqrt 24.5%

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

      7. sqrt-prod 24.5%

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

      8. unpow2 24.5%

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

      9. sqrt-prod 7.4%

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

      10. add-sqr-sqrt 29.0%

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

      11. sqrt-pow1 32.4%

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

      12. metadata-eval 32.4%

          \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

   6. Applied egg-rr 32.4%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 32.1%

         \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right) \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}}^{2} \]

      2. pow2 32.1%

         \[\leadsto {\left(\color{blue}{{\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      3. associate-/r* 32.1%

         \[\leadsto {\left({\left(\sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      4. cbrt-div 32.1%

         \[\leadsto {\left({\color{blue}{\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      5. metadata-eval 32.1%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{\color{blue}{\left(\frac{3}{2}\right)}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      6. sqrt-pow1 27.1%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{{t}^{3}}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      7. unpow3 27.1%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\sqrt{\color{blue}{\left(t \cdot t\right) \cdot t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      8. sqrt-prod 29.7%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{t \cdot t} \cdot \sqrt{t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      9. sqrt-unprod 32.1%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      10. add-cbrt-cube 32.1%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\color{blue}{\sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      11. associate-/r* 32.6%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \]

      12. cbrt-div 32.6%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}}\right)}^{2} \]

      13. metadata-eval 32.6%

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

      14. sqrt-pow1 27.1%

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

      15. unpow3 27.1%

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

      16. sqrt-prod 30.1%

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

      17. sqrt-unprod 32.6%

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

   8. Applied egg-rr 34.1%

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

      1. pow-plus 34.1%

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

      2. metadata-eval 34.1%

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

      3. cube-div 32.6%

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

      4. rem-cube-cbrt 32.7%

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

   10. Simplified 32.7%

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

       1. *-un-lft-identity 32.7%

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

       2. cube-mult 32.7%

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

       3. add-sqr-sqrt 32.8%

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

       4. times-frac 34.4%

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

       5. pow1/2 34.4%

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

       6. pow-flip 34.4%

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

       7. metadata-eval 34.4%

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

   12. Applied egg-rr 34.4%

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

       1. associate-/l/ 34.9%

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

   14. Simplified 34.9%

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

   if 1.60000000000000006e-24 < k

   1. Initial program 51.0%

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

      1. associate-*l* 51.0%

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

      2. *-commutative 51.0%

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

      3. *-commutative 51.0%

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

      4. associate-/r* 55.7%

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

      5. distribute-rgt-in 55.8%

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

      6. unpow2 55.8%

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

      7. times-frac 45.1%

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

      8. sqr-neg 45.1%

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

      9. times-frac 55.8%

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

      10. unpow2 55.8%

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

      11. distribute-rgt-in 55.7%

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

      12. +-commutative 55.7%

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

   3. Simplified 55.8%

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

      1. associate-/l/ 51.0%

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

      2. unpow3 51.0%

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

      3. times-frac 67.9%

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

      4. pow2 67.9%

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

   5. Applied egg-rr 67.9%

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

      1. unpow2 67.9%

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

      2. *-un-lft-identity 67.9%

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

      3. times-frac 73.7%

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

   7. Applied egg-rr 73.7%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-24}:\\ \;\;\;\;{\left({t}^{-0.5} \cdot \frac{\ell}{k \cdot t}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]

Alternative 4: 68.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 28:\\ \;\;\;\;{t_m}^{-1.5} \cdot \frac{{\left(\frac{\ell}{k_m}\right)}^{2}}{{t_m}^{1.5}}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+267}:\\ \;\;\;\;{\left({t_m}^{-0.5} \cdot \frac{\ell}{k_m \cdot t_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k_m \cdot \left(\frac{t_m}{\ell} \cdot \frac{{t_m}^{2}}{\ell}\right)\right) \cdot \left(k_m \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= l 28.0)
    (* (pow t_m -1.5) (/ (pow (/ l k_m) 2.0) (pow t_m 1.5)))
    (if (<= l 1.3e+267)
      (pow (* (pow t_m -0.5) (/ l (* k_m t_m))) 2.0)
      (/
       2.0
       (* (* (sin k_m) (* (/ t_m l) (/ (pow t_m 2.0) l))) (* k_m 2.0)))))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 28.0) {
		tmp = pow(t_m, -1.5) * (pow((l / k_m), 2.0) / pow(t_m, 1.5));
	} else if (l <= 1.3e+267) {
		tmp = pow((pow(t_m, -0.5) * (l / (k_m * t_m))), 2.0);
	} else {
		tmp = 2.0 / ((sin(k_m) * ((t_m / l) * (pow(t_m, 2.0) / l))) * (k_m * 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 28.0d0) then
        tmp = (t_m ** (-1.5d0)) * (((l / k_m) ** 2.0d0) / (t_m ** 1.5d0))
    else if (l <= 1.3d+267) then
        tmp = ((t_m ** (-0.5d0)) * (l / (k_m * t_m))) ** 2.0d0
    else
        tmp = 2.0d0 / ((sin(k_m) * ((t_m / l) * ((t_m ** 2.0d0) / l))) * (k_m * 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 28.0) {
		tmp = Math.pow(t_m, -1.5) * (Math.pow((l / k_m), 2.0) / Math.pow(t_m, 1.5));
	} else if (l <= 1.3e+267) {
		tmp = Math.pow((Math.pow(t_m, -0.5) * (l / (k_m * t_m))), 2.0);
	} else {
		tmp = 2.0 / ((Math.sin(k_m) * ((t_m / l) * (Math.pow(t_m, 2.0) / l))) * (k_m * 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if l <= 28.0:
		tmp = math.pow(t_m, -1.5) * (math.pow((l / k_m), 2.0) / math.pow(t_m, 1.5))
	elif l <= 1.3e+267:
		tmp = math.pow((math.pow(t_m, -0.5) * (l / (k_m * t_m))), 2.0)
	else:
		tmp = 2.0 / ((math.sin(k_m) * ((t_m / l) * (math.pow(t_m, 2.0) / l))) * (k_m * 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (l <= 28.0)
		tmp = Float64((t_m ^ -1.5) * Float64((Float64(l / k_m) ^ 2.0) / (t_m ^ 1.5)));
	elseif (l <= 1.3e+267)
		tmp = Float64((t_m ^ -0.5) * Float64(l / Float64(k_m * t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) * Float64(Float64(t_m / l) * Float64((t_m ^ 2.0) / l))) * Float64(k_m * 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (l <= 28.0)
		tmp = (t_m ^ -1.5) * (((l / k_m) ^ 2.0) / (t_m ^ 1.5));
	elseif (l <= 1.3e+267)
		tmp = ((t_m ^ -0.5) * (l / (k_m * t_m))) ^ 2.0;
	else
		tmp = 2.0 / ((sin(k_m) * ((t_m / l) * ((t_m ^ 2.0) / l))) * (k_m * 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[l, 28.0], N[(N[Power[t$95$m, -1.5], $MachinePrecision] * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.3e+267], N[Power[N[(N[Power[t$95$m, -0.5], $MachinePrecision] * N[(l / N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if l < 28

   1. Initial program 52.4%

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

   3. Simplified 53.9%

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

   4. Taylor expanded in k around 0 49.1%

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

      1. add-sqr-sqrt 39.9%

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

      2. pow2 39.9%

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

      3. sqrt-div 22.6%

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

      4. unpow2 22.6%

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

      5. sqrt-prod 6.8%

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

      6. add-sqr-sqrt 24.3%

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

      7. sqrt-prod 24.3%

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

      8. unpow2 24.3%

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

      9. sqrt-prod 13.7%

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

      10. add-sqr-sqrt 28.9%

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

      11. sqrt-pow1 30.2%

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

      12. metadata-eval 30.2%

          \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

   6. Applied egg-rr 30.2%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 30.2%

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{1.5}} \cdot \frac{\ell}{k \cdot {t}^{1.5}}} \]

      2. associate-/r* 30.6%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}} \cdot \frac{\ell}{k \cdot {t}^{1.5}} \]

      3. associate-/r* 31.1%

         \[\leadsto \frac{\frac{\ell}{k}}{{t}^{1.5}} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}} \]

      4. frac-times 26.9%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{1.5} \cdot {t}^{1.5}}} \]

      5. pow-prod-up 63.5%

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{{t}^{\left(1.5 + 1.5\right)}}} \]

      6. metadata-eval 63.5%

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

   8. Applied egg-rr 63.5%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 63.5%

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

      2. add-sqr-sqrt 26.9%

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

      3. pow-prod-down 26.9%

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

      4. times-frac 31.6%

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

      5. sqrt-pow2 31.7%

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

      6. pow-flip 31.7%

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

      7. metadata-eval 31.7%

         \[\leadsto {t}^{\left(-\color{blue}{1.5}\right)} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\left(\sqrt{t}\right)}^{3}} \]

      8. metadata-eval 31.7%

         \[\leadsto {t}^{\color{blue}{-1.5}} \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\left(\sqrt{t}\right)}^{3}} \]

      9. pow2 31.7%

         \[\leadsto {t}^{-1.5} \cdot \frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{\left(\sqrt{t}\right)}^{3}} \]

      10. sqrt-pow2 31.7%

          \[\leadsto {t}^{-1.5} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}} \]

      11. metadata-eval 31.7%

          \[\leadsto {t}^{-1.5} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t}^{\color{blue}{1.5}}} \]

   10. Applied egg-rr 31.7%

       \[\leadsto \color{blue}{{t}^{-1.5} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t}^{1.5}}} \]

   if 28 < l < 1.30000000000000001e267

   1. Initial program 47.9%

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

   3. Simplified 47.9%

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

   4. Taylor expanded in k around 0 44.8%

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

      1. add-sqr-sqrt 29.5%

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

      2. pow2 29.5%

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

      3. sqrt-div 25.7%

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

      4. unpow2 25.7%

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

      5. sqrt-prod 28.7%

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

      6. add-sqr-sqrt 28.7%

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

      7. sqrt-prod 28.7%

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

      8. unpow2 28.7%

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

      9. sqrt-prod 7.2%

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

      10. add-sqr-sqrt 27.4%

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

      11. sqrt-pow1 30.6%

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

      12. metadata-eval 30.6%

          \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

   6. Applied egg-rr 30.6%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 30.6%

         \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right) \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}}^{2} \]

      2. pow2 30.6%

         \[\leadsto {\left(\color{blue}{{\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      3. associate-/r* 29.0%

         \[\leadsto {\left({\left(\sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      4. cbrt-div 29.0%

         \[\leadsto {\left({\color{blue}{\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      5. metadata-eval 29.0%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{\color{blue}{\left(\frac{3}{2}\right)}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      6. sqrt-pow1 27.2%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{{t}^{3}}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      7. unpow3 27.2%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\sqrt{\color{blue}{\left(t \cdot t\right) \cdot t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      8. sqrt-prod 27.4%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{t \cdot t} \cdot \sqrt{t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      9. sqrt-unprod 29.0%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      10. add-cbrt-cube 29.0%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\color{blue}{\sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      11. associate-/r* 29.0%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \]

      12. cbrt-div 29.0%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}}\right)}^{2} \]

      13. metadata-eval 29.0%

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

      14. sqrt-pow1 27.2%

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

      15. unpow3 27.2%

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

      16. sqrt-prod 27.4%

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

      17. sqrt-unprod 29.0%

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

   8. Applied egg-rr 29.1%

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

      1. pow-plus 29.1%

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

      2. metadata-eval 29.1%

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

      3. cube-div 29.0%

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

      4. rem-cube-cbrt 29.1%

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

   10. Simplified 29.1%

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

       1. *-un-lft-identity 29.1%

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

       2. cube-mult 29.1%

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

       3. add-sqr-sqrt 29.1%

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

       4. times-frac 29.1%

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

       5. pow1/2 29.1%

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

       6. pow-flip 29.1%

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

       7. metadata-eval 29.1%

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

   12. Applied egg-rr 29.1%

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

       1. associate-/l/ 32.2%

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

   14. Simplified 32.2%

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

   if 1.30000000000000001e267 < l

   1. Initial program 45.5%

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

      1. associate-*l* 45.5%

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

      2. *-commutative 45.5%

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

      3. *-commutative 45.5%

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

      4. associate-/r* 46.5%

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

      5. distribute-rgt-in 46.5%

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

      6. unpow2 46.5%

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

      7. times-frac 46.5%

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

      8. sqr-neg 46.5%

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

      9. times-frac 46.5%

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

      10. unpow2 46.5%

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

      11. distribute-rgt-in 46.5%

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

      12. +-commutative 46.5%

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

   3. Simplified 46.5%

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

      1. associate-/l/ 45.5%

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

      2. unpow3 45.5%

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

      3. times-frac 73.2%

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

      4. pow2 73.2%

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

   5. Applied egg-rr 73.2%

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

   6. Taylor expanded in k around 0 74.2%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 28:\\ \;\;\;\;{t}^{-1.5} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{{t}^{1.5}}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+267}:\\ \;\;\;\;{\left({t}^{-0.5} \cdot \frac{\ell}{k \cdot t}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{{t}^{2}}{\ell}\right)\right) \cdot \left(k \cdot 2\right)}\\ \end{array} \]

Alternative 5: 73.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;{\left({t_m}^{-0.5} \cdot \frac{\frac{\ell}{k_m}}{t_m}\right)}^{2}\\ \mathbf{elif}\;t_m \leq 4.8 \cdot 10^{+115}:\\ \;\;\;\;\frac{1}{{t_m}^{2}} \cdot \frac{{\left(\frac{\ell}{k_m}\right)}^{2}}{t_m}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\frac{\ell}{t_m}}{k_m}}{\sqrt{t_m}}\right)}^{2}\\ \end{array} \end{array} \]

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 (<= t_m 7.5e-155)
    (pow (* (pow t_m -0.5) (/ (/ l k_m) t_m)) 2.0)
    (if (<= t_m 4.8e+115)
      (* (/ 1.0 (pow t_m 2.0)) (/ (pow (/ l k_m) 2.0) t_m))
      (pow (/ (/ (/ l t_m) 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) {
	double tmp;
	if (t_m <= 7.5e-155) {
		tmp = pow((pow(t_m, -0.5) * ((l / k_m) / t_m)), 2.0);
	} else if (t_m <= 4.8e+115) {
		tmp = (1.0 / pow(t_m, 2.0)) * (pow((l / k_m), 2.0) / t_m);
	} else {
		tmp = pow((((l / t_m) / k_m) / 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 (t_m <= 7.5d-155) then
        tmp = ((t_m ** (-0.5d0)) * ((l / k_m) / t_m)) ** 2.0d0
    else if (t_m <= 4.8d+115) then
        tmp = (1.0d0 / (t_m ** 2.0d0)) * (((l / k_m) ** 2.0d0) / t_m)
    else
        tmp = (((l / t_m) / k_m) / 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 (t_m <= 7.5e-155) {
		tmp = Math.pow((Math.pow(t_m, -0.5) * ((l / k_m) / t_m)), 2.0);
	} else if (t_m <= 4.8e+115) {
		tmp = (1.0 / Math.pow(t_m, 2.0)) * (Math.pow((l / k_m), 2.0) / t_m);
	} else {
		tmp = Math.pow((((l / t_m) / k_m) / 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 t_m <= 7.5e-155:
		tmp = math.pow((math.pow(t_m, -0.5) * ((l / k_m) / t_m)), 2.0)
	elif t_m <= 4.8e+115:
		tmp = (1.0 / math.pow(t_m, 2.0)) * (math.pow((l / k_m), 2.0) / t_m)
	else:
		tmp = math.pow((((l / t_m) / k_m) / 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 (t_m <= 7.5e-155)
		tmp = Float64((t_m ^ -0.5) * Float64(Float64(l / k_m) / t_m)) ^ 2.0;
	elseif (t_m <= 4.8e+115)
		tmp = Float64(Float64(1.0 / (t_m ^ 2.0)) * Float64((Float64(l / k_m) ^ 2.0) / t_m));
	else
		tmp = Float64(Float64(Float64(l / t_m) / k_m) / 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 (t_m <= 7.5e-155)
		tmp = ((t_m ^ -0.5) * ((l / k_m) / t_m)) ^ 2.0;
	elseif (t_m <= 4.8e+115)
		tmp = (1.0 / (t_m ^ 2.0)) * (((l / k_m) ^ 2.0) / t_m);
	else
		tmp = (((l / t_m) / k_m) / 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[t$95$m, 7.5e-155], N[Power[N[(N[Power[t$95$m, -0.5], $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 4.8e+115], N[(N[(1.0 / N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(l / t$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.5000000000000006e-155

   1. Initial program 50.5%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in k around 0 50.2%

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

      1. add-sqr-sqrt 31.6%

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

      2. pow2 31.6%

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

      3. sqrt-div 9.7%

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

      4. unpow2 9.7%

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

      5. sqrt-prod 2.6%

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

      6. add-sqr-sqrt 10.5%

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

      7. sqrt-prod 10.4%

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

      8. unpow2 10.4%

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

      9. sqrt-prod 4.0%

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

      10. add-sqr-sqrt 10.7%

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

      11. sqrt-pow1 7.2%

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

      12. metadata-eval 7.2%

          \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

   6. Applied egg-rr 7.2%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 7.2%

         \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right) \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}}^{2} \]

      2. pow2 7.2%

         \[\leadsto {\left(\color{blue}{{\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      3. associate-/r* 7.8%

         \[\leadsto {\left({\left(\sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      4. cbrt-div 7.8%

         \[\leadsto {\left({\color{blue}{\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      5. metadata-eval 7.8%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{\color{blue}{\left(\frac{3}{2}\right)}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      6. sqrt-pow1 7.2%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{{t}^{3}}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      7. unpow3 7.2%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\sqrt{\color{blue}{\left(t \cdot t\right) \cdot t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      8. sqrt-prod 7.2%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{t \cdot t} \cdot \sqrt{t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      9. sqrt-unprod 7.8%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      10. add-cbrt-cube 7.8%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\color{blue}{\sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      11. associate-/r* 7.8%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \]

      12. cbrt-div 7.8%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}}\right)}^{2} \]

      13. metadata-eval 7.8%

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

      14. sqrt-pow1 7.2%

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

      15. unpow3 7.2%

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

      16. sqrt-prod 7.2%

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

      17. sqrt-unprod 7.8%

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

   8. Applied egg-rr 8.4%

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

      1. pow-plus 8.4%

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

      2. metadata-eval 8.4%

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

      3. cube-div 7.8%

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

      4. rem-cube-cbrt 7.8%

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

   10. Simplified 7.8%

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

       1. *-un-lft-identity 7.8%

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

       2. cube-mult 7.8%

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

       3. add-sqr-sqrt 7.8%

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

       4. times-frac 8.4%

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

       5. pow1/2 8.4%

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

       6. pow-flip 8.4%

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

       7. metadata-eval 8.4%

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

   12. Applied egg-rr 8.4%

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

   if 7.5000000000000006e-155 < t < 4.8000000000000001e115

   1. Initial program 56.7%

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

   3. Simplified 60.2%

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

   4. Taylor expanded in k around 0 47.8%

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

      1. add-sqr-sqrt 47.8%

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

      2. pow2 47.8%

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

      3. sqrt-div 47.7%

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

      4. unpow2 47.7%

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

      5. sqrt-prod 28.6%

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

      6. add-sqr-sqrt 51.6%

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

      7. sqrt-prod 51.6%

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

      8. unpow2 51.6%

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

      9. sqrt-prod 20.1%

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

      10. add-sqr-sqrt 58.8%

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

      11. sqrt-pow1 60.7%

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

      12. metadata-eval 60.7%

          \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

   6. Applied egg-rr 60.7%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 60.7%

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{1.5}} \cdot \frac{\ell}{k \cdot {t}^{1.5}}} \]

      2. associate-/r* 62.2%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}} \cdot \frac{\ell}{k \cdot {t}^{1.5}} \]

      3. associate-/r* 63.8%

         \[\leadsto \frac{\frac{\ell}{k}}{{t}^{1.5}} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}} \]

      4. frac-times 63.9%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{1.5} \cdot {t}^{1.5}}} \]

      5. pow-prod-up 64.0%

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{{t}^{\left(1.5 + 1.5\right)}}} \]

      6. metadata-eval 64.0%

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

   8. Applied egg-rr 64.0%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 64.0%

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

      2. unpow3 63.9%

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

      3. unpow2 63.9%

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

      4. times-frac 73.8%

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

      5. pow2 73.8%

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

   10. Applied egg-rr 73.8%

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

   if 4.8000000000000001e115 < t

   1. Initial program 44.6%

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

   3. Simplified 44.6%

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

   4. Taylor expanded in k around 0 36.7%

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

      1. add-sqr-sqrt 36.7%

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

      2. pow2 36.7%

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

      3. sqrt-div 36.7%

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

      4. unpow2 36.7%

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

      5. sqrt-prod 23.9%

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

      6. add-sqr-sqrt 40.3%

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

      7. sqrt-prod 40.3%

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

      8. unpow2 40.3%

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

      9. sqrt-prod 28.9%

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

      10. add-sqr-sqrt 48.4%

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

      11. sqrt-pow1 72.9%

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

      12. metadata-eval 72.9%

          \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

   6. Applied egg-rr 72.9%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 72.3%

         \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right) \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}}^{2} \]

      2. pow2 72.3%

         \[\leadsto {\left(\color{blue}{{\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      3. associate-/r* 67.6%

         \[\leadsto {\left({\left(\sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      4. cbrt-div 67.7%

         \[\leadsto {\left({\color{blue}{\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      5. metadata-eval 67.7%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{\color{blue}{\left(\frac{3}{2}\right)}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      6. sqrt-pow1 48.1%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{{t}^{3}}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      7. unpow3 48.1%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\sqrt{\color{blue}{\left(t \cdot t\right) \cdot t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      8. sqrt-prod 55.7%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{t \cdot t} \cdot \sqrt{t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      9. sqrt-unprod 67.6%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      10. add-cbrt-cube 67.6%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\color{blue}{\sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      11. associate-/r* 67.6%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \]

      12. cbrt-div 67.5%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}}\right)}^{2} \]

      13. metadata-eval 67.5%

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

      14. sqrt-pow1 48.1%

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

      15. unpow3 48.1%

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

      16. sqrt-prod 55.5%

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

      17. sqrt-unprod 67.5%

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

   8. Applied egg-rr 72.5%

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

      1. pow-plus 72.5%

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

      2. metadata-eval 72.5%

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

      3. cube-div 67.4%

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

      4. rem-cube-cbrt 67.7%

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

   10. Simplified 67.7%

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

       1. rem-cube-cbrt 67.4%

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

       2. cube-div 72.5%

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

       3. cube-mult 72.5%

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

       4. frac-times 72.5%

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

       5. pow2 72.5%

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

       6. add-sqr-sqrt 72.5%

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

   12. Applied egg-rr 72.5%

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

       1. associate-*l/ 72.4%

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

       2. associate-*r/ 72.5%

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

       3. unpow2 72.5%

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

       4. cube-mult 72.4%

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

       5. rem-cube-cbrt 73.1%

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

       6. associate-/l/ 82.6%

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

       7. associate-/r* 82.6%

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

   14. Simplified 82.6%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;{\left({t}^{-0.5} \cdot \frac{\frac{\ell}{k}}{t}\right)}^{2}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{+115}:\\ \;\;\;\;\frac{1}{{t}^{2}} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\frac{\ell}{t}}{k}}{\sqrt{t}}\right)}^{2}\\ \end{array} \]

Alternative 6: 72.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k_m \leq 1.1 \cdot 10^{-53}:\\ \;\;\;\;{\left({t_m}^{-0.5} \cdot \frac{\ell}{k_m \cdot t_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k_m} \cdot \left(\ell \cdot {t_m}^{-1.5}\right)}{k_m}}{{t_m}^{1.5}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.1e-53)
    (pow (* (pow t_m -0.5) (/ l (* k_m t_m))) 2.0)
    (/ (/ (* (/ l k_m) (* l (pow t_m -1.5))) k_m) (pow t_m 1.5)))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.1e-53) {
		tmp = pow((pow(t_m, -0.5) * (l / (k_m * t_m))), 2.0);
	} else {
		tmp = (((l / k_m) * (l * pow(t_m, -1.5))) / k_m) / pow(t_m, 1.5);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.1d-53) then
        tmp = ((t_m ** (-0.5d0)) * (l / (k_m * t_m))) ** 2.0d0
    else
        tmp = (((l / k_m) * (l * (t_m ** (-1.5d0)))) / k_m) / (t_m ** 1.5d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.1e-53) {
		tmp = Math.pow((Math.pow(t_m, -0.5) * (l / (k_m * t_m))), 2.0);
	} else {
		tmp = (((l / k_m) * (l * Math.pow(t_m, -1.5))) / k_m) / Math.pow(t_m, 1.5);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1.1e-53:
		tmp = math.pow((math.pow(t_m, -0.5) * (l / (k_m * t_m))), 2.0)
	else:
		tmp = (((l / k_m) * (l * math.pow(t_m, -1.5))) / k_m) / math.pow(t_m, 1.5)
	return t_s * tmp

Julia

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1.1e-53)
		tmp = Float64((t_m ^ -0.5) * Float64(l / Float64(k_m * t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(Float64(l / k_m) * Float64(l * (t_m ^ -1.5))) / k_m) / (t_m ^ 1.5));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.1e-53)
		tmp = ((t_m ^ -0.5) * (l / (k_m * t_m))) ^ 2.0;
	else
		tmp = (((l / k_m) * (l * (t_m ^ -1.5))) / k_m) / (t_m ^ 1.5);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.1e-53], N[Power[N[(N[Power[t$95$m, -0.5], $MachinePrecision] * N[(l / N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l * N[Power[t$95$m, -1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision] / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.10000000000000009e-53

   1. Initial program 51.6%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in k around 0 47.3%

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

      1. add-sqr-sqrt 31.8%

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

      2. pow2 31.8%

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

      3. sqrt-div 21.9%

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

      4. unpow2 21.9%

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

      5. sqrt-prod 11.4%

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

      6. add-sqr-sqrt 24.5%

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

      7. sqrt-prod 24.4%

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

      8. unpow2 24.4%

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

      9. sqrt-prod 6.9%

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

      10. add-sqr-sqrt 29.0%

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

      11. sqrt-pow1 32.0%

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

      12. metadata-eval 32.0%

          \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

   6. Applied egg-rr 32.0%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 31.7%

         \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right) \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}}^{2} \]

      2. pow2 31.7%

         \[\leadsto {\left(\color{blue}{{\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      3. associate-/r* 31.7%

         \[\leadsto {\left({\left(\sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      4. cbrt-div 31.7%

         \[\leadsto {\left({\color{blue}{\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      5. metadata-eval 31.7%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{\color{blue}{\left(\frac{3}{2}\right)}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      6. sqrt-pow1 27.1%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{{t}^{3}}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      7. unpow3 27.1%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\sqrt{\color{blue}{\left(t \cdot t\right) \cdot t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      8. sqrt-prod 29.7%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{t \cdot t} \cdot \sqrt{t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      9. sqrt-unprod 31.7%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      10. add-cbrt-cube 31.7%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\color{blue}{\sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      11. associate-/r* 32.2%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \]

      12. cbrt-div 32.2%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}}\right)}^{2} \]

      13. metadata-eval 32.2%

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

      14. sqrt-pow1 27.1%

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

      15. unpow3 27.1%

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

      16. sqrt-prod 30.2%

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

      17. sqrt-unprod 32.2%

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

   8. Applied egg-rr 33.7%

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

      1. pow-plus 33.7%

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

      2. metadata-eval 33.7%

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

      3. cube-div 32.2%

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

      4. rem-cube-cbrt 32.3%

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

   10. Simplified 32.3%

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

       1. *-un-lft-identity 32.3%

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

       2. cube-mult 32.3%

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

       3. add-sqr-sqrt 32.4%

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

       4. times-frac 34.0%

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

       5. pow1/2 34.0%

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

       6. pow-flip 34.0%

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

       7. metadata-eval 34.0%

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

   12. Applied egg-rr 34.0%

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

       1. associate-/l/ 34.5%

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

   14. Simplified 34.5%

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

   if 1.10000000000000009e-53 < k

   1. Initial program 49.5%

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

   3. Simplified 49.5%

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

   4. Taylor expanded in k around 0 48.2%

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

      1. add-sqr-sqrt 48.0%

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

      2. pow2 48.0%

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

      3. sqrt-div 24.9%

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

      4. unpow2 24.9%

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

      5. sqrt-prod 13.3%

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

      6. add-sqr-sqrt 25.2%

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

      7. sqrt-prod 25.2%

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

      8. unpow2 25.2%

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

      9. sqrt-prod 24.2%

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

      10. add-sqr-sqrt 24.2%

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

      11. sqrt-pow1 23.8%

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

      12. metadata-eval 23.8%

          \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

   6. Applied egg-rr 23.8%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 23.8%

         \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right) \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}}^{2} \]

      2. pow2 23.8%

         \[\leadsto {\left(\color{blue}{{\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      3. associate-/r* 23.8%

         \[\leadsto {\left({\left(\sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      4. cbrt-div 23.7%

         \[\leadsto {\left({\color{blue}{\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      5. metadata-eval 23.7%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{\color{blue}{\left(\frac{3}{2}\right)}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      6. sqrt-pow1 20.9%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{{t}^{3}}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      7. unpow3 20.9%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\sqrt{\color{blue}{\left(t \cdot t\right) \cdot t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      8. sqrt-prod 21.0%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{t \cdot t} \cdot \sqrt{t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      9. sqrt-unprod 23.7%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      10. add-cbrt-cube 23.8%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\color{blue}{\sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      11. associate-/r* 23.8%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \]

      12. cbrt-div 23.7%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}}\right)}^{2} \]

      13. metadata-eval 23.7%

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

      14. sqrt-pow1 20.9%

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

      15. unpow3 20.9%

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

      16. sqrt-prod 21.0%

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

      17. sqrt-unprod 23.7%

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

   8. Applied egg-rr 23.8%

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

      1. pow-plus 23.8%

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

      2. metadata-eval 23.8%

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

      3. cube-div 23.7%

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

      4. rem-cube-cbrt 23.8%

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

   10. Simplified 23.8%

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

       1. unpow2 23.8%

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

       2. rem-cube-cbrt 23.8%

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

       3. cube-div 23.8%

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

       4. clear-num 23.8%

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

       5. un-div-inv 23.8%

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

       6. cube-div 23.8%

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

       7. rem-cube-cbrt 23.8%

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

       8. div-inv 23.8%

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

       9. sqrt-pow2 23.8%

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

       10. pow-flip 23.8%

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

       11. metadata-eval 23.8%

           \[\leadsto \frac{\frac{\ell}{k} \cdot {t}^{\left(-\color{blue}{1.5}\right)}}{\frac{{\left(\sqrt{t}\right)}^{3}}{\frac{\ell}{k}}} \]

       12. metadata-eval 23.8%

           \[\leadsto \frac{\frac{\ell}{k} \cdot {t}^{\color{blue}{-1.5}}}{\frac{{\left(\sqrt{t}\right)}^{3}}{\frac{\ell}{k}}} \]

       13. div-inv 23.8%

           \[\leadsto \frac{\frac{\ell}{k} \cdot {t}^{-1.5}}{\color{blue}{{\left(\sqrt{t}\right)}^{3} \cdot \frac{1}{\frac{\ell}{k}}}} \]

       14. sqrt-pow2 23.8%

           \[\leadsto \frac{\frac{\ell}{k} \cdot {t}^{-1.5}}{\color{blue}{{t}^{\left(\frac{3}{2}\right)}} \cdot \frac{1}{\frac{\ell}{k}}} \]

       15. metadata-eval 23.8%

           \[\leadsto \frac{\frac{\ell}{k} \cdot {t}^{-1.5}}{{t}^{\color{blue}{1.5}} \cdot \frac{1}{\frac{\ell}{k}}} \]

       16. clear-num 23.8%

           \[\leadsto \frac{\frac{\ell}{k} \cdot {t}^{-1.5}}{{t}^{1.5} \cdot \color{blue}{\frac{k}{\ell}}} \]

   12. Applied egg-rr 23.8%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot {t}^{-1.5}}{{t}^{1.5} \cdot \frac{k}{\ell}}} \]
   13. Step-by-step derivation

       1. *-commutative 23.8%

          \[\leadsto \frac{\frac{\ell}{k} \cdot {t}^{-1.5}}{\color{blue}{\frac{k}{\ell} \cdot {t}^{1.5}}} \]

       2. associate-/r* 26.5%

          \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k} \cdot {t}^{-1.5}}{\frac{k}{\ell}}}{{t}^{1.5}}} \]

       3. associate-/l* 26.5%

          \[\leadsto \frac{\color{blue}{\frac{\left(\frac{\ell}{k} \cdot {t}^{-1.5}\right) \cdot \ell}{k}}}{{t}^{1.5}} \]

       4. associate-*r/ 26.5%

          \[\leadsto \frac{\color{blue}{\left(\frac{\ell}{k} \cdot {t}^{-1.5}\right) \cdot \frac{\ell}{k}}}{{t}^{1.5}} \]

       5. associate-*l* 26.5%

          \[\leadsto \frac{\color{blue}{\frac{\ell}{k} \cdot \left({t}^{-1.5} \cdot \frac{\ell}{k}\right)}}{{t}^{1.5}} \]

       6. associate-*r/ 26.5%

          \[\leadsto \frac{\frac{\ell}{k} \cdot \color{blue}{\frac{{t}^{-1.5} \cdot \ell}{k}}}{{t}^{1.5}} \]

       7. associate-*r/ 26.5%

          \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{k} \cdot \left({t}^{-1.5} \cdot \ell\right)}{k}}}{{t}^{1.5}} \]

   14. Simplified 26.5%

       \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{k} \cdot \left({t}^{-1.5} \cdot \ell\right)}{k}}{{t}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-53}:\\ \;\;\;\;{\left({t}^{-0.5} \cdot \frac{\ell}{k \cdot t}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k} \cdot \left(\ell \cdot {t}^{-1.5}\right)}{k}}{{t}^{1.5}}\\ \end{array} \]

Alternative 7: 71.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 8.8 \cdot 10^{-110}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{k_m}}{t_m \cdot \sqrt{t_m}}\right)}^{2}\\ \mathbf{elif}\;t_m \leq 4.4 \cdot 10^{+96}:\\ \;\;\;\;{\left(\frac{\ell}{k_m}\right)}^{2} \cdot {t_m}^{-3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\frac{\ell}{t_m}}{k_m}}{\sqrt{t_m}}\right)}^{2}\\ \end{array} \end{array} \]

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 (<= t_m 8.8e-110)
    (pow (/ (/ l k_m) (* t_m (sqrt t_m))) 2.0)
    (if (<= t_m 4.4e+96)
      (* (pow (/ l k_m) 2.0) (pow t_m -3.0))
      (pow (/ (/ (/ l t_m) 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) {
	double tmp;
	if (t_m <= 8.8e-110) {
		tmp = pow(((l / k_m) / (t_m * sqrt(t_m))), 2.0);
	} else if (t_m <= 4.4e+96) {
		tmp = pow((l / k_m), 2.0) * pow(t_m, -3.0);
	} else {
		tmp = pow((((l / t_m) / k_m) / 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 (t_m <= 8.8d-110) then
        tmp = ((l / k_m) / (t_m * sqrt(t_m))) ** 2.0d0
    else if (t_m <= 4.4d+96) then
        tmp = ((l / k_m) ** 2.0d0) * (t_m ** (-3.0d0))
    else
        tmp = (((l / t_m) / k_m) / 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 (t_m <= 8.8e-110) {
		tmp = Math.pow(((l / k_m) / (t_m * Math.sqrt(t_m))), 2.0);
	} else if (t_m <= 4.4e+96) {
		tmp = Math.pow((l / k_m), 2.0) * Math.pow(t_m, -3.0);
	} else {
		tmp = Math.pow((((l / t_m) / k_m) / 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 t_m <= 8.8e-110:
		tmp = math.pow(((l / k_m) / (t_m * math.sqrt(t_m))), 2.0)
	elif t_m <= 4.4e+96:
		tmp = math.pow((l / k_m), 2.0) * math.pow(t_m, -3.0)
	else:
		tmp = math.pow((((l / t_m) / k_m) / 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 (t_m <= 8.8e-110)
		tmp = Float64(Float64(l / k_m) / Float64(t_m * sqrt(t_m))) ^ 2.0;
	elseif (t_m <= 4.4e+96)
		tmp = Float64((Float64(l / k_m) ^ 2.0) * (t_m ^ -3.0));
	else
		tmp = Float64(Float64(Float64(l / t_m) / k_m) / 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 (t_m <= 8.8e-110)
		tmp = ((l / k_m) / (t_m * sqrt(t_m))) ^ 2.0;
	elseif (t_m <= 4.4e+96)
		tmp = ((l / k_m) ^ 2.0) * (t_m ^ -3.0);
	else
		tmp = (((l / t_m) / k_m) / 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[t$95$m, 8.8e-110], N[Power[N[(N[(l / k$95$m), $MachinePrecision] / N[(t$95$m * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 4.4e+96], N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[t$95$m, -3.0], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(l / t$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 8.7999999999999997e-110

   1. Initial program 48.7%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in k around 0 48.4%

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

      1. add-sqr-sqrt 31.2%

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

      2. pow2 31.2%

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

      3. sqrt-div 10.8%

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

      4. unpow2 10.8%

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

      5. sqrt-prod 4.3%

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

      6. add-sqr-sqrt 11.6%

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

      7. sqrt-prod 11.6%

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

      8. unpow2 11.6%

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

      9. sqrt-prod 4.4%

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

      10. add-sqr-sqrt 11.8%

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

      11. sqrt-pow1 8.7%

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

      12. metadata-eval 8.7%

          \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

   6. Applied egg-rr 8.7%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 8.7%

         \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right) \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}}^{2} \]

      2. pow2 8.7%

         \[\leadsto {\left(\color{blue}{{\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      3. associate-/r* 9.8%

         \[\leadsto {\left({\left(\sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      4. cbrt-div 9.8%

         \[\leadsto {\left({\color{blue}{\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      5. metadata-eval 9.8%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{\color{blue}{\left(\frac{3}{2}\right)}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      6. sqrt-pow1 8.5%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{{t}^{3}}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      7. unpow3 8.5%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\sqrt{\color{blue}{\left(t \cdot t\right) \cdot t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      8. sqrt-prod 9.2%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{t \cdot t} \cdot \sqrt{t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      9. sqrt-unprod 9.8%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      10. add-cbrt-cube 9.8%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\color{blue}{\sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      11. associate-/r* 10.3%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \]

      12. cbrt-div 10.3%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}}\right)}^{2} \]

      13. metadata-eval 10.3%

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

      14. sqrt-pow1 8.5%

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

      15. unpow3 8.5%

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

      16. sqrt-prod 9.7%

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

      17. sqrt-unprod 10.3%

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

   8. Applied egg-rr 10.9%

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

      1. pow-plus 10.9%

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

      2. metadata-eval 10.9%

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

      3. cube-div 10.3%

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

      4. rem-cube-cbrt 10.3%

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

   10. Simplified 10.3%

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

       1. cube-mult 10.3%

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

       2. add-sqr-sqrt 10.3%

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

   12. Applied egg-rr 10.3%

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

   if 8.7999999999999997e-110 < t < 4.3999999999999998e96

   1. Initial program 62.7%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in k around 0 52.4%

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

      1. add-sqr-sqrt 52.4%

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

      2. pow2 52.4%

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

      3. sqrt-div 52.3%

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

      4. unpow2 52.3%

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

      5. sqrt-prod 27.8%

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

      6. add-sqr-sqrt 57.6%

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

      7. sqrt-prod 57.6%

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

      8. unpow2 57.6%

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

      9. sqrt-prod 23.2%

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

      10. add-sqr-sqrt 67.6%

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

      11. sqrt-pow1 67.5%

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

      12. metadata-eval 67.5%

          \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

   6. Applied egg-rr 67.5%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 67.5%

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{1.5}} \cdot \frac{\ell}{k \cdot {t}^{1.5}}} \]

      2. associate-/r* 67.5%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}} \cdot \frac{\ell}{k \cdot {t}^{1.5}} \]

      3. associate-/r* 67.6%

         \[\leadsto \frac{\frac{\ell}{k}}{{t}^{1.5}} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}} \]

      4. frac-times 74.8%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{1.5} \cdot {t}^{1.5}}} \]

      5. pow-prod-up 74.9%

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{{t}^{\left(1.5 + 1.5\right)}}} \]

      6. metadata-eval 74.9%

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

   8. Applied egg-rr 74.9%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 74.7%

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

      2. pow-prod-down 74.7%

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

      3. frac-times 67.5%

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

      4. unpow2 67.5%

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

      5. expm1-log1p-u 67.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\frac{\ell}{k}}{{\left(\sqrt{t}\right)}^{3}}\right)}^{2}\right)\right)} \]

      6. expm1-udef 59.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\frac{\ell}{k}}{{\left(\sqrt{t}\right)}^{3}}\right)}^{2}\right)} - 1} \]

   10. Applied egg-rr 59.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k}\right)}^{2} \cdot {t}^{-3}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 74.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k}\right)}^{2} \cdot {t}^{-3}\right)\right)} \]

       2. expm1-log1p 75.1%

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

   12. Simplified 75.1%

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

   if 4.3999999999999998e96 < t

   1. Initial program 48.7%

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

   3. Simplified 48.7%

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

   4. Taylor expanded in k around 0 39.5%

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

      1. add-sqr-sqrt 39.5%

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

      2. pow2 39.5%

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

      3. sqrt-div 39.5%

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

      4. unpow2 39.5%

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

      5. sqrt-prod 25.9%

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

      6. add-sqr-sqrt 42.7%

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

      7. sqrt-prod 42.7%

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

      8. unpow2 42.7%

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

      9. sqrt-prod 27.9%

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

      10. add-sqr-sqrt 49.9%

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

      11. sqrt-pow1 73.7%

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

      12. metadata-eval 73.7%

          \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

   6. Applied egg-rr 73.7%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 73.1%

         \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right) \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}}^{2} \]

      2. pow2 73.1%

         \[\leadsto {\left(\color{blue}{{\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      3. associate-/r* 69.0%

         \[\leadsto {\left({\left(\sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      4. cbrt-div 69.0%

         \[\leadsto {\left({\color{blue}{\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      5. metadata-eval 69.0%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{\color{blue}{\left(\frac{3}{2}\right)}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      6. sqrt-pow1 49.6%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{{t}^{3}}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      7. unpow3 49.6%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\sqrt{\color{blue}{\left(t \cdot t\right) \cdot t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      8. sqrt-prod 58.4%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{t \cdot t} \cdot \sqrt{t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      9. sqrt-unprod 69.0%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      10. add-cbrt-cube 69.0%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\color{blue}{\sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      11. associate-/r* 69.0%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \]

      12. cbrt-div 68.9%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}}\right)}^{2} \]

      13. metadata-eval 68.9%

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

      14. sqrt-pow1 49.6%

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

      15. unpow3 49.6%

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

      16. sqrt-prod 58.3%

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

      17. sqrt-unprod 68.9%

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

   8. Applied egg-rr 73.3%

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

      1. pow-plus 73.3%

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

      2. metadata-eval 73.3%

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

      3. cube-div 68.8%

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

      4. rem-cube-cbrt 69.2%

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

   10. Simplified 69.2%

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

       1. rem-cube-cbrt 68.8%

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

       2. cube-div 73.3%

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

       3. cube-mult 73.3%

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

       4. frac-times 73.3%

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

       5. pow2 73.3%

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

       6. add-sqr-sqrt 73.3%

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

   12. Applied egg-rr 73.3%

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

       1. associate-*l/ 73.2%

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

       2. associate-*r/ 73.3%

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

       3. unpow2 73.3%

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

       4. cube-mult 73.2%

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

       5. rem-cube-cbrt 73.9%

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

       6. associate-/l/ 82.4%

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

       7. associate-/r* 82.3%

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

   14. Simplified 82.3%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.8 \cdot 10^{-110}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{k}}{t \cdot \sqrt{t}}\right)}^{2}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+96}:\\ \;\;\;\;{\left(\frac{\ell}{k}\right)}^{2} \cdot {t}^{-3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\frac{\ell}{t}}{k}}{\sqrt{t}}\right)}^{2}\\ \end{array} \]

Alternative 8: 73.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 8.8 \cdot 10^{-110}:\\ \;\;\;\;{\left({t_m}^{-0.5} \cdot \frac{\frac{\ell}{k_m}}{t_m}\right)}^{2}\\ \mathbf{elif}\;t_m \leq 4.4 \cdot 10^{+96}:\\ \;\;\;\;{\left(\frac{\ell}{k_m}\right)}^{2} \cdot {t_m}^{-3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\frac{\ell}{t_m}}{k_m}}{\sqrt{t_m}}\right)}^{2}\\ \end{array} \end{array} \]

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 (<= t_m 8.8e-110)
    (pow (* (pow t_m -0.5) (/ (/ l k_m) t_m)) 2.0)
    (if (<= t_m 4.4e+96)
      (* (pow (/ l k_m) 2.0) (pow t_m -3.0))
      (pow (/ (/ (/ l t_m) 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) {
	double tmp;
	if (t_m <= 8.8e-110) {
		tmp = pow((pow(t_m, -0.5) * ((l / k_m) / t_m)), 2.0);
	} else if (t_m <= 4.4e+96) {
		tmp = pow((l / k_m), 2.0) * pow(t_m, -3.0);
	} else {
		tmp = pow((((l / t_m) / k_m) / 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 (t_m <= 8.8d-110) then
        tmp = ((t_m ** (-0.5d0)) * ((l / k_m) / t_m)) ** 2.0d0
    else if (t_m <= 4.4d+96) then
        tmp = ((l / k_m) ** 2.0d0) * (t_m ** (-3.0d0))
    else
        tmp = (((l / t_m) / k_m) / 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 (t_m <= 8.8e-110) {
		tmp = Math.pow((Math.pow(t_m, -0.5) * ((l / k_m) / t_m)), 2.0);
	} else if (t_m <= 4.4e+96) {
		tmp = Math.pow((l / k_m), 2.0) * Math.pow(t_m, -3.0);
	} else {
		tmp = Math.pow((((l / t_m) / k_m) / 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 t_m <= 8.8e-110:
		tmp = math.pow((math.pow(t_m, -0.5) * ((l / k_m) / t_m)), 2.0)
	elif t_m <= 4.4e+96:
		tmp = math.pow((l / k_m), 2.0) * math.pow(t_m, -3.0)
	else:
		tmp = math.pow((((l / t_m) / k_m) / 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 (t_m <= 8.8e-110)
		tmp = Float64((t_m ^ -0.5) * Float64(Float64(l / k_m) / t_m)) ^ 2.0;
	elseif (t_m <= 4.4e+96)
		tmp = Float64((Float64(l / k_m) ^ 2.0) * (t_m ^ -3.0));
	else
		tmp = Float64(Float64(Float64(l / t_m) / k_m) / 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 (t_m <= 8.8e-110)
		tmp = ((t_m ^ -0.5) * ((l / k_m) / t_m)) ^ 2.0;
	elseif (t_m <= 4.4e+96)
		tmp = ((l / k_m) ^ 2.0) * (t_m ^ -3.0);
	else
		tmp = (((l / t_m) / k_m) / 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[t$95$m, 8.8e-110], N[Power[N[(N[Power[t$95$m, -0.5], $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 4.4e+96], N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[Power[t$95$m, -3.0], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(l / t$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 8.7999999999999997e-110

   1. Initial program 48.7%

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

   3. Simplified 49.1%

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

   4. Taylor expanded in k around 0 48.4%

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

      1. add-sqr-sqrt 31.2%

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

      2. pow2 31.2%

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

      3. sqrt-div 10.8%

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

      4. unpow2 10.8%

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

      5. sqrt-prod 4.3%

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

      6. add-sqr-sqrt 11.6%

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

      7. sqrt-prod 11.6%

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

      8. unpow2 11.6%

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

      9. sqrt-prod 4.4%

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

      10. add-sqr-sqrt 11.8%

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

      11. sqrt-pow1 8.7%

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

      12. metadata-eval 8.7%

          \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

   6. Applied egg-rr 8.7%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 8.7%

         \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right) \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}}^{2} \]

      2. pow2 8.7%

         \[\leadsto {\left(\color{blue}{{\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      3. associate-/r* 9.8%

         \[\leadsto {\left({\left(\sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      4. cbrt-div 9.8%

         \[\leadsto {\left({\color{blue}{\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      5. metadata-eval 9.8%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{\color{blue}{\left(\frac{3}{2}\right)}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      6. sqrt-pow1 8.5%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{{t}^{3}}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      7. unpow3 8.5%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\sqrt{\color{blue}{\left(t \cdot t\right) \cdot t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      8. sqrt-prod 9.2%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{t \cdot t} \cdot \sqrt{t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      9. sqrt-unprod 9.8%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      10. add-cbrt-cube 9.8%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\color{blue}{\sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      11. associate-/r* 10.3%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \]

      12. cbrt-div 10.3%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}}\right)}^{2} \]

      13. metadata-eval 10.3%

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

      14. sqrt-pow1 8.5%

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

      15. unpow3 8.5%

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

      16. sqrt-prod 9.7%

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

      17. sqrt-unprod 10.3%

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

   8. Applied egg-rr 10.9%

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

      1. pow-plus 10.9%

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

      2. metadata-eval 10.9%

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

      3. cube-div 10.3%

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

      4. rem-cube-cbrt 10.3%

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

   10. Simplified 10.3%

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

       1. *-un-lft-identity 10.3%

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

       2. cube-mult 10.3%

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

       3. add-sqr-sqrt 10.3%

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

       4. times-frac 10.9%

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

       5. pow1/2 10.9%

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

       6. pow-flip 10.9%

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

       7. metadata-eval 10.9%

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

   12. Applied egg-rr 10.9%

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

   if 8.7999999999999997e-110 < t < 4.3999999999999998e96

   1. Initial program 62.7%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in k around 0 52.4%

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

      1. add-sqr-sqrt 52.4%

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

      2. pow2 52.4%

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

      3. sqrt-div 52.3%

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

      4. unpow2 52.3%

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

      5. sqrt-prod 27.8%

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

      6. add-sqr-sqrt 57.6%

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

      7. sqrt-prod 57.6%

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

      8. unpow2 57.6%

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

      9. sqrt-prod 23.2%

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

      10. add-sqr-sqrt 67.6%

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

      11. sqrt-pow1 67.5%

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

      12. metadata-eval 67.5%

          \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

   6. Applied egg-rr 67.5%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 67.5%

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{1.5}} \cdot \frac{\ell}{k \cdot {t}^{1.5}}} \]

      2. associate-/r* 67.5%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}} \cdot \frac{\ell}{k \cdot {t}^{1.5}} \]

      3. associate-/r* 67.6%

         \[\leadsto \frac{\frac{\ell}{k}}{{t}^{1.5}} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}} \]

      4. frac-times 74.8%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{1.5} \cdot {t}^{1.5}}} \]

      5. pow-prod-up 74.9%

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{{t}^{\left(1.5 + 1.5\right)}}} \]

      6. metadata-eval 74.9%

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

   8. Applied egg-rr 74.9%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 74.7%

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

      2. pow-prod-down 74.7%

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

      3. frac-times 67.5%

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

      4. unpow2 67.5%

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

      5. expm1-log1p-u 67.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\frac{\ell}{k}}{{\left(\sqrt{t}\right)}^{3}}\right)}^{2}\right)\right)} \]

      6. expm1-udef 59.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\frac{\ell}{k}}{{\left(\sqrt{t}\right)}^{3}}\right)}^{2}\right)} - 1} \]

   10. Applied egg-rr 59.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k}\right)}^{2} \cdot {t}^{-3}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 74.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k}\right)}^{2} \cdot {t}^{-3}\right)\right)} \]

       2. expm1-log1p 75.1%

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

   12. Simplified 75.1%

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

   if 4.3999999999999998e96 < t

   1. Initial program 48.7%

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

   3. Simplified 48.7%

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

   4. Taylor expanded in k around 0 39.5%

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

      1. add-sqr-sqrt 39.5%

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

      2. pow2 39.5%

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

      3. sqrt-div 39.5%

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

      4. unpow2 39.5%

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

      5. sqrt-prod 25.9%

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

      6. add-sqr-sqrt 42.7%

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

      7. sqrt-prod 42.7%

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

      8. unpow2 42.7%

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

      9. sqrt-prod 27.9%

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

      10. add-sqr-sqrt 49.9%

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

      11. sqrt-pow1 73.7%

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

      12. metadata-eval 73.7%

          \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

   6. Applied egg-rr 73.7%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 73.1%

         \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right) \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}}^{2} \]

      2. pow2 73.1%

         \[\leadsto {\left(\color{blue}{{\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      3. associate-/r* 69.0%

         \[\leadsto {\left({\left(\sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      4. cbrt-div 69.0%

         \[\leadsto {\left({\color{blue}{\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      5. metadata-eval 69.0%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{\color{blue}{\left(\frac{3}{2}\right)}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      6. sqrt-pow1 49.6%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{{t}^{3}}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      7. unpow3 49.6%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\sqrt{\color{blue}{\left(t \cdot t\right) \cdot t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      8. sqrt-prod 58.4%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{t \cdot t} \cdot \sqrt{t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      9. sqrt-unprod 69.0%

         \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      10. add-cbrt-cube 69.0%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\color{blue}{\sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

      11. associate-/r* 69.0%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \]

      12. cbrt-div 68.9%

          \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}}\right)}^{2} \]

      13. metadata-eval 68.9%

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

      14. sqrt-pow1 49.6%

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

      15. unpow3 49.6%

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

      16. sqrt-prod 58.3%

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

      17. sqrt-unprod 68.9%

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

   8. Applied egg-rr 73.3%

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

      1. pow-plus 73.3%

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

      2. metadata-eval 73.3%

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

      3. cube-div 68.8%

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

      4. rem-cube-cbrt 69.2%

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

   10. Simplified 69.2%

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

       1. rem-cube-cbrt 68.8%

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

       2. cube-div 73.3%

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

       3. cube-mult 73.3%

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

       4. frac-times 73.3%

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

       5. pow2 73.3%

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

       6. add-sqr-sqrt 73.3%

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

   12. Applied egg-rr 73.3%

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

       1. associate-*l/ 73.2%

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

       2. associate-*r/ 73.3%

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

       3. unpow2 73.3%

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

       4. cube-mult 73.2%

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

       5. rem-cube-cbrt 73.9%

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

       6. associate-/l/ 82.4%

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

       7. associate-/r* 82.3%

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

   14. Simplified 82.3%

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

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.8 \cdot 10^{-110}:\\ \;\;\;\;{\left({t}^{-0.5} \cdot \frac{\frac{\ell}{k}}{t}\right)}^{2}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+96}:\\ \;\;\;\;{\left(\frac{\ell}{k}\right)}^{2} \cdot {t}^{-3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\frac{\ell}{t}}{k}}{\sqrt{t}}\right)}^{2}\\ \end{array} \]

Alternative 9: 67.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{\ell}{k_m}}{t_m} \cdot \frac{\frac{\ell}{k_m}}{{t_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k_m}{\frac{\ell}{{t_m}^{1.5}}}\right)}^{-2}\\ \end{array} \end{array} \]

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 (<= l 2e-71)
    (* (/ (/ l k_m) t_m) (/ (/ l k_m) (pow t_m 2.0)))
    (pow (/ k_m (/ l (pow t_m 1.5))) -2.0))))

C

k_m = fabs(k);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 2e-71) {
		tmp = ((l / k_m) / t_m) * ((l / k_m) / pow(t_m, 2.0));
	} else {
		tmp = pow((k_m / (l / pow(t_m, 1.5))), -2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 2d-71) then
        tmp = ((l / k_m) / t_m) * ((l / k_m) / (t_m ** 2.0d0))
    else
        tmp = (k_m / (l / (t_m ** 1.5d0))) ** (-2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 2e-71) {
		tmp = ((l / k_m) / t_m) * ((l / k_m) / Math.pow(t_m, 2.0));
	} else {
		tmp = Math.pow((k_m / (l / Math.pow(t_m, 1.5))), -2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if l <= 2e-71:
		tmp = ((l / k_m) / t_m) * ((l / k_m) / math.pow(t_m, 2.0))
	else:
		tmp = math.pow((k_m / (l / math.pow(t_m, 1.5))), -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 (l <= 2e-71)
		tmp = Float64(Float64(Float64(l / k_m) / t_m) * Float64(Float64(l / k_m) / (t_m ^ 2.0)));
	else
		tmp = Float64(k_m / Float64(l / (t_m ^ 1.5))) ^ -2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (l <= 2e-71)
		tmp = ((l / k_m) / t_m) * ((l / k_m) / (t_m ^ 2.0));
	else
		tmp = (k_m / (l / (t_m ^ 1.5))) ^ -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[l, 2e-71], N[(N[(N[(l / k$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(k$95$m / N[(l / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.9999999999999998e-71

   1. Initial program 51.1%

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

   3. Simplified 52.3%

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

   4. Taylor expanded in k around 0 48.3%

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

      1. add-sqr-sqrt 40.5%

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

      2. pow2 40.5%

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

      3. sqrt-div 23.1%

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

      4. unpow2 23.1%

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

      5. sqrt-prod 5.6%

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

      6. add-sqr-sqrt 25.0%

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

      7. sqrt-prod 25.0%

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

      8. unpow2 25.0%

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

      9. sqrt-prod 15.1%

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

      10. add-sqr-sqrt 30.1%

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

      11. sqrt-pow1 30.9%

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

      12. metadata-eval 30.9%

          \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

   6. Applied egg-rr 30.9%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 30.9%

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{1.5}} \cdot \frac{\ell}{k \cdot {t}^{1.5}}} \]

      2. associate-/r* 31.4%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}} \cdot \frac{\ell}{k \cdot {t}^{1.5}} \]

      3. associate-/r* 32.0%

         \[\leadsto \frac{\frac{\ell}{k}}{{t}^{1.5}} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}} \]

      4. frac-times 27.9%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{1.5} \cdot {t}^{1.5}}} \]

      5. pow-prod-up 63.9%

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{{t}^{\left(1.5 + 1.5\right)}}} \]

      6. metadata-eval 63.9%

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

   8. Applied egg-rr 63.9%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]
   9. Step-by-step derivation

      1. unpow3 63.9%

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

      2. unpow2 63.9%

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

      3. times-frac 68.2%

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

   10. Applied egg-rr 68.2%

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

   if 1.9999999999999998e-71 < l

   1. Initial program 50.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in k around 0 46.2%

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

      1. add-sqr-sqrt 27.9%

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

      2. pow2 27.9%

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

      3. sqrt-div 22.0%

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

      4. unpow2 22.0%

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

      5. sqrt-prod 24.0%

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

      6. add-sqr-sqrt 24.0%

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

      7. sqrt-prod 24.0%

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

      8. unpow2 24.0%

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

      9. sqrt-prod 4.8%

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

      10. add-sqr-sqrt 23.2%

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

      11. sqrt-pow1 27.5%

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

      12. metadata-eval 27.5%

          \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

   6. Applied egg-rr 27.5%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 27.5%

         \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{1.5}} \cdot \frac{\ell}{k \cdot {t}^{1.5}}} \]

      2. associate-/r* 26.5%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}} \cdot \frac{\ell}{k \cdot {t}^{1.5}} \]

      3. associate-/r* 26.5%

         \[\leadsto \frac{\frac{\ell}{k}}{{t}^{1.5}} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}} \]

      4. frac-times 24.0%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{1.5} \cdot {t}^{1.5}}} \]

      5. pow-prod-up 51.1%

         \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{{t}^{\left(1.5 + 1.5\right)}}} \]

      6. metadata-eval 51.1%

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

   8. Applied egg-rr 51.1%

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

   9. Taylor expanded in l around 0 46.2%

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

       1. unpow2 46.2%

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

       2. associate-/l* 47.3%

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

       3. *-rgt-identity 47.3%

          \[\leadsto \frac{\color{blue}{\ell \cdot 1}}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}} \]

       4. *-commutative 47.3%

          \[\leadsto \frac{\ell \cdot 1}{\frac{\color{blue}{{t}^{3} \cdot {k}^{2}}}{\ell}} \]

       5. metadata-eval 47.3%

          \[\leadsto \frac{\ell \cdot 1}{\frac{{t}^{\color{blue}{\left(2 \cdot 1.5\right)}} \cdot {k}^{2}}{\ell}} \]

       6. pow-sqr 24.0%

          \[\leadsto \frac{\ell \cdot 1}{\frac{\color{blue}{\left({t}^{1.5} \cdot {t}^{1.5}\right)} \cdot {k}^{2}}{\ell}} \]

       7. unpow2 24.0%

          \[\leadsto \frac{\ell \cdot 1}{\frac{\left({t}^{1.5} \cdot {t}^{1.5}\right) \cdot \color{blue}{\left(k \cdot k\right)}}{\ell}} \]

       8. swap-sqr 27.5%

          \[\leadsto \frac{\ell \cdot 1}{\frac{\color{blue}{\left({t}^{1.5} \cdot k\right) \cdot \left({t}^{1.5} \cdot k\right)}}{\ell}} \]

       9. associate-*r/ 27.5%

          \[\leadsto \frac{\ell \cdot 1}{\color{blue}{\left({t}^{1.5} \cdot k\right) \cdot \frac{{t}^{1.5} \cdot k}{\ell}}} \]

       10. associate-*r/ 26.4%

           \[\leadsto \frac{\ell \cdot 1}{\left({t}^{1.5} \cdot k\right) \cdot \color{blue}{\left({t}^{1.5} \cdot \frac{k}{\ell}\right)}} \]

       11. times-frac 26.5%

           \[\leadsto \color{blue}{\frac{\ell}{{t}^{1.5} \cdot k} \cdot \frac{1}{{t}^{1.5} \cdot \frac{k}{\ell}}} \]

       12. *-lft-identity 26.5%

           \[\leadsto \frac{\color{blue}{1 \cdot \ell}}{{t}^{1.5} \cdot k} \cdot \frac{1}{{t}^{1.5} \cdot \frac{k}{\ell}} \]

       13. associate-/l* 26.5%

           \[\leadsto \color{blue}{\frac{1}{\frac{{t}^{1.5} \cdot k}{\ell}}} \cdot \frac{1}{{t}^{1.5} \cdot \frac{k}{\ell}} \]

       14. associate-*r/ 26.5%

           \[\leadsto \frac{1}{\color{blue}{{t}^{1.5} \cdot \frac{k}{\ell}}} \cdot \frac{1}{{t}^{1.5} \cdot \frac{k}{\ell}} \]

       15. unpow-1 26.5%

           \[\leadsto \color{blue}{{\left({t}^{1.5} \cdot \frac{k}{\ell}\right)}^{-1}} \cdot \frac{1}{{t}^{1.5} \cdot \frac{k}{\ell}} \]

       16. unpow-1 26.5%

           \[\leadsto {\left({t}^{1.5} \cdot \frac{k}{\ell}\right)}^{-1} \cdot \color{blue}{{\left({t}^{1.5} \cdot \frac{k}{\ell}\right)}^{-1}} \]

       17. pow-sqr 26.5%

           \[\leadsto \color{blue}{{\left({t}^{1.5} \cdot \frac{k}{\ell}\right)}^{\left(2 \cdot -1\right)}} \]

       18. *-commutative 26.5%

           \[\leadsto {\color{blue}{\left(\frac{k}{\ell} \cdot {t}^{1.5}\right)}}^{\left(2 \cdot -1\right)} \]

       19. associate-*l/ 27.5%

           \[\leadsto {\color{blue}{\left(\frac{k \cdot {t}^{1.5}}{\ell}\right)}}^{\left(2 \cdot -1\right)} \]

       20. associate-/l* 27.5%

           \[\leadsto {\color{blue}{\left(\frac{k}{\frac{\ell}{{t}^{1.5}}}\right)}}^{\left(2 \cdot -1\right)} \]

       21. metadata-eval 27.5%

           \[\leadsto {\left(\frac{k}{\frac{\ell}{{t}^{1.5}}}\right)}^{\color{blue}{-2}} \]

   11. Simplified 27.5%

       \[\leadsto \color{blue}{{\left(\frac{k}{\frac{\ell}{{t}^{1.5}}}\right)}^{-2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{\ell}{k}}{{t}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{\frac{\ell}{{t}^{1.5}}}\right)}^{-2}\\ \end{array} \]

Alternative 10: 71.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot {\left({t_m}^{-0.5} \cdot \frac{\ell}{k_m \cdot t_m}\right)}^{2} \end{array} \]

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 (pow (* (pow t_m -0.5) (/ l (* k_m 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 * pow((pow(t_m, -0.5) * (l / (k_m * 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 * (((t_m ** (-0.5d0)) * (l / (k_m * 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 * Math.pow((Math.pow(t_m, -0.5) * (l / (k_m * 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 * math.pow((math.pow(t_m, -0.5) * (l / (k_m * 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((t_m ^ -0.5) * Float64(l / Float64(k_m * 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 * (((t_m ^ -0.5) * (l / (k_m * 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[Power[N[(N[Power[t$95$m, -0.5], $MachinePrecision] * N[(l / N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.0%

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

3. Simplified 52.2%

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

4. Taylor expanded in k around 0 47.6%

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

   1. add-sqr-sqrt 36.2%

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

   2. pow2 36.2%

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

   3. sqrt-div 22.7%

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

   4. unpow2 22.7%

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

   5. sqrt-prod 11.9%

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

   6. add-sqr-sqrt 24.7%

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

   7. sqrt-prod 24.6%

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

   8. unpow2 24.6%

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

   9. sqrt-prod 11.6%

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

   10. add-sqr-sqrt 27.7%

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

   11. sqrt-pow1 29.8%

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

   12. metadata-eval 29.8%

       \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

6. Applied egg-rr 29.8%

   \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
7. Step-by-step derivation

   1. add-cube-cbrt 29.6%

      \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right) \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}}^{2} \]

   2. pow2 29.6%

      \[\leadsto {\left(\color{blue}{{\left(\sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2}} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

   3. associate-/r* 29.6%

      \[\leadsto {\left({\left(\sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

   4. cbrt-div 29.6%

      \[\leadsto {\left({\color{blue}{\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}\right)}}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

   5. metadata-eval 29.6%

      \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{\color{blue}{\left(\frac{3}{2}\right)}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

   6. sqrt-pow1 25.4%

      \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{{t}^{3}}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

   7. unpow3 25.4%

      \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\sqrt{\color{blue}{\left(t \cdot t\right) \cdot t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

   8. sqrt-prod 27.4%

      \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\sqrt{t \cdot t} \cdot \sqrt{t}}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

   9. sqrt-unprod 29.6%

      \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

   10. add-cbrt-cube 29.6%

       \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\color{blue}{\sqrt{t}}}\right)}^{2} \cdot \sqrt[3]{\frac{\ell}{k \cdot {t}^{1.5}}}\right)}^{2} \]

   11. associate-/r* 29.9%

       \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}}}\right)}^{2} \]

   12. cbrt-div 29.9%

       \[\leadsto {\left({\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt{t}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{\frac{\ell}{k}}}{\sqrt[3]{{t}^{1.5}}}}\right)}^{2} \]

   13. metadata-eval 29.9%

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

   14. sqrt-pow1 25.4%

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

   15. unpow3 25.4%

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

   16. sqrt-prod 27.7%

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

   17. sqrt-unprod 29.9%

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

8. Applied egg-rr 31.1%

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

   1. pow-plus 31.1%

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

   2. metadata-eval 31.1%

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

   3. cube-div 29.9%

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

   4. rem-cube-cbrt 30.0%

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

10. Simplified 30.0%

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

    1. *-un-lft-identity 30.0%

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

    2. cube-mult 30.0%

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

    3. add-sqr-sqrt 30.1%

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

    4. times-frac 31.3%

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

    5. pow1/2 31.3%

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

    6. pow-flip 31.3%

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

    7. metadata-eval 31.3%

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

12. Applied egg-rr 31.3%

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

    1. associate-/l/ 31.6%

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

14. Simplified 31.6%

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

15. Final simplification 31.6%

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

Alternative 11: 67.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot {\left(\frac{\ell}{k_m \cdot {t_m}^{1.5}}\right)}^{2} \end{array} \]

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 (pow (/ l (* k_m (pow t_m 1.5))) 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 * pow((l / (k_m * pow(t_m, 1.5))), 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 * ((l / (k_m * (t_m ** 1.5d0))) ** 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 * Math.pow((l / (k_m * Math.pow(t_m, 1.5))), 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 * math.pow((l / (k_m * math.pow(t_m, 1.5))), 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(l / Float64(k_m * (t_m ^ 1.5))) ^ 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 * ((l / (k_m * (t_m ^ 1.5))) ^ 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[Power[N[(l / N[(k$95$m * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.0%

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

3. Simplified 52.2%

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

4. Taylor expanded in k around 0 47.6%

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

   1. add-sqr-sqrt 36.2%

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

   2. pow2 36.2%

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

   3. sqrt-div 22.7%

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

   4. unpow2 22.7%

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

   5. sqrt-prod 11.9%

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

   6. add-sqr-sqrt 24.7%

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

   7. sqrt-prod 24.6%

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

   8. unpow2 24.6%

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

   9. sqrt-prod 11.6%

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

   10. add-sqr-sqrt 27.7%

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

   11. sqrt-pow1 29.8%

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

   12. metadata-eval 29.8%

       \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

6. Applied egg-rr 29.8%

   \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]

7. Final simplification 29.8%

   \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2} \]

Alternative 12: 68.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot {\left(\frac{\frac{\ell}{k_m}}{{t_m}^{1.5}}\right)}^{2} \end{array} \]

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 (pow (/ (/ l k_m) (pow t_m 1.5)) 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 * pow(((l / k_m) / pow(t_m, 1.5)), 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 * (((l / k_m) / (t_m ** 1.5d0)) ** 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 * Math.pow(((l / k_m) / Math.pow(t_m, 1.5)), 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 * math.pow(((l / k_m) / math.pow(t_m, 1.5)), 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(Float64(l / k_m) / (t_m ^ 1.5)) ^ 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 * (((l / k_m) / (t_m ^ 1.5)) ^ 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[Power[N[(N[(l / k$95$m), $MachinePrecision] / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.0%

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

3. Simplified 52.2%

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

4. Taylor expanded in k around 0 47.6%

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

   1. expm1-log1p-u 37.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)\right)} \]

   2. expm1-udef 35.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\right)} - 1} \]

6. Applied egg-rr 27.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 29.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

   2. expm1-log1p 29.8%

      \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]

   3. associate-/r* 30.1%

      \[\leadsto {\color{blue}{\left(\frac{\frac{\ell}{k}}{{t}^{1.5}}\right)}}^{2} \]

8. Simplified 30.1%

   \[\leadsto \color{blue}{{\left(\frac{\frac{\ell}{k}}{{t}^{1.5}}\right)}^{2}} \]

9. Final simplification 30.1%

   \[\leadsto {\left(\frac{\frac{\ell}{k}}{{t}^{1.5}}\right)}^{2} \]

Alternative 13: 65.8% accurate, 3.8× speedup

Math

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

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 (* (/ (/ l k_m) t_m) (/ (/ l k_m) (pow 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 * (((l / k_m) / t_m) * ((l / k_m) / pow(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 * (((l / k_m) / t_m) * ((l / k_m) / (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 * (((l / k_m) / t_m) * ((l / k_m) / Math.pow(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 * (((l / k_m) / t_m) * ((l / k_m) / math.pow(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(Float64(Float64(l / k_m) / t_m) * Float64(Float64(l / k_m) / (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 * (((l / k_m) / t_m) * ((l / k_m) / (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[(N[(N[(l / k$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[(l / k$95$m), $MachinePrecision] / N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.0%

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

3. Simplified 52.2%

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

4. Taylor expanded in k around 0 47.6%

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

   1. add-sqr-sqrt 36.2%

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

   2. pow2 36.2%

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

   3. sqrt-div 22.7%

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

   4. unpow2 22.7%

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

   5. sqrt-prod 11.9%

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

   6. add-sqr-sqrt 24.7%

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

   7. sqrt-prod 24.6%

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

   8. unpow2 24.6%

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

   9. sqrt-prod 11.6%

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

   10. add-sqr-sqrt 27.7%

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

   11. sqrt-pow1 29.8%

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

   12. metadata-eval 29.8%

       \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

6. Applied egg-rr 29.8%

   \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
7. Step-by-step derivation

   1. unpow2 29.8%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{1.5}} \cdot \frac{\ell}{k \cdot {t}^{1.5}}} \]

   2. associate-/r* 29.7%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}} \cdot \frac{\ell}{k \cdot {t}^{1.5}} \]

   3. associate-/r* 30.1%

      \[\leadsto \frac{\frac{\ell}{k}}{{t}^{1.5}} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}} \]

   4. frac-times 26.6%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{1.5} \cdot {t}^{1.5}}} \]

   5. pow-prod-up 59.5%

      \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{{t}^{\left(1.5 + 1.5\right)}}} \]

   6. metadata-eval 59.5%

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

8. Applied egg-rr 59.5%

   \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]
9. Step-by-step derivation

   1. unpow3 59.5%

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

   2. unpow2 59.5%

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

   3. times-frac 63.7%

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

10. Applied egg-rr 63.7%

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

11. Final simplification 63.7%

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

Alternative 14: 59.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \frac{\frac{\ell}{k_m} \cdot \frac{\ell}{k_m}}{{t_m}^{3}} \end{array} \]

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 (/ (* (/ l k_m) (/ l k_m)) (pow t_m 3.0))))

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 * (((l / k_m) * (l / k_m)) / pow(t_m, 3.0));
}

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 * (((l / k_m) * (l / k_m)) / (t_m ** 3.0d0))
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 * (((l / k_m) * (l / k_m)) / Math.pow(t_m, 3.0));
}

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 * (((l / k_m) * (l / k_m)) / math.pow(t_m, 3.0))

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(Float64(Float64(l / k_m) * Float64(l / k_m)) / (t_m ^ 3.0)))
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 * (((l / k_m) * (l / k_m)) / (t_m ^ 3.0));
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[(N[(N[(l / k$95$m), $MachinePrecision] * N[(l / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.0%

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

3. Simplified 52.2%

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

4. Taylor expanded in k around 0 47.6%

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

   1. add-sqr-sqrt 36.2%

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

   2. pow2 36.2%

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

   3. sqrt-div 22.7%

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

   4. unpow2 22.7%

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

   5. sqrt-prod 11.9%

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

   6. add-sqr-sqrt 24.7%

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

   7. sqrt-prod 24.6%

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

   8. unpow2 24.6%

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

   9. sqrt-prod 11.6%

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

   10. add-sqr-sqrt 27.7%

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

   11. sqrt-pow1 29.8%

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

   12. metadata-eval 29.8%

       \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2} \]

6. Applied egg-rr 29.8%

   \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]
7. Step-by-step derivation

   1. unpow2 29.8%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot {t}^{1.5}} \cdot \frac{\ell}{k \cdot {t}^{1.5}}} \]

   2. associate-/r* 29.7%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}} \cdot \frac{\ell}{k \cdot {t}^{1.5}} \]

   3. associate-/r* 30.1%

      \[\leadsto \frac{\frac{\ell}{k}}{{t}^{1.5}} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{1.5}}} \]

   4. frac-times 26.6%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{1.5} \cdot {t}^{1.5}}} \]

   5. pow-prod-up 59.5%

      \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{{t}^{\left(1.5 + 1.5\right)}}} \]

   6. metadata-eval 59.5%

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

8. Applied egg-rr 59.5%

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

9. Final simplification 59.5%

   \[\leadsto \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}} \]

Reproduce

herbie shell --seed 2023325 
(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))))