Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.9% → 86.2%

Time: 1.2min

Alternatives: 19

Speedup: 35.0×

• 39.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.9% 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: 86.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.6:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k\_m \cdot \sin k\_m} \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 1.85 \cdot 10^{+121}:\\ \;\;\;\;\frac{\frac{2}{{k\_m}^{2}}}{\tan k\_m \cdot \left(\sin k\_m \cdot t\_m\right)} \cdot \left(\ell \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\sqrt{2} \cdot \frac{t\_m}{k\_m}\right) \cdot \frac{\ell}{{t\_m}^{1.5}}\right)}^{2} \cdot \frac{1}{\sin k\_m \cdot \tan k\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.6)
    (pow
     (* (/ (* l (sqrt 2.0)) (* k_m (sin k_m))) (sqrt (/ (cos k_m) t_m)))
     2.0)
    (if (<= k_m 1.85e+121)
      (* (/ (/ 2.0 (pow k_m 2.0)) (* (tan k_m) (* (sin k_m) t_m))) (* l l))
      (*
       (pow (* (* (sqrt 2.0) (/ t_m k_m)) (/ l (pow t_m 1.5))) 2.0)
       (/ 1.0 (* (sin k_m) (tan 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 <= 1.6) {
		tmp = pow((((l * sqrt(2.0)) / (k_m * sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	} else if (k_m <= 1.85e+121) {
		tmp = ((2.0 / pow(k_m, 2.0)) / (tan(k_m) * (sin(k_m) * t_m))) * (l * l);
	} else {
		tmp = pow(((sqrt(2.0) * (t_m / k_m)) * (l / pow(t_m, 1.5))), 2.0) * (1.0 / (sin(k_m) * tan(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 <= 1.6d0) then
        tmp = (((l * sqrt(2.0d0)) / (k_m * sin(k_m))) * sqrt((cos(k_m) / t_m))) ** 2.0d0
    else if (k_m <= 1.85d+121) then
        tmp = ((2.0d0 / (k_m ** 2.0d0)) / (tan(k_m) * (sin(k_m) * t_m))) * (l * l)
    else
        tmp = (((sqrt(2.0d0) * (t_m / k_m)) * (l / (t_m ** 1.5d0))) ** 2.0d0) * (1.0d0 / (sin(k_m) * tan(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 <= 1.6) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / (k_m * Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else if (k_m <= 1.85e+121) {
		tmp = ((2.0 / Math.pow(k_m, 2.0)) / (Math.tan(k_m) * (Math.sin(k_m) * t_m))) * (l * l);
	} else {
		tmp = Math.pow(((Math.sqrt(2.0) * (t_m / k_m)) * (l / Math.pow(t_m, 1.5))), 2.0) * (1.0 / (Math.sin(k_m) * Math.tan(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 <= 1.6:
		tmp = math.pow((((l * math.sqrt(2.0)) / (k_m * math.sin(k_m))) * math.sqrt((math.cos(k_m) / t_m))), 2.0)
	elif k_m <= 1.85e+121:
		tmp = ((2.0 / math.pow(k_m, 2.0)) / (math.tan(k_m) * (math.sin(k_m) * t_m))) * (l * l)
	else:
		tmp = math.pow(((math.sqrt(2.0) * (t_m / k_m)) * (l / math.pow(t_m, 1.5))), 2.0) * (1.0 / (math.sin(k_m) * math.tan(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 <= 1.6)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / Float64(k_m * sin(k_m))) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	elseif (k_m <= 1.85e+121)
		tmp = Float64(Float64(Float64(2.0 / (k_m ^ 2.0)) / Float64(tan(k_m) * Float64(sin(k_m) * t_m))) * Float64(l * l));
	else
		tmp = Float64((Float64(Float64(sqrt(2.0) * Float64(t_m / k_m)) * Float64(l / (t_m ^ 1.5))) ^ 2.0) * Float64(1.0 / Float64(sin(k_m) * tan(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 <= 1.6)
		tmp = (((l * sqrt(2.0)) / (k_m * sin(k_m))) * sqrt((cos(k_m) / t_m))) ^ 2.0;
	elseif (k_m <= 1.85e+121)
		tmp = ((2.0 / (k_m ^ 2.0)) / (tan(k_m) * (sin(k_m) * t_m))) * (l * l);
	else
		tmp = (((sqrt(2.0) * (t_m / k_m)) * (l / (t_m ^ 1.5))) ^ 2.0) * (1.0 / (sin(k_m) * tan(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, 1.6], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 1.85e+121], N[(N[(N[(2.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.6000000000000001

   1. Initial program 36.9%

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

      1. *-commutative 36.9%

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

      2. associate-/r* 36.9%

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

   3. Simplified 42.3%

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

      1. add-sqr-sqrt 26.2%

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

      2. sqrt-div 19.7%

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

      3. sqrt-div 19.7%

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

   6. Applied egg-rr 30.3%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 30.3%

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

      2. associate-/r* 30.3%

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

      3. associate-/r/ 31.3%

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

      4. associate-/r/ 31.3%

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

      5. *-commutative 31.3%

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

   8. Simplified 31.3%

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

   9. Taylor expanded in t around 0 47.8%

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

   if 1.6000000000000001 < k < 1.85000000000000006e121

   1. Initial program 28.8%

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

   3. Simplified 39.0%

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

      1. add-log-exp 17.3%

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

      2. exp-prod 21.0%

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

      3. associate-*r* 21.0%

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

      4. *-commutative 21.0%

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

   6. Applied egg-rr 21.0%

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

   7. Taylor expanded in k around inf 78.2%

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

      1. associate-*r* 78.2%

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

   9. Simplified 78.2%

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

       1. *-un-lft-identity 78.2%

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

       2. associate-/r* 81.1%

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

       3. *-commutative 81.1%

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

   11. Applied egg-rr 81.1%

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

   if 1.85000000000000006e121 < k

   1. Initial program 32.9%

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

      1. *-commutative 32.9%

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

      2. associate-/r* 32.9%

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

   3. Simplified 41.0%

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

      1. add-sqr-sqrt 40.9%

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

      2. sqrt-div 16.4%

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

      3. sqrt-div 16.4%

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

   6. Applied egg-rr 16.8%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 16.8%

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

      2. associate-/r* 16.8%

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

      3. associate-/r/ 16.8%

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

      4. associate-/r/ 16.8%

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

      5. *-commutative 16.8%

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

   8. Simplified 16.8%

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

      1. div-inv 16.8%

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

      2. unpow-prod-down 16.8%

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

      3. associate-*l/ 16.8%

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

      4. pow1/2 16.8%

         \[\leadsto {\left(\frac{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \ell}{{t}^{1.5}}\right)}^{2} \cdot {\left(\frac{1}{\color{blue}{{\left(\sin k \cdot \tan k\right)}^{0.5}}}\right)}^{2} \]

      5. pow-flip 16.8%

         \[\leadsto {\left(\frac{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \ell}{{t}^{1.5}}\right)}^{2} \cdot {\color{blue}{\left({\left(\sin k \cdot \tan k\right)}^{\left(-0.5\right)}\right)}}^{2} \]

      6. *-commutative 16.8%

         \[\leadsto {\left(\frac{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \ell}{{t}^{1.5}}\right)}^{2} \cdot {\left({\color{blue}{\left(\tan k \cdot \sin k\right)}}^{\left(-0.5\right)}\right)}^{2} \]

      7. metadata-eval 16.8%

         \[\leadsto {\left(\frac{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \ell}{{t}^{1.5}}\right)}^{2} \cdot {\left({\left(\tan k \cdot \sin k\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   10. Applied egg-rr 16.8%

       \[\leadsto \color{blue}{{\left(\frac{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \ell}{{t}^{1.5}}\right)}^{2} \cdot {\left({\left(\tan k \cdot \sin k\right)}^{-0.5}\right)}^{2}} \]
   11. Step-by-step derivation

       1. associate-/l* 16.8%

          \[\leadsto {\color{blue}{\left(\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \frac{\ell}{{t}^{1.5}}\right)}}^{2} \cdot {\left({\left(\tan k \cdot \sin k\right)}^{-0.5}\right)}^{2} \]

       2. *-commutative 16.8%

          \[\leadsto {\left(\color{blue}{\left(\frac{\sqrt{2}}{k} \cdot t\right)} \cdot \frac{\ell}{{t}^{1.5}}\right)}^{2} \cdot {\left({\left(\tan k \cdot \sin k\right)}^{-0.5}\right)}^{2} \]

       3. associate-*l/ 16.8%

          \[\leadsto {\left(\color{blue}{\frac{\sqrt{2} \cdot t}{k}} \cdot \frac{\ell}{{t}^{1.5}}\right)}^{2} \cdot {\left({\left(\tan k \cdot \sin k\right)}^{-0.5}\right)}^{2} \]

       4. associate-/l* 16.8%

          \[\leadsto {\left(\color{blue}{\left(\sqrt{2} \cdot \frac{t}{k}\right)} \cdot \frac{\ell}{{t}^{1.5}}\right)}^{2} \cdot {\left({\left(\tan k \cdot \sin k\right)}^{-0.5}\right)}^{2} \]

       5. unpow2 16.8%

          \[\leadsto {\left(\left(\sqrt{2} \cdot \frac{t}{k}\right) \cdot \frac{\ell}{{t}^{1.5}}\right)}^{2} \cdot \color{blue}{\left({\left(\tan k \cdot \sin k\right)}^{-0.5} \cdot {\left(\tan k \cdot \sin k\right)}^{-0.5}\right)} \]

       6. pow-sqr 33.1%

          \[\leadsto {\left(\left(\sqrt{2} \cdot \frac{t}{k}\right) \cdot \frac{\ell}{{t}^{1.5}}\right)}^{2} \cdot \color{blue}{{\left(\tan k \cdot \sin k\right)}^{\left(2 \cdot -0.5\right)}} \]

       7. metadata-eval 33.1%

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

       8. unpow-1 33.1%

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

       9. *-commutative 33.1%

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

   12. Simplified 33.1%

       \[\leadsto \color{blue}{{\left(\left(\sqrt{2} \cdot \frac{t}{k}\right) \cdot \frac{\ell}{{t}^{1.5}}\right)}^{2} \cdot \frac{1}{\sin k \cdot \tan k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.85 \cdot 10^{+121}:\\ \;\;\;\;\frac{\frac{2}{{k}^{2}}}{\tan k \cdot \left(\sin k \cdot t\right)} \cdot \left(\ell \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\sqrt{2} \cdot \frac{t}{k}\right) \cdot \frac{\ell}{{t}^{1.5}}\right)}^{2} \cdot \frac{1}{\sin k \cdot \tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt{2}}{k\_m}\\ t_3 := t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.058:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k\_m \cdot \sin k\_m} \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_2 \cdot \frac{t\_m}{{\left(t\_3 \cdot \sqrt[3]{\log \left(e^{\sin k\_m}\right) \cdot \tan k\_m}\right)}^{2}}\right) \cdot \left(t\_2 \cdot \frac{t\_m}{t\_3}\right)}{\sqrt[3]{\sin k\_m \cdot \tan k\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (sqrt 2.0) k_m)) (t_3 (* t_m (pow (cbrt l) -2.0))))
   (*
    t_s
    (if (<= k_m 0.058)
      (pow
       (* (/ (* l (sqrt 2.0)) (* k_m (sin k_m))) (sqrt (/ (cos k_m) t_m)))
       2.0)
      (/
       (*
        (*
         t_2
         (/ t_m (pow (* t_3 (cbrt (* (log (exp (sin k_m))) (tan k_m)))) 2.0)))
        (* t_2 (/ t_m t_3)))
       (cbrt (* (sin k_m) (tan k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = sqrt(2.0) / k_m;
	double t_3 = t_m * pow(cbrt(l), -2.0);
	double tmp;
	if (k_m <= 0.058) {
		tmp = pow((((l * sqrt(2.0)) / (k_m * sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = ((t_2 * (t_m / pow((t_3 * cbrt((log(exp(sin(k_m))) * tan(k_m)))), 2.0))) * (t_2 * (t_m / t_3))) / cbrt((sin(k_m) * tan(k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.sqrt(2.0) / k_m;
	double t_3 = t_m * Math.pow(Math.cbrt(l), -2.0);
	double tmp;
	if (k_m <= 0.058) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / (k_m * Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = ((t_2 * (t_m / Math.pow((t_3 * Math.cbrt((Math.log(Math.exp(Math.sin(k_m))) * Math.tan(k_m)))), 2.0))) * (t_2 * (t_m / t_3))) / Math.cbrt((Math.sin(k_m) * Math.tan(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(sqrt(2.0) / k_m)
	t_3 = Float64(t_m * (cbrt(l) ^ -2.0))
	tmp = 0.0
	if (k_m <= 0.058)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / Float64(k_m * sin(k_m))) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(t_2 * Float64(t_m / (Float64(t_3 * cbrt(Float64(log(exp(sin(k_m))) * tan(k_m)))) ^ 2.0))) * Float64(t_2 * Float64(t_m / t_3))) / cbrt(Float64(sin(k_m) * tan(k_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 0.058], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(t$95$2 * N[(t$95$m / N[Power[N[(t$95$3 * N[Power[N[(N[Log[N[Exp[N[Sin[k$95$m], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(t$95$m / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0580000000000000029

   1. Initial program 37.0%

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

      1. *-commutative 37.0%

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

      2. associate-/r* 37.1%

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

   3. Simplified 42.5%

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

      1. add-sqr-sqrt 26.3%

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

      2. sqrt-div 19.8%

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

      3. sqrt-div 19.8%

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

   6. Applied egg-rr 30.5%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 30.5%

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

      2. associate-/r* 30.5%

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

      3. associate-/r/ 31.5%

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

      4. associate-/r/ 31.4%

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

      5. *-commutative 31.4%

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

   8. Simplified 31.4%

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

   9. Taylor expanded in t around 0 48.0%

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

   if 0.0580000000000000029 < k

   1. Initial program 30.6%

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

      1. *-commutative 30.6%

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

      2. associate-/r* 30.7%

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

   3. Simplified 39.6%

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

      1. add-sqr-sqrt 39.6%

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

      2. add-cube-cbrt 39.5%

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

      3. times-frac 39.5%

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

   6. Applied egg-rr 77.6%

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

      1. associate-/r/ 77.6%

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

      2. associate-/r* 77.6%

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

      3. associate-/r/ 77.7%

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

   8. Simplified 77.7%

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

      1. *-commutative 77.7%

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

      2. cbrt-prod 77.8%

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

   10. Applied egg-rr 77.8%

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

       1. associate-*r/ 77.7%

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

   12. Applied egg-rr 79.7%

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

       1. add-log-exp 79.7%

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

   14. Applied egg-rr 79.7%

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

Alternative 3: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t_3 := \frac{\sqrt{2}}{k\_m}\\ t_4 := t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.6:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k\_m \cdot \sin k\_m} \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_3 \cdot \frac{t\_m}{t\_4}\right) \cdot \left(t\_3 \cdot \frac{t\_m}{{\left(t\_4 \cdot t\_2\right)}^{2}}\right)}{t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (cbrt (* (sin k_m) (tan k_m))))
        (t_3 (/ (sqrt 2.0) k_m))
        (t_4 (* t_m (pow (cbrt l) -2.0))))
   (*
    t_s
    (if (<= k_m 1.6)
      (pow
       (* (/ (* l (sqrt 2.0)) (* k_m (sin k_m))) (sqrt (/ (cos k_m) t_m)))
       2.0)
      (/ (* (* t_3 (/ t_m t_4)) (* t_3 (/ t_m (pow (* t_4 t_2) 2.0)))) t_2)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = cbrt((sin(k_m) * tan(k_m)));
	double t_3 = sqrt(2.0) / k_m;
	double t_4 = t_m * pow(cbrt(l), -2.0);
	double tmp;
	if (k_m <= 1.6) {
		tmp = pow((((l * sqrt(2.0)) / (k_m * sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = ((t_3 * (t_m / t_4)) * (t_3 * (t_m / pow((t_4 * t_2), 2.0)))) / t_2;
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double t_3 = Math.sqrt(2.0) / k_m;
	double t_4 = t_m * Math.pow(Math.cbrt(l), -2.0);
	double tmp;
	if (k_m <= 1.6) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / (k_m * Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = ((t_3 * (t_m / t_4)) * (t_3 * (t_m / Math.pow((t_4 * t_2), 2.0)))) / t_2;
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = cbrt(Float64(sin(k_m) * tan(k_m)))
	t_3 = Float64(sqrt(2.0) / k_m)
	t_4 = Float64(t_m * (cbrt(l) ^ -2.0))
	tmp = 0.0
	if (k_m <= 1.6)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / Float64(k_m * sin(k_m))) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(t_3 * Float64(t_m / t_4)) * Float64(t_3 * Float64(t_m / (Float64(t_4 * t_2) ^ 2.0)))) / t_2);
	end
	return Float64(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[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.6], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(t$95$3 * N[(t$95$m / t$95$4), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[(t$95$m / N[Power[N[(t$95$4 * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.6000000000000001

   1. Initial program 36.9%

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

      1. *-commutative 36.9%

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

      2. associate-/r* 36.9%

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

   3. Simplified 42.3%

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

      1. add-sqr-sqrt 26.2%

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

      2. sqrt-div 19.7%

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

      3. sqrt-div 19.7%

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

   6. Applied egg-rr 30.3%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 30.3%

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

      2. associate-/r* 30.3%

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

      3. associate-/r/ 31.3%

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

      4. associate-/r/ 31.3%

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

      5. *-commutative 31.3%

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

   8. Simplified 31.3%

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

   9. Taylor expanded in t around 0 47.8%

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

   if 1.6000000000000001 < k

   1. Initial program 31.0%

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

      1. *-commutative 31.0%

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

      2. associate-/r* 31.0%

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

   3. Simplified 40.1%

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

      1. add-sqr-sqrt 40.1%

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

      2. add-cube-cbrt 40.0%

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

      3. times-frac 40.0%

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

   6. Applied egg-rr 77.3%

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

      1. associate-/r/ 77.4%

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

      2. associate-/r* 77.4%

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

      3. associate-/r/ 77.4%

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

   8. Simplified 77.4%

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

      1. *-commutative 77.4%

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

      2. cbrt-prod 77.5%

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

   10. Applied egg-rr 77.5%

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

       1. associate-*r/ 77.4%

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

   12. Applied egg-rr 79.4%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{2}}{k} \cdot \frac{t}{t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{t}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t_3 := \frac{\sqrt{2}}{k\_m}\\ t_4 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.6:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k\_m \cdot \sin k\_m} \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_3 \cdot \frac{t\_m}{{\left(\left(t\_m \cdot t\_4\right) \cdot t\_2\right)}^{2}}\right) \cdot \frac{t\_3}{t\_4}}{t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (cbrt (* (sin k_m) (tan k_m))))
        (t_3 (/ (sqrt 2.0) k_m))
        (t_4 (pow (cbrt l) -2.0)))
   (*
    t_s
    (if (<= k_m 1.6)
      (pow
       (* (/ (* l (sqrt 2.0)) (* k_m (sin k_m))) (sqrt (/ (cos k_m) t_m)))
       2.0)
      (/ (* (* t_3 (/ t_m (pow (* (* t_m t_4) t_2) 2.0))) (/ t_3 t_4)) t_2)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = cbrt((sin(k_m) * tan(k_m)));
	double t_3 = sqrt(2.0) / k_m;
	double t_4 = pow(cbrt(l), -2.0);
	double tmp;
	if (k_m <= 1.6) {
		tmp = pow((((l * sqrt(2.0)) / (k_m * sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = ((t_3 * (t_m / pow(((t_m * t_4) * t_2), 2.0))) * (t_3 / t_4)) / t_2;
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double t_3 = Math.sqrt(2.0) / k_m;
	double t_4 = Math.pow(Math.cbrt(l), -2.0);
	double tmp;
	if (k_m <= 1.6) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / (k_m * Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = ((t_3 * (t_m / Math.pow(((t_m * t_4) * t_2), 2.0))) * (t_3 / t_4)) / t_2;
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = cbrt(Float64(sin(k_m) * tan(k_m)))
	t_3 = Float64(sqrt(2.0) / k_m)
	t_4 = cbrt(l) ^ -2.0
	tmp = 0.0
	if (k_m <= 1.6)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / Float64(k_m * sin(k_m))) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(t_3 * Float64(t_m / (Float64(Float64(t_m * t_4) * t_2) ^ 2.0))) * Float64(t_3 / t_4)) / t_2);
	end
	return Float64(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[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.6], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(t$95$3 * N[(t$95$m / N[Power[N[(N[(t$95$m * t$95$4), $MachinePrecision] * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 / t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.6000000000000001

   1. Initial program 36.9%

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

      1. *-commutative 36.9%

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

      2. associate-/r* 36.9%

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

   3. Simplified 42.3%

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

      1. add-sqr-sqrt 26.2%

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

      2. sqrt-div 19.7%

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

      3. sqrt-div 19.7%

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

   6. Applied egg-rr 30.3%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 30.3%

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

      2. associate-/r* 30.3%

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

      3. associate-/r/ 31.3%

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

      4. associate-/r/ 31.3%

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

      5. *-commutative 31.3%

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

   8. Simplified 31.3%

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

   9. Taylor expanded in t around 0 47.8%

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

   if 1.6000000000000001 < k

   1. Initial program 31.0%

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

      1. *-commutative 31.0%

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

      2. associate-/r* 31.0%

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

   3. Simplified 40.1%

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

      1. add-sqr-sqrt 40.1%

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

      2. add-cube-cbrt 40.0%

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

      3. times-frac 40.0%

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

   6. Applied egg-rr 77.3%

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

      1. associate-/r/ 77.4%

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

      2. associate-/r* 77.4%

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

      3. associate-/r/ 77.4%

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

   8. Simplified 77.4%

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

      1. *-commutative 77.4%

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

      2. cbrt-prod 77.5%

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

   10. Applied egg-rr 77.5%

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

       1. associate-*r/ 77.4%

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

   12. Applied egg-rr 79.4%

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

       1. associate-/r* 79.3%

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

       2. pow1 79.3%

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

       3. pow1 79.3%

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

       4. pow-div 79.3%

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

       5. metadata-eval 79.3%

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

       6. metadata-eval 79.3%

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

       7. div-inv 79.4%

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

   14. Applied egg-rr 79.4%

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

Alternative 5: 67.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t_3 := \frac{\sqrt{2}}{k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.6:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k\_m \cdot \sin k\_m} \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_3 \cdot \frac{t\_m}{{\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot t\_2\right)}^{2}}\right) \cdot \left(t\_3 \cdot {\ell}^{0.6666666666666666}\right)}{t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (cbrt (* (sin k_m) (tan k_m)))) (t_3 (/ (sqrt 2.0) k_m)))
   (*
    t_s
    (if (<= k_m 1.6)
      (pow
       (* (/ (* l (sqrt 2.0)) (* k_m (sin k_m))) (sqrt (/ (cos k_m) t_m)))
       2.0)
      (/
       (*
        (* t_3 (/ t_m (pow (* (* t_m (pow (cbrt l) -2.0)) t_2) 2.0)))
        (* t_3 (pow l 0.6666666666666666)))
       t_2)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = cbrt((sin(k_m) * tan(k_m)));
	double t_3 = sqrt(2.0) / k_m;
	double tmp;
	if (k_m <= 1.6) {
		tmp = pow((((l * sqrt(2.0)) / (k_m * sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = ((t_3 * (t_m / pow(((t_m * pow(cbrt(l), -2.0)) * t_2), 2.0))) * (t_3 * pow(l, 0.6666666666666666))) / t_2;
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double t_3 = Math.sqrt(2.0) / k_m;
	double tmp;
	if (k_m <= 1.6) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / (k_m * Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = ((t_3 * (t_m / Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * t_2), 2.0))) * (t_3 * Math.pow(l, 0.6666666666666666))) / t_2;
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = cbrt(Float64(sin(k_m) * tan(k_m)))
	t_3 = Float64(sqrt(2.0) / k_m)
	tmp = 0.0
	if (k_m <= 1.6)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / Float64(k_m * sin(k_m))) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(t_3 * Float64(t_m / (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * t_2) ^ 2.0))) * Float64(t_3 * (l ^ 0.6666666666666666))) / t_2);
	end
	return Float64(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[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.6], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(t$95$3 * N[(t$95$m / N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[Power[l, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.6000000000000001

   1. Initial program 36.9%

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

      1. *-commutative 36.9%

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

      2. associate-/r* 36.9%

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

   3. Simplified 42.3%

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

      1. add-sqr-sqrt 26.2%

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

      2. sqrt-div 19.7%

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

      3. sqrt-div 19.7%

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

   6. Applied egg-rr 30.3%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 30.3%

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

      2. associate-/r* 30.3%

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

      3. associate-/r/ 31.3%

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

      4. associate-/r/ 31.3%

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

      5. *-commutative 31.3%

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

   8. Simplified 31.3%

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

   9. Taylor expanded in t around 0 47.8%

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

   if 1.6000000000000001 < k

   1. Initial program 31.0%

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

      1. *-commutative 31.0%

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

      2. associate-/r* 31.0%

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

   3. Simplified 40.1%

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

      1. add-sqr-sqrt 40.1%

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

      2. add-cube-cbrt 40.0%

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

      3. times-frac 40.0%

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

   6. Applied egg-rr 77.3%

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

      1. associate-/r/ 77.4%

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

      2. associate-/r* 77.4%

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

      3. associate-/r/ 77.4%

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

   8. Simplified 77.4%

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

      1. *-commutative 77.4%

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

      2. cbrt-prod 77.5%

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

   10. Applied egg-rr 77.5%

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

       1. associate-*r/ 77.4%

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

   12. Applied egg-rr 79.4%

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

       1. associate-/r* 79.3%

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

       2. pow1 79.3%

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

       3. pow1 79.3%

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

       4. pow-div 79.3%

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

       5. metadata-eval 79.3%

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

       6. metadata-eval 79.3%

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

       7. pow-flip 79.4%

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

       8. metadata-eval 79.4%

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

       9. pow2 79.4%

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

       10. pow1/3 41.8%

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

       11. pow1/3 41.5%

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

       12. pow-prod-up 41.5%

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

       13. metadata-eval 41.5%

           \[\leadsto \frac{\left(\frac{\sqrt{2}}{k} \cdot \frac{t}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot {\ell}^{\color{blue}{0.6666666666666666}}\right)}{\sqrt[3]{\sin k \cdot \tan k}} \]

   14. Applied egg-rr 41.5%

       \[\leadsto \frac{\left(\frac{\sqrt{2}}{k} \cdot \frac{t}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \color{blue}{{\ell}^{0.6666666666666666}}\right)}{\sqrt[3]{\sin k \cdot \tan k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 86.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \ell \cdot \sqrt{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-295}:\\ \;\;\;\;{\left(\frac{t\_2}{{k\_m}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+236}:\\ \;\;\;\;\frac{2}{{k\_m}^{2}} \cdot \frac{{\ell}^{2}}{\sin k\_m \cdot \left(t\_m \cdot \tan k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t\_2}{k\_m \cdot \sin k\_m} \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* l (sqrt 2.0))))
   (*
    t_s
    (if (<= (* l l) 1e-295)
      (pow (* (/ t_2 (pow k_m 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
      (if (<= (* l l) 2e+236)
        (*
         (/ 2.0 (pow k_m 2.0))
         (/ (pow l 2.0) (* (sin k_m) (* t_m (tan k_m)))))
        (pow (* (/ t_2 (* k_m (sin k_m))) (sqrt (/ (cos 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 t_2 = l * sqrt(2.0);
	double tmp;
	if ((l * l) <= 1e-295) {
		tmp = pow(((t_2 / pow(k_m, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 2e+236) {
		tmp = (2.0 / pow(k_m, 2.0)) * (pow(l, 2.0) / (sin(k_m) * (t_m * tan(k_m))));
	} else {
		tmp = pow(((t_2 / (k_m * sin(k_m))) * sqrt((cos(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) :: t_2
    real(8) :: tmp
    t_2 = l * sqrt(2.0d0)
    if ((l * l) <= 1d-295) then
        tmp = ((t_2 / (k_m ** 2.0d0)) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else if ((l * l) <= 2d+236) then
        tmp = (2.0d0 / (k_m ** 2.0d0)) * ((l ** 2.0d0) / (sin(k_m) * (t_m * tan(k_m))))
    else
        tmp = ((t_2 / (k_m * sin(k_m))) * sqrt((cos(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 t_2 = l * Math.sqrt(2.0);
	double tmp;
	if ((l * l) <= 1e-295) {
		tmp = Math.pow(((t_2 / Math.pow(k_m, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else if ((l * l) <= 2e+236) {
		tmp = (2.0 / Math.pow(k_m, 2.0)) * (Math.pow(l, 2.0) / (Math.sin(k_m) * (t_m * Math.tan(k_m))));
	} else {
		tmp = Math.pow(((t_2 / (k_m * Math.sin(k_m))) * Math.sqrt((Math.cos(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):
	t_2 = l * math.sqrt(2.0)
	tmp = 0
	if (l * l) <= 1e-295:
		tmp = math.pow(((t_2 / math.pow(k_m, 2.0)) * math.sqrt((1.0 / t_m))), 2.0)
	elif (l * l) <= 2e+236:
		tmp = (2.0 / math.pow(k_m, 2.0)) * (math.pow(l, 2.0) / (math.sin(k_m) * (t_m * math.tan(k_m))))
	else:
		tmp = math.pow(((t_2 / (k_m * math.sin(k_m))) * math.sqrt((math.cos(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)
	t_2 = Float64(l * sqrt(2.0))
	tmp = 0.0
	if (Float64(l * l) <= 1e-295)
		tmp = Float64(Float64(t_2 / (k_m ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	elseif (Float64(l * l) <= 2e+236)
		tmp = Float64(Float64(2.0 / (k_m ^ 2.0)) * Float64((l ^ 2.0) / Float64(sin(k_m) * Float64(t_m * tan(k_m)))));
	else
		tmp = Float64(Float64(t_2 / Float64(k_m * sin(k_m))) * sqrt(Float64(cos(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)
	t_2 = l * sqrt(2.0);
	tmp = 0.0;
	if ((l * l) <= 1e-295)
		tmp = ((t_2 / (k_m ^ 2.0)) * sqrt((1.0 / t_m))) ^ 2.0;
	elseif ((l * l) <= 2e+236)
		tmp = (2.0 / (k_m ^ 2.0)) * ((l ^ 2.0) / (sin(k_m) * (t_m * tan(k_m))));
	else
		tmp = ((t_2 / (k_m * sin(k_m))) * sqrt((cos(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_] := Block[{t$95$2 = N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-295], N[Power[N[(N[(t$95$2 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 2e+236], N[(N[(2.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Sin[k$95$m], $MachinePrecision] * N[(t$95$m * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(t$95$2 / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 1.00000000000000006e-295

   1. Initial program 14.7%

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

      1. *-commutative 14.7%

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

      2. associate-/r* 14.7%

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

   3. Simplified 28.1%

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

      1. add-sqr-sqrt 28.0%

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

      2. sqrt-div 13.3%

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

      3. sqrt-div 13.3%

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

   6. Applied egg-rr 15.9%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 15.9%

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

      2. associate-/r* 15.9%

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

      3. associate-/r/ 18.4%

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

      4. associate-/r/ 18.4%

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

      5. *-commutative 18.4%

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

   8. Simplified 18.4%

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

   9. Taylor expanded in k around 0 34.3%

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

       1. *-commutative 34.3%

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

   11. Simplified 34.3%

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

   if 1.00000000000000006e-295 < (*.f64 l l) < 2.00000000000000011e236

   1. Initial program 43.1%

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

   3. Simplified 52.7%

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

      1. add-log-exp 27.6%

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

      2. exp-prod 37.1%

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

      3. associate-*r* 37.1%

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

      4. *-commutative 37.1%

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

   6. Applied egg-rr 37.1%

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

   7. Taylor expanded in k around inf 90.5%

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

      1. associate-*r* 90.4%

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

   9. Simplified 90.4%

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

       1. associate-*l/ 90.5%

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

       2. pow2 90.5%

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

       3. *-commutative 90.5%

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

   11. Applied egg-rr 90.5%

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

       1. times-frac 92.3%

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

       2. associate-*r* 92.2%

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

   13. Simplified 92.2%

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

   if 2.00000000000000011e236 < (*.f64 l l)

   1. Initial program 45.2%

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

      1. *-commutative 45.2%

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

      2. associate-/r* 45.2%

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

   3. Simplified 45.3%

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

      1. add-sqr-sqrt 22.0%

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

      2. sqrt-div 22.0%

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

      3. sqrt-div 22.0%

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

   6. Applied egg-rr 28.3%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 28.3%

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

      2. associate-/r* 28.3%

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

      3. associate-/r/ 28.3%

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

      4. associate-/r/ 28.3%

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

      5. *-commutative 28.3%

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

   8. Simplified 28.3%

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

   9. Taylor expanded in t around 0 49.8%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-295}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+236}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{\sin k \cdot \left(t \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.06 \cdot 10^{-14}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k\_m}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{k\_m}^{2}}}{\tan k\_m \cdot \left(\sin k\_m \cdot t\_m\right)} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.06e-14)
    (pow (* (/ (* l (sqrt 2.0)) (pow k_m 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
    (* (/ (/ 2.0 (pow k_m 2.0)) (* (tan k_m) (* (sin k_m) t_m))) (* l l)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.06e-14) {
		tmp = pow((((l * sqrt(2.0)) / pow(k_m, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = ((2.0 / pow(k_m, 2.0)) / (tan(k_m) * (sin(k_m) * t_m))) * (l * l);
	}
	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 <= 1.06d-14) then
        tmp = (((l * sqrt(2.0d0)) / (k_m ** 2.0d0)) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = ((2.0d0 / (k_m ** 2.0d0)) / (tan(k_m) * (sin(k_m) * t_m))) * (l * l)
    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 <= 1.06e-14) {
		tmp = Math.pow((((l * Math.sqrt(2.0)) / Math.pow(k_m, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = ((2.0 / Math.pow(k_m, 2.0)) / (Math.tan(k_m) * (Math.sin(k_m) * t_m))) * (l * l);
	}
	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 <= 1.06e-14:
		tmp = math.pow((((l * math.sqrt(2.0)) / math.pow(k_m, 2.0)) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = ((2.0 / math.pow(k_m, 2.0)) / (math.tan(k_m) * (math.sin(k_m) * t_m))) * (l * l)
	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 <= 1.06e-14)
		tmp = Float64(Float64(Float64(l * sqrt(2.0)) / (k_m ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(2.0 / (k_m ^ 2.0)) / Float64(tan(k_m) * Float64(sin(k_m) * t_m))) * Float64(l * l));
	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 <= 1.06e-14)
		tmp = (((l * sqrt(2.0)) / (k_m ^ 2.0)) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = ((2.0 / (k_m ^ 2.0)) / (tan(k_m) * (sin(k_m) * t_m))) * (l * l);
	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, 1.06e-14], N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(2.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.06e-14

   1. Initial program 37.6%

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

      1. *-commutative 37.6%

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

      2. associate-/r* 37.6%

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

   3. Simplified 42.6%

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

      1. add-sqr-sqrt 26.7%

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

      2. sqrt-div 20.1%

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

      3. sqrt-div 20.1%

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

   6. Applied egg-rr 30.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 30.4%

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

      2. associate-/r* 30.4%

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

      3. associate-/r/ 31.4%

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

      4. associate-/r/ 31.4%

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

      5. *-commutative 31.4%

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

   8. Simplified 31.4%

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

   9. Taylor expanded in k around 0 42.2%

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

       1. *-commutative 42.2%

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

   11. Simplified 42.2%

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

   if 1.06e-14 < k

   1. Initial program 29.4%

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

   3. Simplified 40.7%

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

      1. add-log-exp 28.6%

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

      2. exp-prod 26.2%

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

      3. associate-*r* 26.2%

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

      4. *-commutative 26.2%

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

   6. Applied egg-rr 26.2%

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

   7. Taylor expanded in k around inf 68.8%

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

      1. associate-*r* 68.7%

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

   9. Simplified 68.7%

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

       1. *-un-lft-identity 68.7%

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

       2. associate-/r* 70.0%

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

       3. *-commutative 70.0%

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

   11. Applied egg-rr 70.0%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.06 \cdot 10^{-14}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{k}^{2}}}{\tan k \cdot \left(\sin k \cdot t\right)} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.4 \cdot 10^{-14}:\\ \;\;\;\;{\left(\left(\ell \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\frac{1}{t\_m}}}{{k\_m}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{k\_m}^{2}}}{\tan k\_m \cdot \left(\sin k\_m \cdot t\_m\right)} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.4e-14)
    (pow (* (* l (sqrt 2.0)) (/ (sqrt (/ 1.0 t_m)) (pow k_m 2.0))) 2.0)
    (* (/ (/ 2.0 (pow k_m 2.0)) (* (tan k_m) (* (sin k_m) t_m))) (* l l)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.4e-14) {
		tmp = pow(((l * sqrt(2.0)) * (sqrt((1.0 / t_m)) / pow(k_m, 2.0))), 2.0);
	} else {
		tmp = ((2.0 / pow(k_m, 2.0)) / (tan(k_m) * (sin(k_m) * t_m))) * (l * l);
	}
	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 <= 1.4d-14) then
        tmp = ((l * sqrt(2.0d0)) * (sqrt((1.0d0 / t_m)) / (k_m ** 2.0d0))) ** 2.0d0
    else
        tmp = ((2.0d0 / (k_m ** 2.0d0)) / (tan(k_m) * (sin(k_m) * t_m))) * (l * l)
    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 <= 1.4e-14) {
		tmp = Math.pow(((l * Math.sqrt(2.0)) * (Math.sqrt((1.0 / t_m)) / Math.pow(k_m, 2.0))), 2.0);
	} else {
		tmp = ((2.0 / Math.pow(k_m, 2.0)) / (Math.tan(k_m) * (Math.sin(k_m) * t_m))) * (l * l);
	}
	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 <= 1.4e-14:
		tmp = math.pow(((l * math.sqrt(2.0)) * (math.sqrt((1.0 / t_m)) / math.pow(k_m, 2.0))), 2.0)
	else:
		tmp = ((2.0 / math.pow(k_m, 2.0)) / (math.tan(k_m) * (math.sin(k_m) * t_m))) * (l * l)
	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 <= 1.4e-14)
		tmp = Float64(Float64(l * sqrt(2.0)) * Float64(sqrt(Float64(1.0 / t_m)) / (k_m ^ 2.0))) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(2.0 / (k_m ^ 2.0)) / Float64(tan(k_m) * Float64(sin(k_m) * t_m))) * Float64(l * l));
	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 <= 1.4e-14)
		tmp = ((l * sqrt(2.0)) * (sqrt((1.0 / t_m)) / (k_m ^ 2.0))) ^ 2.0;
	else
		tmp = ((2.0 / (k_m ^ 2.0)) / (tan(k_m) * (sin(k_m) * t_m))) * (l * l);
	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, 1.4e-14], N[Power[N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(2.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.4e-14

   1. Initial program 37.6%

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

      1. *-commutative 37.6%

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

      2. associate-/r* 37.6%

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

   3. Simplified 42.6%

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

      1. add-sqr-sqrt 26.7%

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

      2. sqrt-div 20.1%

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

      3. sqrt-div 20.1%

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

   6. Applied egg-rr 30.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}} \cdot \frac{\frac{\sqrt{2}}{\frac{k}{t}}}{\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}}} \]
   7. Step-by-step derivation

      1. unpow2 30.4%

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

      2. associate-/r* 30.4%

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

      3. associate-/r/ 31.4%

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

      4. associate-/r/ 31.4%

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

      5. *-commutative 31.4%

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

   8. Simplified 31.4%

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

   9. Taylor expanded in k around 0 42.2%

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

       1. associate-*l/ 41.3%

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

       2. associate-/l* 42.2%

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

       3. *-commutative 42.2%

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

   11. Simplified 42.2%

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

   if 1.4e-14 < k

   1. Initial program 29.4%

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

   3. Simplified 40.7%

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

      1. add-log-exp 28.6%

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

      2. exp-prod 26.2%

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

      3. associate-*r* 26.2%

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

      4. *-commutative 26.2%

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

   6. Applied egg-rr 26.2%

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

   7. Taylor expanded in k around inf 68.8%

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

      1. associate-*r* 68.7%

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

   9. Simplified 68.7%

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

       1. *-un-lft-identity 68.7%

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

       2. associate-/r* 70.0%

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

       3. *-commutative 70.0%

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

   11. Applied egg-rr 70.0%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-14}:\\ \;\;\;\;{\left(\left(\ell \cdot \sqrt{2}\right) \cdot \frac{\sqrt{\frac{1}{t}}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{k}^{2}}}{\tan k \cdot \left(\sin k \cdot t\right)} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.1% accurate, 1.3× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (/ (/ 2.0 (pow k_m 2.0)) (* (tan k_m) (* (sin k_m) t_m))) (* l l))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.3%

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

3. Simplified 43.7%

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

   1. add-log-exp 25.8%

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

   2. exp-prod 32.7%

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

   3. associate-*r* 32.7%

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

   4. *-commutative 32.7%

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

6. Applied egg-rr 32.7%

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

7. Taylor expanded in k around inf 74.4%

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

   1. associate-*r* 74.4%

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

9. Simplified 74.4%

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

    1. *-un-lft-identity 74.4%

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

    2. associate-/r* 74.7%

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

    3. *-commutative 74.7%

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

11. Applied egg-rr 74.7%

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

12. Final simplification 74.7%

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

Alternative 10: 74.1% accurate, 1.3× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (/ (/ 2.0 (pow k_m 2.0)) (* (sin k_m) (* t_m (tan k_m)))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.3%

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

3. Simplified 43.7%

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

   1. add-log-exp 25.8%

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

   2. exp-prod 32.7%

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

   3. associate-*r* 32.7%

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

   4. *-commutative 32.7%

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

6. Applied egg-rr 32.7%

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

7. Taylor expanded in k around inf 74.4%

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

   1. associate-*r* 74.4%

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

9. Simplified 74.4%

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

    1. div-inv 74.4%

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

    2. *-commutative 74.4%

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

11. Applied egg-rr 74.4%

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

    1. associate-*r/ 74.4%

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

    2. metadata-eval 74.4%

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

    3. associate-/r* 74.7%

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

    4. associate-*r* 74.7%

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

13. Simplified 74.7%

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

14. Final simplification 74.7%

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

Alternative 11: 74.0% accurate, 1.3× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (/ 2.0 (* (* (sin k_m) (tan k_m)) (* t_m (pow k_m 2.0)))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.3%

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

3. Simplified 43.7%

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

   1. add-log-exp 25.8%

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

   2. exp-prod 32.7%

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

   3. associate-*r* 32.7%

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

   4. *-commutative 32.7%

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

6. Applied egg-rr 32.7%

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

7. Taylor expanded in k around inf 74.4%

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

   1. associate-*r* 74.4%

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

9. Simplified 74.4%

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

10. Final simplification 74.4%

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

Alternative 12: 74.0% accurate, 2.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (/ 2.0 (* (* k_m k_m) (* (tan k_m) (* (sin k_m) t_m)))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.3%

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

3. Simplified 43.7%

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

   1. add-log-exp 25.8%

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

   2. exp-prod 32.7%

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

   3. associate-*r* 32.7%

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

   4. *-commutative 32.7%

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

6. Applied egg-rr 32.7%

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

7. Taylor expanded in k around inf 74.4%

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

   1. associate-*r* 74.4%

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

9. Simplified 74.4%

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

    1. unpow2 74.4%

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

11. Applied egg-rr 74.4%

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

12. Final simplification 74.4%

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

Alternative 13: 65.0% accurate, 2.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (/ 2.0 (* (pow k_m 2.0) (* k_m (* (sin k_m) t_m)))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.3%

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

3. Simplified 43.7%

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

   1. add-log-exp 25.8%

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

   2. exp-prod 32.7%

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

   3. associate-*r* 32.7%

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

   4. *-commutative 32.7%

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

6. Applied egg-rr 32.7%

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

7. Taylor expanded in k around inf 74.4%

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

   1. associate-*r* 74.4%

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

9. Simplified 74.4%

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

10. Taylor expanded in k around 0 64.8%

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

11. Final simplification 64.8%

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

Alternative 14: 64.9% accurate, 2.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (/ 2.0 (* (pow k_m 2.0) (* (tan k_m) (* k_m t_m)))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.3%

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

3. Simplified 43.7%

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

   1. add-log-exp 25.8%

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

   2. exp-prod 32.7%

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

   3. associate-*r* 32.7%

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

   4. *-commutative 32.7%

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

6. Applied egg-rr 32.7%

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

7. Taylor expanded in k around inf 74.4%

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

   1. associate-*r* 74.4%

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

9. Simplified 74.4%

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

10. Taylor expanded in k around 0 64.5%

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

11. Final simplification 64.5%

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

Alternative 15: 62.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{4}}}{t\_m}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* 2.0 (/ (/ (pow l 2.0) (pow k_m 4.0)) t_m))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * ((pow(l, 2.0) / pow(k_m, 4.0)) / t_m));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (2.0d0 * (((l ** 2.0d0) / (k_m ** 4.0d0)) / t_m))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * (2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 4.0)) / t_m));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * (2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 4.0)) / t_m))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 4.0)) / t_m)))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 * (((l ^ 2.0) / (k_m ^ 4.0)) / t_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.3%

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

3. Simplified 43.7%

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

5. Taylor expanded in k around 0 62.9%

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

   1. expm1-log1p-u 48.4%

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

   2. expm1-undefine 32.1%

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

   3. *-commutative 32.1%

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

7. Applied egg-rr 32.1%

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

   1. expm1-define 48.4%

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

   2. *-commutative 48.4%

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

9. Simplified 48.4%

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

10. Taylor expanded in k around 0 62.9%

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

    1. associate-/r* 63.2%

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

12. Simplified 63.2%

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

Alternative 16: 62.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k\_m}^{4}}}{t\_m}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (/ (/ 2.0 (pow k_m 4.0)) t_m))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * ((2.0 / pow(k_m, 4.0)) / t_m));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * ((l * l) * ((2.0d0 / (k_m ** 4.0d0)) / t_m))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * ((2.0 / Math.pow(k_m, 4.0)) / t_m));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * ((l * l) * ((2.0 / math.pow(k_m, 4.0)) / t_m))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64(l * l) * Float64(Float64(2.0 / (k_m ^ 4.0)) / t_m)))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * ((l * l) * ((2.0 / (k_m ^ 4.0)) / t_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.3%

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

3. Simplified 43.7%

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

   1. add-cube-cbrt 43.7%

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

   2. pow3 43.7%

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

   3. cbrt-prod 43.7%

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

   4. rem-cbrt-cube 61.2%

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

   5. associate-*r* 61.2%

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

   6. *-commutative 61.2%

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

6. Applied egg-rr 61.2%

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

7. Taylor expanded in k around 0 62.9%

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

   1. associate-/r* 62.9%

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

9. Simplified 62.9%

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

10. Final simplification 62.9%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k}^{4}}}{t} \]
11. Add Preprocessing

Alternative 17: 62.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{2}{t\_m \cdot {k\_m}^{4}}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (/ 2.0 (* t_m (pow k_m 4.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * (2.0 / (t_m * pow(k_m, 4.0))));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * ((l * l) * (2.0d0 / (t_m * (k_m ** 4.0d0))))
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * (2.0 / (t_m * Math.pow(k_m, 4.0))));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * ((l * l) * (2.0 / (t_m * math.pow(k_m, 4.0))))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64(l * l) * Float64(2.0 / Float64(t_m * (k_m ^ 4.0)))))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * ((l * l) * (2.0 / (t_m * (k_m ^ 4.0))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.3%

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

3. Simplified 43.7%

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

5. Taylor expanded in k around 0 62.9%

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

6. Final simplification 62.9%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}} \]
7. Add Preprocessing

Alternative 18: 55.9% accurate, 35.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.6 \cdot 10^{-29}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{0}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot 0\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (if (<= k_m 1.6e-29) (* (* l l) (/ 2.0 0.0)) (* (* l l) 0.0))))

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-29) {
		tmp = (l * l) * (2.0 / 0.0);
	} else {
		tmp = (l * l) * 0.0;
	}
	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-29) then
        tmp = (l * l) * (2.0d0 / 0.0d0)
    else
        tmp = (l * l) * 0.0d0
    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-29) {
		tmp = (l * l) * (2.0 / 0.0);
	} else {
		tmp = (l * l) * 0.0;
	}
	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-29:
		tmp = (l * l) * (2.0 / 0.0)
	else:
		tmp = (l * l) * 0.0
	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-29)
		tmp = Float64(Float64(l * l) * Float64(2.0 / 0.0));
	else
		tmp = Float64(Float64(l * l) * 0.0);
	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-29)
		tmp = (l * l) * (2.0 / 0.0);
	else
		tmp = (l * l) * 0.0;
	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-29], N[(N[(l * l), $MachinePrecision] * N[(2.0 / 0.0), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * 0.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.6e-29

   1. Initial program 37.9%

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

   3. Simplified 44.8%

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

      1. add-log-exp 24.7%

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

      2. exp-prod 35.4%

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

      3. associate-*r* 35.4%

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

      4. *-commutative 35.4%

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

   6. Applied egg-rr 35.4%

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

   7. Taylor expanded in t around 0 27.2%

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

      1. metadata-eval 27.2%

         \[\leadsto \frac{2}{\color{blue}{0}} \cdot \left(\ell \cdot \ell\right) \]

   9. Applied egg-rr 27.2%

      \[\leadsto \frac{2}{\color{blue}{0}} \cdot \left(\ell \cdot \ell\right) \]

   if 1.6e-29 < k

   1. Initial program 29.1%

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

   3. Simplified 41.3%

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

      1. add-log-exp 28.5%

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

      2. exp-prod 26.2%

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

      3. associate-*r* 26.2%

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

      4. *-commutative 26.2%

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

   6. Applied egg-rr 26.2%

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

   7. Taylor expanded in t around 0 5.1%

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

      1. metadata-eval 5.1%

         \[\leadsto \frac{2}{\color{blue}{0}} \cdot \left(\ell \cdot \ell\right) \]

   9. Applied egg-rr 5.1%

      \[\leadsto \frac{2}{\color{blue}{0}} \cdot \left(\ell \cdot \ell\right) \]
   10. Step-by-step derivation

       1. clear-num 5.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{0}{2}}} \cdot \left(\ell \cdot \ell\right) \]

       2. metadata-eval 5.1%

          \[\leadsto \frac{1}{\color{blue}{0}} \cdot \left(\ell \cdot \ell\right) \]

       3. inv-pow 5.1%

          \[\leadsto \color{blue}{{0}^{-1}} \cdot \left(\ell \cdot \ell\right) \]

   11. Applied egg-rr 5.1%

       \[\leadsto \color{blue}{{0}^{-1}} \cdot \left(\ell \cdot \ell\right) \]
   12. Step-by-step derivation

       1. pow-base-0 42.3%

          \[\leadsto \color{blue}{0} \cdot \left(\ell \cdot \ell\right) \]

   13. Simplified 42.3%

       \[\leadsto \color{blue}{0} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-29}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{0}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot 0\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 28.0% accurate, 84.2× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot 0\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m) :precision binary64 (* t_s (* (* l l) 0.0)))

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 * l) * 0.0);
}

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 * l) * 0.0d0)
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 * l) * 0.0);
}

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 * l) * 0.0)

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 * l) * 0.0))
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 * l) * 0.0);
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[(l * l), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.3%

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

3. Simplified 43.7%

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

   1. add-log-exp 25.8%

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

   2. exp-prod 32.7%

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

   3. associate-*r* 32.7%

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

   4. *-commutative 32.7%

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

6. Applied egg-rr 32.7%

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

7. Taylor expanded in t around 0 20.7%

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

   1. metadata-eval 20.7%

      \[\leadsto \frac{2}{\color{blue}{0}} \cdot \left(\ell \cdot \ell\right) \]

9. Applied egg-rr 20.7%

   \[\leadsto \frac{2}{\color{blue}{0}} \cdot \left(\ell \cdot \ell\right) \]
10. Step-by-step derivation

    1. clear-num 20.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{0}{2}}} \cdot \left(\ell \cdot \ell\right) \]

    2. metadata-eval 20.7%

       \[\leadsto \frac{1}{\color{blue}{0}} \cdot \left(\ell \cdot \ell\right) \]

    3. inv-pow 20.7%

       \[\leadsto \color{blue}{{0}^{-1}} \cdot \left(\ell \cdot \ell\right) \]

11. Applied egg-rr 20.7%

    \[\leadsto \color{blue}{{0}^{-1}} \cdot \left(\ell \cdot \ell\right) \]
12. Step-by-step derivation

    1. pow-base-0 27.2%

       \[\leadsto \color{blue}{0} \cdot \left(\ell \cdot \ell\right) \]

13. Simplified 27.2%

    \[\leadsto \color{blue}{0} \cdot \left(\ell \cdot \ell\right) \]

14. Final simplification 27.2%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot 0 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024116 
(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))))