Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.8% → 87.9%

Time: 20.5s

Alternatives: 14

Speedup: 3.8×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.8% 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: 87.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.7 \cdot 10^{-11}:\\ \;\;\;\;2 \cdot \left|{\left(\sqrt{\cos k\_m} \cdot \frac{\frac{\frac{\ell}{\sin k\_m}}{\sqrt{t\_m}}}{k\_m}\right)}^{2}\right|\\ \mathbf{elif}\;k\_m \leq 4 \cdot 10^{+155}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k\_m}^{2}} \cdot \frac{\cos k\_m}{t\_m \cdot {\sin k\_m}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m}{t\_m} \cdot {\left(t\_2 \cdot \left(\sqrt[3]{\tan k\_m} \cdot \sqrt[3]{\sin k\_m}\right)\right)}^{2}\right) \cdot \left(\frac{k\_m}{t\_m} \cdot \left(t\_2 \cdot \sqrt[3]{\sin k\_m \cdot \tan k\_m}\right)\right)}\\ \end{array} \end{array} \end{array} \]

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 (* t_m (pow (cbrt l) -2.0))))
   (*
    t_s
    (if (<= k_m 4.7e-11)
      (*
       2.0
       (fabs
        (pow (* (sqrt (cos k_m)) (/ (/ (/ l (sin k_m)) (sqrt t_m)) k_m)) 2.0)))
      (if (<= k_m 4e+155)
        (*
         2.0
         (*
          (/ (pow l 2.0) (pow k_m 2.0))
          (/ (cos k_m) (* t_m (pow (sin k_m) 2.0)))))
        (/
         2.0
         (*
          (*
           (/ k_m t_m)
           (pow (* t_2 (* (cbrt (tan k_m)) (cbrt (sin k_m)))) 2.0))
          (* (/ k_m t_m) (* t_2 (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 = t_m * pow(cbrt(l), -2.0);
	double tmp;
	if (k_m <= 4.7e-11) {
		tmp = 2.0 * fabs(pow((sqrt(cos(k_m)) * (((l / sin(k_m)) / sqrt(t_m)) / k_m)), 2.0));
	} else if (k_m <= 4e+155) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) * (cos(k_m) / (t_m * pow(sin(k_m), 2.0))));
	} else {
		tmp = 2.0 / (((k_m / t_m) * pow((t_2 * (cbrt(tan(k_m)) * cbrt(sin(k_m)))), 2.0)) * ((k_m / t_m) * (t_2 * 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 = t_m * Math.pow(Math.cbrt(l), -2.0);
	double tmp;
	if (k_m <= 4.7e-11) {
		tmp = 2.0 * Math.abs(Math.pow((Math.sqrt(Math.cos(k_m)) * (((l / Math.sin(k_m)) / Math.sqrt(t_m)) / k_m)), 2.0));
	} else if (k_m <= 4e+155) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) * (Math.cos(k_m) / (t_m * Math.pow(Math.sin(k_m), 2.0))));
	} else {
		tmp = 2.0 / (((k_m / t_m) * Math.pow((t_2 * (Math.cbrt(Math.tan(k_m)) * Math.cbrt(Math.sin(k_m)))), 2.0)) * ((k_m / t_m) * (t_2 * 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(t_m * (cbrt(l) ^ -2.0))
	tmp = 0.0
	if (k_m <= 4.7e-11)
		tmp = Float64(2.0 * abs((Float64(sqrt(cos(k_m)) * Float64(Float64(Float64(l / sin(k_m)) / sqrt(t_m)) / k_m)) ^ 2.0)));
	elseif (k_m <= 4e+155)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) * Float64(cos(k_m) / Float64(t_m * (sin(k_m) ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / t_m) * (Float64(t_2 * Float64(cbrt(tan(k_m)) * cbrt(sin(k_m)))) ^ 2.0)) * Float64(Float64(k_m / t_m) * Float64(t_2 * 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[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 4.7e-11], N[(2.0 * N[Abs[N[Power[N[(N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision] * N[(N[(N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 4e+155], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[Power[N[(t$95$2 * N[(N[Power[N[Tan[k$95$m], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(t$95$2 * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 4.69999999999999993e-11

   1. Initial program 35.6%

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

   3. Simplified 40.9%

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

      1. associate-*l/ 41.3%

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

      2. add-sqr-sqrt 19.7%

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

      3. associate-/r* 19.7%

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

   6. Applied egg-rr 21.8%

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

   7. Taylor expanded in t around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. times-frac 0.0%

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

      3. *-commutative 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 72.4%

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

      7. *-commutative 72.4%

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

      8. associate-*l* 72.4%

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

      9. neg-mul-1 72.4%

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

   9. Simplified 72.4%

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

   11. Applied egg-rr 28.9%

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

       1. unpow2 28.9%

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

       2. rem-sqrt-square 37.2%

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

       3. fabs-mul 37.2%

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

       4. metadata-eval 37.2%

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

       5. associate-*l/ 37.2%

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

       6. associate-/l* 37.2%

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

       7. associate-/r* 37.2%

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

   13. Simplified 37.2%

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

   if 4.69999999999999993e-11 < k < 4.00000000000000003e155

   1. Initial program 12.4%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in t around 0 87.9%

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

      1. associate-/l* 88.0%

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

      2. associate-/l* 87.8%

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

   7. Simplified 87.8%

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

   8. Taylor expanded in k around inf 88.0%

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

      1. times-frac 85.3%

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

   10. Simplified 85.3%

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

   if 4.00000000000000003e155 < k

   1. Initial program 40.6%

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

      1. *-commutative 40.6%

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

      2. associate-/r* 40.6%

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

   3. Simplified 49.7%

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

      1. add-sqr-sqrt 49.7%

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

      2. add-cube-cbrt 49.7%

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

      3. times-frac 49.7%

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

   6. Applied egg-rr 79.1%

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

      1. associate-/l/ 79.1%

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

      2. associate-/l/ 79.1%

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

   8. Simplified 79.1%

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

      1. frac-times 79.1%

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

      2. pow1/2 79.1%

         \[\leadsto \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{2}}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)} \]

      3. pow1/2 79.1%

         \[\leadsto \frac{{2}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)} \]

      4. pow-prod-up 79.1%

         \[\leadsto \frac{\color{blue}{{2}^{\left(0.5 + 0.5\right)}}}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)} \]

      5. metadata-eval 79.1%

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

      6. metadata-eval 79.1%

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

   10. Applied egg-rr 79.1%

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

       1. *-commutative 79.1%

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

       2. cbrt-prod 79.1%

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

   12. Applied egg-rr 79.1%

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

Alternative 2: 87.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.2 \cdot 10^{-10}:\\ \;\;\;\;2 \cdot \left|{\left(\sqrt{\cos k\_m} \cdot \frac{\frac{\frac{\ell}{\sin k\_m}}{\sqrt{t\_m}}}{k\_m}\right)}^{2}\right|\\ \mathbf{elif}\;k\_m \leq 4 \cdot 10^{+155}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k\_m}^{2}} \cdot \frac{\cos k\_m}{t\_m \cdot {\sin k\_m}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m}{t\_m} \cdot {\left(\left(t\_m \cdot t\_2\right) \cdot \left(\sqrt[3]{\tan k\_m} \cdot \sqrt[3]{\sin k\_m}\right)\right)}^{2}\right) \cdot \left(k\_m \cdot \left(t\_2 \cdot \sqrt[3]{\sin k\_m \cdot \tan k\_m}\right)\right)}\\ \end{array} \end{array} \end{array} \]

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 (pow (cbrt l) -2.0)))
   (*
    t_s
    (if (<= k_m 1.2e-10)
      (*
       2.0
       (fabs
        (pow (* (sqrt (cos k_m)) (/ (/ (/ l (sin k_m)) (sqrt t_m)) k_m)) 2.0)))
      (if (<= k_m 4e+155)
        (*
         2.0
         (*
          (/ (pow l 2.0) (pow k_m 2.0))
          (/ (cos k_m) (* t_m (pow (sin k_m) 2.0)))))
        (/
         2.0
         (*
          (*
           (/ k_m t_m)
           (pow (* (* t_m t_2) (* (cbrt (tan k_m)) (cbrt (sin k_m)))) 2.0))
          (* k_m (* t_2 (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 = pow(cbrt(l), -2.0);
	double tmp;
	if (k_m <= 1.2e-10) {
		tmp = 2.0 * fabs(pow((sqrt(cos(k_m)) * (((l / sin(k_m)) / sqrt(t_m)) / k_m)), 2.0));
	} else if (k_m <= 4e+155) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) * (cos(k_m) / (t_m * pow(sin(k_m), 2.0))));
	} else {
		tmp = 2.0 / (((k_m / t_m) * pow(((t_m * t_2) * (cbrt(tan(k_m)) * cbrt(sin(k_m)))), 2.0)) * (k_m * (t_2 * 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.pow(Math.cbrt(l), -2.0);
	double tmp;
	if (k_m <= 1.2e-10) {
		tmp = 2.0 * Math.abs(Math.pow((Math.sqrt(Math.cos(k_m)) * (((l / Math.sin(k_m)) / Math.sqrt(t_m)) / k_m)), 2.0));
	} else if (k_m <= 4e+155) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) * (Math.cos(k_m) / (t_m * Math.pow(Math.sin(k_m), 2.0))));
	} else {
		tmp = 2.0 / (((k_m / t_m) * Math.pow(((t_m * t_2) * (Math.cbrt(Math.tan(k_m)) * Math.cbrt(Math.sin(k_m)))), 2.0)) * (k_m * (t_2 * 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 = cbrt(l) ^ -2.0
	tmp = 0.0
	if (k_m <= 1.2e-10)
		tmp = Float64(2.0 * abs((Float64(sqrt(cos(k_m)) * Float64(Float64(Float64(l / sin(k_m)) / sqrt(t_m)) / k_m)) ^ 2.0)));
	elseif (k_m <= 4e+155)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) * Float64(cos(k_m) / Float64(t_m * (sin(k_m) ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / t_m) * (Float64(Float64(t_m * t_2) * Float64(cbrt(tan(k_m)) * cbrt(sin(k_m)))) ^ 2.0)) * Float64(k_m * Float64(t_2 * 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[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.2e-10], N[(2.0 * N[Abs[N[Power[N[(N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision] * N[(N[(N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 4e+155], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[Power[N[(N[(t$95$m * t$95$2), $MachinePrecision] * N[(N[Power[N[Tan[k$95$m], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(k$95$m * N[(t$95$2 * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.2e-10

   1. Initial program 35.6%

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

   3. Simplified 40.9%

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

      1. associate-*l/ 41.3%

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

      2. add-sqr-sqrt 19.7%

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

      3. associate-/r* 19.7%

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

   6. Applied egg-rr 21.8%

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

   7. Taylor expanded in t around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. times-frac 0.0%

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

      3. *-commutative 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 72.4%

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

      7. *-commutative 72.4%

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

      8. associate-*l* 72.4%

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

      9. neg-mul-1 72.4%

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

   9. Simplified 72.4%

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

   11. Applied egg-rr 28.9%

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

       1. unpow2 28.9%

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

       2. rem-sqrt-square 37.2%

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

       3. fabs-mul 37.2%

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

       4. metadata-eval 37.2%

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

       5. associate-*l/ 37.2%

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

       6. associate-/l* 37.2%

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

       7. associate-/r* 37.2%

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

   13. Simplified 37.2%

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

   if 1.2e-10 < k < 4.00000000000000003e155

   1. Initial program 12.4%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in t around 0 87.9%

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

      1. associate-/l* 88.0%

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

      2. associate-/l* 87.8%

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

   7. Simplified 87.8%

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

   8. Taylor expanded in k around inf 88.0%

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

      1. times-frac 85.3%

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

   10. Simplified 85.3%

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

   if 4.00000000000000003e155 < k

   1. Initial program 40.6%

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

      1. *-commutative 40.6%

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

      2. associate-/r* 40.6%

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

   3. Simplified 49.7%

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

      1. add-sqr-sqrt 49.7%

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

      2. add-cube-cbrt 49.7%

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

      3. times-frac 49.7%

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

   6. Applied egg-rr 79.1%

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

      1. associate-/l/ 79.1%

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

      2. associate-/l/ 79.1%

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

   8. Simplified 79.1%

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

      1. frac-times 79.1%

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

      2. pow1/2 79.1%

         \[\leadsto \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{2}}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)} \]

      3. pow1/2 79.1%

         \[\leadsto \frac{{2}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)} \]

      4. pow-prod-up 79.1%

         \[\leadsto \frac{\color{blue}{{2}^{\left(0.5 + 0.5\right)}}}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)} \]

      5. metadata-eval 79.1%

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

      6. metadata-eval 79.1%

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

   10. Applied egg-rr 79.1%

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

       1. *-commutative 79.1%

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

       2. cbrt-prod 79.1%

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

   12. Applied egg-rr 79.1%

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

       1. associate-*l/ 73.8%

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

       2. associate-*l* 73.8%

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

   14. Applied egg-rr 73.8%

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

       1. associate-/l* 79.1%

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

       2. *-commutative 79.1%

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

       3. associate-/l* 79.1%

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

       4. *-commutative 79.1%

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

       5. *-inverses 79.1%

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

   16. Simplified 79.1%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-10}:\\ \;\;\;\;2 \cdot \left|{\left(\sqrt{\cos k} \cdot \frac{\frac{\frac{\ell}{\sin k}}{\sqrt{t}}}{k}\right)}^{2}\right|\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+155}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{t} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\sin k}\right)\right)}^{2}\right) \cdot \left(k \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.2 \cdot 10^{-10}:\\ \;\;\;\;2 \cdot \left|{\left(\sqrt{\cos k\_m} \cdot \frac{\frac{\frac{\ell}{\sin k\_m}}{\sqrt{t\_m}}}{k\_m}\right)}^{2}\right|\\ \mathbf{elif}\;k\_m \leq 4 \cdot 10^{+155}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k\_m}^{2}} \cdot \frac{\cos k\_m}{t\_m \cdot {\sin k\_m}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k\_m}{t\_m} \cdot t\_2\right) \cdot \left(\frac{k\_m}{t\_m} \cdot {t\_2}^{2}\right)}\\ \end{array} \end{array} \end{array} \]

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 (* (* t_m (pow (cbrt l) -2.0)) (cbrt (* (sin k_m) (tan k_m))))))
   (*
    t_s
    (if (<= k_m 1.2e-10)
      (*
       2.0
       (fabs
        (pow (* (sqrt (cos k_m)) (/ (/ (/ l (sin k_m)) (sqrt t_m)) k_m)) 2.0)))
      (if (<= k_m 4e+155)
        (*
         2.0
         (*
          (/ (pow l 2.0) (pow k_m 2.0))
          (/ (cos k_m) (* t_m (pow (sin k_m) 2.0)))))
        (/ 2.0 (* (* (/ k_m t_m) t_2) (* (/ k_m t_m) (pow t_2 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 = (t_m * pow(cbrt(l), -2.0)) * cbrt((sin(k_m) * tan(k_m)));
	double tmp;
	if (k_m <= 1.2e-10) {
		tmp = 2.0 * fabs(pow((sqrt(cos(k_m)) * (((l / sin(k_m)) / sqrt(t_m)) / k_m)), 2.0));
	} else if (k_m <= 4e+155) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) * (cos(k_m) / (t_m * pow(sin(k_m), 2.0))));
	} else {
		tmp = 2.0 / (((k_m / t_m) * t_2) * ((k_m / t_m) * pow(t_2, 2.0)));
	}
	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 = (t_m * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double tmp;
	if (k_m <= 1.2e-10) {
		tmp = 2.0 * Math.abs(Math.pow((Math.sqrt(Math.cos(k_m)) * (((l / Math.sin(k_m)) / Math.sqrt(t_m)) / k_m)), 2.0));
	} else if (k_m <= 4e+155) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) * (Math.cos(k_m) / (t_m * Math.pow(Math.sin(k_m), 2.0))));
	} else {
		tmp = 2.0 / (((k_m / t_m) * t_2) * ((k_m / t_m) * Math.pow(t_2, 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(Float64(t_m * (cbrt(l) ^ -2.0)) * cbrt(Float64(sin(k_m) * tan(k_m))))
	tmp = 0.0
	if (k_m <= 1.2e-10)
		tmp = Float64(2.0 * abs((Float64(sqrt(cos(k_m)) * Float64(Float64(Float64(l / sin(k_m)) / sqrt(t_m)) / k_m)) ^ 2.0)));
	elseif (k_m <= 4e+155)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) * Float64(cos(k_m) / Float64(t_m * (sin(k_m) ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k_m / t_m) * t_2) * Float64(Float64(k_m / t_m) * (t_2 ^ 2.0))));
	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[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.2e-10], N[(2.0 * N[Abs[N[Power[N[(N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision] * N[(N[(N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 4e+155], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.2e-10

   1. Initial program 35.6%

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

   3. Simplified 40.9%

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

      1. associate-*l/ 41.3%

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

      2. add-sqr-sqrt 19.7%

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

      3. associate-/r* 19.7%

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

   6. Applied egg-rr 21.8%

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

   7. Taylor expanded in t around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. times-frac 0.0%

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

      3. *-commutative 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 72.4%

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

      7. *-commutative 72.4%

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

      8. associate-*l* 72.4%

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

      9. neg-mul-1 72.4%

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

   9. Simplified 72.4%

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

   11. Applied egg-rr 28.9%

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

       1. unpow2 28.9%

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

       2. rem-sqrt-square 37.2%

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

       3. fabs-mul 37.2%

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

       4. metadata-eval 37.2%

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

       5. associate-*l/ 37.2%

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

       6. associate-/l* 37.2%

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

       7. associate-/r* 37.2%

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

   13. Simplified 37.2%

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

   if 1.2e-10 < k < 4.00000000000000003e155

   1. Initial program 12.4%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in t around 0 87.9%

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

      1. associate-/l* 88.0%

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

      2. associate-/l* 87.8%

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

   7. Simplified 87.8%

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

   8. Taylor expanded in k around inf 88.0%

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

      1. times-frac 85.3%

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

   10. Simplified 85.3%

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

   if 4.00000000000000003e155 < k

   1. Initial program 40.6%

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

      1. *-commutative 40.6%

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

      2. associate-/r* 40.6%

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

   3. Simplified 49.7%

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

      1. add-sqr-sqrt 49.7%

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

      2. add-cube-cbrt 49.7%

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

      3. times-frac 49.7%

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

   6. Applied egg-rr 79.1%

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

      1. associate-/l/ 79.1%

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

      2. associate-/l/ 79.1%

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

   8. Simplified 79.1%

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

      1. frac-times 79.1%

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

      2. pow1/2 79.1%

         \[\leadsto \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{2}}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)} \]

      3. pow1/2 79.1%

         \[\leadsto \frac{{2}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)} \]

      4. pow-prod-up 79.1%

         \[\leadsto \frac{\color{blue}{{2}^{\left(0.5 + 0.5\right)}}}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)} \]

      5. metadata-eval 79.1%

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

      6. metadata-eval 79.1%

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

   10. Applied egg-rr 79.1%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-10}:\\ \;\;\;\;2 \cdot \left|{\left(\sqrt{\cos k} \cdot \frac{\frac{\frac{\ell}{\sin k}}{\sqrt{t}}}{k}\right)}^{2}\right|\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+155}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{t} \cdot \left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right) \cdot \left(\frac{k}{t} \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.2 \cdot 10^{-10}:\\ \;\;\;\;2 \cdot \left|{\left(\sqrt{\cos k\_m} \cdot \frac{\frac{\frac{\ell}{\sin k\_m}}{\sqrt{t\_m}}}{k\_m}\right)}^{2}\right|\\ \mathbf{elif}\;k\_m \leq 4.1 \cdot 10^{+155}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k\_m}^{2}} \cdot \frac{\cos k\_m}{t\_m \cdot {\sin k\_m}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k\_m}{t\_m} \cdot \left(\frac{k\_m}{t\_m} \cdot \left(\left(\sin k\_m \cdot \tan k\_m\right) \cdot {\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)\right)}\\ \end{array} \end{array} \]

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 1.2e-10)
    (*
     2.0
     (fabs
      (pow (* (sqrt (cos k_m)) (/ (/ (/ l (sin k_m)) (sqrt t_m)) k_m)) 2.0)))
    (if (<= k_m 4.1e+155)
      (*
       2.0
       (*
        (/ (pow l 2.0) (pow k_m 2.0))
        (/ (cos k_m) (* t_m (pow (sin k_m) 2.0)))))
      (/
       2.0
       (*
        (/ k_m t_m)
        (*
         (/ k_m t_m)
         (*
          (* (sin k_m) (tan k_m))
          (pow (* t_m (pow (cbrt l) -2.0)) 3.0)))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.2e-10) {
		tmp = 2.0 * fabs(pow((sqrt(cos(k_m)) * (((l / sin(k_m)) / sqrt(t_m)) / k_m)), 2.0));
	} else if (k_m <= 4.1e+155) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) * (cos(k_m) / (t_m * pow(sin(k_m), 2.0))));
	} else {
		tmp = 2.0 / ((k_m / t_m) * ((k_m / t_m) * ((sin(k_m) * tan(k_m)) * pow((t_m * pow(cbrt(l), -2.0)), 3.0))));
	}
	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 tmp;
	if (k_m <= 1.2e-10) {
		tmp = 2.0 * Math.abs(Math.pow((Math.sqrt(Math.cos(k_m)) * (((l / Math.sin(k_m)) / Math.sqrt(t_m)) / k_m)), 2.0));
	} else if (k_m <= 4.1e+155) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) * (Math.cos(k_m) / (t_m * Math.pow(Math.sin(k_m), 2.0))));
	} else {
		tmp = 2.0 / ((k_m / t_m) * ((k_m / t_m) * ((Math.sin(k_m) * Math.tan(k_m)) * Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0))));
	}
	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 <= 1.2e-10)
		tmp = Float64(2.0 * abs((Float64(sqrt(cos(k_m)) * Float64(Float64(Float64(l / sin(k_m)) / sqrt(t_m)) / k_m)) ^ 2.0)));
	elseif (k_m <= 4.1e+155)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) * Float64(cos(k_m) / Float64(t_m * (sin(k_m) ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m / t_m) * Float64(Float64(k_m / t_m) * Float64(Float64(sin(k_m) * tan(k_m)) * (Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0)))));
	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_] := N[(t$95$s * If[LessEqual[k$95$m, 1.2e-10], N[(2.0 * N[Abs[N[Power[N[(N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision] * N[(N[(N[(l / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 4.1e+155], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.2e-10

   1. Initial program 35.6%

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

   3. Simplified 40.9%

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

      1. associate-*l/ 41.3%

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

      2. add-sqr-sqrt 19.7%

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

      3. associate-/r* 19.7%

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

   6. Applied egg-rr 21.8%

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

   7. Taylor expanded in t around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. times-frac 0.0%

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

      3. *-commutative 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 72.4%

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

      7. *-commutative 72.4%

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

      8. associate-*l* 72.4%

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

      9. neg-mul-1 72.4%

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

   9. Simplified 72.4%

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

   11. Applied egg-rr 28.9%

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

       1. unpow2 28.9%

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

       2. rem-sqrt-square 37.2%

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

       3. fabs-mul 37.2%

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

       4. metadata-eval 37.2%

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

       5. associate-*l/ 37.2%

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

       6. associate-/l* 37.2%

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

       7. associate-/r* 37.2%

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

   13. Simplified 37.2%

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

   if 1.2e-10 < k < 4.0999999999999998e155

   1. Initial program 12.4%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in t around 0 87.9%

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

      1. associate-/l* 88.0%

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

      2. associate-/l* 87.8%

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

   7. Simplified 87.8%

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

   8. Taylor expanded in k around inf 88.0%

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

      1. times-frac 85.3%

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

   10. Simplified 85.3%

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

   if 4.0999999999999998e155 < k

   1. Initial program 40.6%

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

      1. *-commutative 40.6%

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

      2. associate-/r* 40.6%

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

   3. Simplified 49.7%

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

      1. add-sqr-sqrt 49.7%

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

      2. add-cube-cbrt 49.7%

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

      3. times-frac 49.7%

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

   6. Applied egg-rr 79.1%

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

      1. associate-/l/ 79.1%

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

      2. associate-/l/ 79.1%

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

   8. Simplified 79.1%

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

      1. frac-times 79.1%

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

      2. pow1/2 79.1%

         \[\leadsto \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{2}}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)} \]

      3. pow1/2 79.1%

         \[\leadsto \frac{{2}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)} \]

      4. pow-prod-up 79.1%

         \[\leadsto \frac{\color{blue}{{2}^{\left(0.5 + 0.5\right)}}}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)} \]

      5. metadata-eval 79.1%

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

      6. metadata-eval 79.1%

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

   10. Applied egg-rr 79.1%

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

       1. associate-*l* 76.4%

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

       2. *-commutative 76.4%

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

       3. associate-*r* 67.9%

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

   12. Simplified 68.2%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-10}:\\ \;\;\;\;2 \cdot \left|{\left(\sqrt{\cos k} \cdot \frac{\frac{\frac{\ell}{\sin k}}{\sqrt{t}}}{k}\right)}^{2}\right|\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{+155}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{t} \cdot \left(\frac{k}{t} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.2 \cdot 10^{-10}:\\ \;\;\;\;\left|-2 \cdot {\left(\sqrt{\cos k\_m} \cdot \frac{\frac{\ell}{\sin k\_m \cdot \sqrt{t\_m}}}{k\_m}\right)}^{2}\right|\\ \mathbf{elif}\;k\_m \leq 9.6 \cdot 10^{+155}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k\_m}^{2}} \cdot \frac{\cos k\_m}{t\_m \cdot {\sin k\_m}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k\_m}{t\_m} \cdot \left(\frac{k\_m}{t\_m} \cdot \left(\left(\sin k\_m \cdot \tan k\_m\right) \cdot {\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)\right)}\\ \end{array} \end{array} \]

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 1.2e-10)
    (fabs
     (*
      -2.0
      (pow (* (sqrt (cos k_m)) (/ (/ l (* (sin k_m) (sqrt t_m))) k_m)) 2.0)))
    (if (<= k_m 9.6e+155)
      (*
       2.0
       (*
        (/ (pow l 2.0) (pow k_m 2.0))
        (/ (cos k_m) (* t_m (pow (sin k_m) 2.0)))))
      (/
       2.0
       (*
        (/ k_m t_m)
        (*
         (/ k_m t_m)
         (*
          (* (sin k_m) (tan k_m))
          (pow (* t_m (pow (cbrt l) -2.0)) 3.0)))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1.2e-10) {
		tmp = fabs((-2.0 * pow((sqrt(cos(k_m)) * ((l / (sin(k_m) * sqrt(t_m))) / k_m)), 2.0)));
	} else if (k_m <= 9.6e+155) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) * (cos(k_m) / (t_m * pow(sin(k_m), 2.0))));
	} else {
		tmp = 2.0 / ((k_m / t_m) * ((k_m / t_m) * ((sin(k_m) * tan(k_m)) * pow((t_m * pow(cbrt(l), -2.0)), 3.0))));
	}
	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 tmp;
	if (k_m <= 1.2e-10) {
		tmp = Math.abs((-2.0 * Math.pow((Math.sqrt(Math.cos(k_m)) * ((l / (Math.sin(k_m) * Math.sqrt(t_m))) / k_m)), 2.0)));
	} else if (k_m <= 9.6e+155) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 2.0)) * (Math.cos(k_m) / (t_m * Math.pow(Math.sin(k_m), 2.0))));
	} else {
		tmp = 2.0 / ((k_m / t_m) * ((k_m / t_m) * ((Math.sin(k_m) * Math.tan(k_m)) * Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0))));
	}
	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 <= 1.2e-10)
		tmp = abs(Float64(-2.0 * (Float64(sqrt(cos(k_m)) * Float64(Float64(l / Float64(sin(k_m) * sqrt(t_m))) / k_m)) ^ 2.0)));
	elseif (k_m <= 9.6e+155)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) * Float64(cos(k_m) / Float64(t_m * (sin(k_m) ^ 2.0)))));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m / t_m) * Float64(Float64(k_m / t_m) * Float64(Float64(sin(k_m) * tan(k_m)) * (Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0)))));
	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_] := N[(t$95$s * If[LessEqual[k$95$m, 1.2e-10], N[Abs[N[(-2.0 * N[Power[N[(N[Sqrt[N[Cos[k$95$m], $MachinePrecision]], $MachinePrecision] * N[(N[(l / N[(N[Sin[k$95$m], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[k$95$m, 9.6e+155], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.2e-10

   1. Initial program 35.6%

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

   3. Simplified 40.9%

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

      1. associate-*l/ 41.3%

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

      2. add-sqr-sqrt 19.7%

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

      3. associate-/r* 19.7%

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

   6. Applied egg-rr 21.8%

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

   7. Taylor expanded in t around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. times-frac 0.0%

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

      3. *-commutative 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 72.4%

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

      7. *-commutative 72.4%

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

      8. associate-*l* 72.4%

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

      9. neg-mul-1 72.4%

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

   9. Simplified 72.4%

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

   11. Applied egg-rr 28.9%

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

       1. unpow2 28.9%

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

       2. rem-sqrt-square 37.2%

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

       3. associate-*l/ 37.2%

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

       4. associate-/l* 37.2%

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

   13. Simplified 37.2%

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

   if 1.2e-10 < k < 9.60000000000000083e155

   1. Initial program 12.4%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in t around 0 87.9%

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

      1. associate-/l* 88.0%

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

      2. associate-/l* 87.8%

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

   7. Simplified 87.8%

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

   8. Taylor expanded in k around inf 88.0%

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

      1. times-frac 85.3%

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

   10. Simplified 85.3%

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

   if 9.60000000000000083e155 < k

   1. Initial program 40.6%

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

      1. *-commutative 40.6%

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

      2. associate-/r* 40.6%

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

   3. Simplified 49.7%

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

      1. add-sqr-sqrt 49.7%

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

      2. add-cube-cbrt 49.7%

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

      3. times-frac 49.7%

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

   6. Applied egg-rr 79.1%

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

      1. associate-/l/ 79.1%

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

      2. associate-/l/ 79.1%

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

   8. Simplified 79.1%

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

      1. frac-times 79.1%

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

      2. pow1/2 79.1%

         \[\leadsto \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{2}}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)} \]

      3. pow1/2 79.1%

         \[\leadsto \frac{{2}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)} \]

      4. pow-prod-up 79.1%

         \[\leadsto \frac{\color{blue}{{2}^{\left(0.5 + 0.5\right)}}}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)} \]

      5. metadata-eval 79.1%

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

      6. metadata-eval 79.1%

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

   10. Applied egg-rr 79.1%

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

       1. associate-*l* 76.4%

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

       2. *-commutative 76.4%

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

       3. associate-*r* 67.9%

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

   12. Simplified 68.2%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-10}:\\ \;\;\;\;\left|-2 \cdot {\left(\sqrt{\cos k} \cdot \frac{\frac{\ell}{\sin k \cdot \sqrt{t}}}{k}\right)}^{2}\right|\\ \mathbf{elif}\;k \leq 9.6 \cdot 10^{+155}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{t} \cdot \left(\frac{k}{t} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 3.1 \cdot 10^{-156}:\\ \;\;\;\;{\left(\ell \cdot \left(\frac{\sqrt{2}}{{k\_m}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)\right)}^{2}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+131}:\\ \;\;\;\;-2 \cdot \left(\frac{\cos k\_m}{{k\_m}^{2}} \cdot \frac{{\ell}^{2}}{t\_m \cdot \left(-{\sin k\_m}^{2}\right)}\right)\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+159}:\\ \;\;\;\;{\left(\ell \cdot \left(\frac{\sqrt{2}}{k\_m \cdot \sin k\_m} \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k\_m \cdot \frac{\frac{k\_m}{t\_m} \cdot \left(\left(\sin k\_m \cdot \tan k\_m\right) \cdot {\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}{t\_m}}\\ \end{array} \end{array} \]

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 (<= l 3.1e-156)
    (pow (* l (* (/ (sqrt 2.0) (pow k_m 2.0)) (sqrt (/ 1.0 t_m)))) 2.0)
    (if (<= l 1.6e+131)
      (*
       -2.0
       (*
        (/ (cos k_m) (pow k_m 2.0))
        (/ (pow l 2.0) (* t_m (- (pow (sin k_m) 2.0))))))
      (if (<= l 2.3e+159)
        (pow
         (* l (* (/ (sqrt 2.0) (* k_m (sin k_m))) (sqrt (/ (cos k_m) t_m))))
         2.0)
        (/
         2.0
         (*
          k_m
          (/
           (*
            (/ k_m t_m)
            (* (* (sin k_m) (tan k_m)) (pow (* t_m (pow (cbrt l) -2.0)) 3.0)))
           t_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 3.1e-156) {
		tmp = pow((l * ((sqrt(2.0) / pow(k_m, 2.0)) * sqrt((1.0 / t_m)))), 2.0);
	} else if (l <= 1.6e+131) {
		tmp = -2.0 * ((cos(k_m) / pow(k_m, 2.0)) * (pow(l, 2.0) / (t_m * -pow(sin(k_m), 2.0))));
	} else if (l <= 2.3e+159) {
		tmp = pow((l * ((sqrt(2.0) / (k_m * sin(k_m))) * sqrt((cos(k_m) / t_m)))), 2.0);
	} else {
		tmp = 2.0 / (k_m * (((k_m / t_m) * ((sin(k_m) * tan(k_m)) * pow((t_m * pow(cbrt(l), -2.0)), 3.0))) / t_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 tmp;
	if (l <= 3.1e-156) {
		tmp = Math.pow((l * ((Math.sqrt(2.0) / Math.pow(k_m, 2.0)) * Math.sqrt((1.0 / t_m)))), 2.0);
	} else if (l <= 1.6e+131) {
		tmp = -2.0 * ((Math.cos(k_m) / Math.pow(k_m, 2.0)) * (Math.pow(l, 2.0) / (t_m * -Math.pow(Math.sin(k_m), 2.0))));
	} else if (l <= 2.3e+159) {
		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 = 2.0 / (k_m * (((k_m / t_m) * ((Math.sin(k_m) * Math.tan(k_m)) * Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0))) / t_m));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (l <= 3.1e-156)
		tmp = Float64(l * Float64(Float64(sqrt(2.0) / (k_m ^ 2.0)) * sqrt(Float64(1.0 / t_m)))) ^ 2.0;
	elseif (l <= 1.6e+131)
		tmp = Float64(-2.0 * Float64(Float64(cos(k_m) / (k_m ^ 2.0)) * Float64((l ^ 2.0) / Float64(t_m * Float64(-(sin(k_m) ^ 2.0))))));
	elseif (l <= 2.3e+159)
		tmp = Float64(l * Float64(Float64(sqrt(2.0) / Float64(k_m * sin(k_m))) * sqrt(Float64(cos(k_m) / t_m)))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(k_m * Float64(Float64(Float64(k_m / t_m) * Float64(Float64(sin(k_m) * tan(k_m)) * (Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0))) / t_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_] := N[(t$95$s * If[LessEqual[l, 3.1e-156], N[Power[N[(l * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[l, 1.6e+131], N[(-2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * (-N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.3e+159], N[Power[N[(l * N[(N[(N[Sqrt[2.0], $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]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(k$95$m * N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if l < 3.0999999999999998e-156

   1. Initial program 33.3%

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

   3. Simplified 43.9%

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

      1. add-sqr-sqrt 35.9%

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

      2. pow2 35.9%

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

   6. Applied egg-rr 29.4%

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

   7. Taylor expanded in k around 0 40.3%

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

   if 3.0999999999999998e-156 < l < 1.6000000000000001e131

   1. Initial program 34.5%

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

   3. Simplified 44.1%

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

      1. associate-*l/ 45.3%

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

      2. add-sqr-sqrt 17.5%

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

      3. associate-/r* 17.5%

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

   6. Applied egg-rr 20.0%

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

   7. Taylor expanded in t around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. times-frac 0.0%

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

      3. *-commutative 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 87.8%

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

      7. *-commutative 87.8%

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

      8. associate-*l* 87.8%

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

      9. neg-mul-1 87.8%

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

   9. Simplified 87.8%

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

   if 1.6000000000000001e131 < l < 2.29999999999999995e159

   1. Initial program 35.1%

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

   3. Simplified 35.1%

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

      1. add-sqr-sqrt 6.5%

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

      2. pow2 6.5%

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

   6. Applied egg-rr 5.8%

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

   7. Taylor expanded in t around 0 42.4%

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

   if 2.29999999999999995e159 < l

   1. Initial program 30.3%

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

      1. *-commutative 30.3%

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

      2. associate-/r* 30.3%

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

   3. Simplified 30.3%

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

      1. add-sqr-sqrt 30.3%

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

      2. add-cube-cbrt 30.3%

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

      3. times-frac 30.3%

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

   6. Applied egg-rr 83.5%

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

      1. associate-/l/ 83.5%

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

      2. associate-/l/ 83.5%

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

   8. Simplified 83.5%

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

      1. frac-times 83.5%

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

      2. pow1/2 83.5%

         \[\leadsto \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{2}}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)} \]

      3. pow1/2 83.5%

         \[\leadsto \frac{{2}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)} \]

      4. pow-prod-up 83.6%

         \[\leadsto \frac{\color{blue}{{2}^{\left(0.5 + 0.5\right)}}}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2} \cdot \frac{k}{t}\right) \cdot \left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{k}{t}\right)} \]

      5. metadata-eval 83.6%

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

      6. metadata-eval 83.6%

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

   10. Applied egg-rr 83.8%

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

       1. pow1 83.8%

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

       2. associate-*l* 81.1%

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

       3. associate-*l* 81.1%

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

       4. associate-*r* 81.1%

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

   12. Applied egg-rr 81.1%

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

       1. unpow1 81.1%

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

       2. associate-*l/ 72.9%

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

       3. associate-/l* 81.1%

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

   14. Simplified 72.8%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.1 \cdot 10^{-156}:\\ \;\;\;\;{\left(\ell \cdot \left(\frac{\sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)\right)}^{2}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+131}:\\ \;\;\;\;-2 \cdot \left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot \left(-{\sin k}^{2}\right)}\right)\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+159}:\\ \;\;\;\;{\left(\ell \cdot \left(\frac{\sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \frac{\frac{k}{t} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-315}:\\ \;\;\;\;{\left(\ell \cdot \left(\frac{\sqrt{2}}{{k\_m}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+258}:\\ \;\;\;\;-2 \cdot \left(\frac{\cos k\_m}{{k\_m}^{2}} \cdot \frac{{\ell}^{2}}{t\_m \cdot \left(-{\sin k\_m}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \left(\frac{\sqrt{2}}{k\_m \cdot \sin k\_m} \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)\right)}^{2}\\ \end{array} \end{array} \]

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 (<= (* l l) 2e-315)
    (pow (* l (* (/ (sqrt 2.0) (pow k_m 2.0)) (sqrt (/ 1.0 t_m)))) 2.0)
    (if (<= (* l l) 1e+258)
      (*
       -2.0
       (*
        (/ (cos k_m) (pow k_m 2.0))
        (/ (pow l 2.0) (* t_m (- (pow (sin k_m) 2.0))))))
      (pow
       (* l (* (/ (sqrt 2.0) (* 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 tmp;
	if ((l * l) <= 2e-315) {
		tmp = pow((l * ((sqrt(2.0) / pow(k_m, 2.0)) * sqrt((1.0 / t_m)))), 2.0);
	} else if ((l * l) <= 1e+258) {
		tmp = -2.0 * ((cos(k_m) / pow(k_m, 2.0)) * (pow(l, 2.0) / (t_m * -pow(sin(k_m), 2.0))));
	} else {
		tmp = pow((l * ((sqrt(2.0) / (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) :: tmp
    if ((l * l) <= 2d-315) then
        tmp = (l * ((sqrt(2.0d0) / (k_m ** 2.0d0)) * sqrt((1.0d0 / t_m)))) ** 2.0d0
    else if ((l * l) <= 1d+258) then
        tmp = (-2.0d0) * ((cos(k_m) / (k_m ** 2.0d0)) * ((l ** 2.0d0) / (t_m * -(sin(k_m) ** 2.0d0))))
    else
        tmp = (l * ((sqrt(2.0d0) / (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 tmp;
	if ((l * l) <= 2e-315) {
		tmp = Math.pow((l * ((Math.sqrt(2.0) / Math.pow(k_m, 2.0)) * Math.sqrt((1.0 / t_m)))), 2.0);
	} else if ((l * l) <= 1e+258) {
		tmp = -2.0 * ((Math.cos(k_m) / Math.pow(k_m, 2.0)) * (Math.pow(l, 2.0) / (t_m * -Math.pow(Math.sin(k_m), 2.0))));
	} else {
		tmp = Math.pow((l * ((Math.sqrt(2.0) / (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):
	tmp = 0
	if (l * l) <= 2e-315:
		tmp = math.pow((l * ((math.sqrt(2.0) / math.pow(k_m, 2.0)) * math.sqrt((1.0 / t_m)))), 2.0)
	elif (l * l) <= 1e+258:
		tmp = -2.0 * ((math.cos(k_m) / math.pow(k_m, 2.0)) * (math.pow(l, 2.0) / (t_m * -math.pow(math.sin(k_m), 2.0))))
	else:
		tmp = math.pow((l * ((math.sqrt(2.0) / (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)
	tmp = 0.0
	if (Float64(l * l) <= 2e-315)
		tmp = Float64(l * Float64(Float64(sqrt(2.0) / (k_m ^ 2.0)) * sqrt(Float64(1.0 / t_m)))) ^ 2.0;
	elseif (Float64(l * l) <= 1e+258)
		tmp = Float64(-2.0 * Float64(Float64(cos(k_m) / (k_m ^ 2.0)) * Float64((l ^ 2.0) / Float64(t_m * Float64(-(sin(k_m) ^ 2.0))))));
	else
		tmp = Float64(l * Float64(Float64(sqrt(2.0) / 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)
	tmp = 0.0;
	if ((l * l) <= 2e-315)
		tmp = (l * ((sqrt(2.0) / (k_m ^ 2.0)) * sqrt((1.0 / t_m)))) ^ 2.0;
	elseif ((l * l) <= 1e+258)
		tmp = -2.0 * ((cos(k_m) / (k_m ^ 2.0)) * ((l ^ 2.0) / (t_m * -(sin(k_m) ^ 2.0))));
	else
		tmp = (l * ((sqrt(2.0) / (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_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 2e-315], N[Power[N[(l * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+258], N[(-2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * (-N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(l * N[(N[(N[Sqrt[2.0], $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]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 2.0000000019e-315

   1. Initial program 20.2%

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

   3. Simplified 31.8%

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

      1. add-sqr-sqrt 30.1%

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

      2. pow2 30.1%

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

   6. Applied egg-rr 30.4%

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

   7. Taylor expanded in k around 0 44.9%

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

   if 2.0000000019e-315 < (*.f64 l l) < 1.00000000000000006e258

   1. Initial program 39.5%

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

   3. Simplified 51.9%

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

      1. associate-*l/ 52.5%

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

      2. add-sqr-sqrt 24.0%

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

      3. associate-/r* 24.0%

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

   6. Applied egg-rr 24.0%

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

   7. Taylor expanded in t around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. times-frac 0.0%

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

      3. *-commutative 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 88.7%

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

      7. *-commutative 88.7%

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

      8. associate-*l* 88.7%

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

      9. neg-mul-1 88.7%

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

   9. Simplified 88.7%

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

   if 1.00000000000000006e258 < (*.f64 l l)

   1. Initial program 33.6%

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

   3. Simplified 33.5%

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

      1. add-sqr-sqrt 17.1%

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

      2. pow2 17.1%

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

   6. Applied egg-rr 20.0%

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

   7. Taylor expanded in t around 0 46.6%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-315}:\\ \;\;\;\;{\left(\ell \cdot \left(\frac{\sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)\right)}^{2}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+258}:\\ \;\;\;\;-2 \cdot \left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot \left(-{\sin k}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \left(\frac{\sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 5.2 \cdot 10^{-37}:\\ \;\;\;\;{\left(\ell \cdot \left(\frac{\sqrt{2}}{{k\_m}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k\_m}^{2}} \cdot \frac{\cos k\_m}{t\_m \cdot {\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \]

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 5.2e-37)
    (pow (* l (* (/ (sqrt 2.0) (pow k_m 2.0)) (sqrt (/ 1.0 t_m)))) 2.0)
    (*
     2.0
     (*
      (/ (pow l 2.0) (pow k_m 2.0))
      (/ (cos k_m) (* t_m (pow (sin k_m) 2.0))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5.2e-37) {
		tmp = pow((l * ((sqrt(2.0) / pow(k_m, 2.0)) * sqrt((1.0 / t_m)))), 2.0);
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 2.0)) * (cos(k_m) / (t_m * pow(sin(k_m), 2.0))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5.2d-37) then
        tmp = (l * ((sqrt(2.0d0) / (k_m ** 2.0d0)) * sqrt((1.0d0 / t_m)))) ** 2.0d0
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k_m ** 2.0d0)) * (cos(k_m) / (t_m * (sin(k_m) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5.2e-37) {
		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(l, 2.0) / Math.pow(k_m, 2.0)) * (Math.cos(k_m) / (t_m * Math.pow(Math.sin(k_m), 2.0))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 5.2e-37:
		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(l, 2.0) / math.pow(k_m, 2.0)) * (math.cos(k_m) / (t_m * math.pow(math.sin(k_m), 2.0))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 5.2e-37)
		tmp = Float64(l * Float64(Float64(sqrt(2.0) / (k_m ^ 2.0)) * sqrt(Float64(1.0 / t_m)))) ^ 2.0;
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 2.0)) * Float64(cos(k_m) / Float64(t_m * (sin(k_m) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 5.2e-37)
		tmp = (l * ((sqrt(2.0) / (k_m ^ 2.0)) * sqrt((1.0 / t_m)))) ^ 2.0;
	else
		tmp = 2.0 * (((l ^ 2.0) / (k_m ^ 2.0)) * (cos(k_m) / (t_m * (sin(k_m) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 5.2e-37], N[Power[N[(l * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k$95$m], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.19999999999999959e-37

   1. Initial program 36.3%

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

   3. Simplified 41.2%

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

      1. add-sqr-sqrt 28.0%

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

      2. pow2 28.0%

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

   6. Applied egg-rr 24.8%

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

   7. Taylor expanded in k around 0 39.5%

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

   if 5.19999999999999959e-37 < k

   1. Initial program 25.7%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in t around 0 71.6%

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

      1. associate-/l* 71.6%

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

      2. associate-/l* 71.5%

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

   7. Simplified 71.5%

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

   8. Taylor expanded in k around inf 71.6%

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

      1. times-frac 70.4%

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

   10. Simplified 70.4%

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

Alternative 9: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 5.2 \cdot 10^{-37}:\\ \;\;\;\;{\left(\ell \cdot \left(\frac{\sqrt{2}}{{k\_m}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{{k\_m}^{2} \cdot \left(t\_m \cdot {\sin k\_m}^{2}\right)}{\cos k\_m}}\\ \end{array} \end{array} \]

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 5.2e-37)
    (pow (* l (* (/ (sqrt 2.0) (pow k_m 2.0)) (sqrt (/ 1.0 t_m)))) 2.0)
    (*
     (* l l)
     (/ 2.0 (/ (* (pow k_m 2.0) (* t_m (pow (sin k_m) 2.0))) (cos k_m)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5.2e-37) {
		tmp = pow((l * ((sqrt(2.0) / pow(k_m, 2.0)) * sqrt((1.0 / t_m)))), 2.0);
	} else {
		tmp = (l * l) * (2.0 / ((pow(k_m, 2.0) * (t_m * pow(sin(k_m), 2.0))) / cos(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5.2d-37) then
        tmp = (l * ((sqrt(2.0d0) / (k_m ** 2.0d0)) * sqrt((1.0d0 / t_m)))) ** 2.0d0
    else
        tmp = (l * l) * (2.0d0 / (((k_m ** 2.0d0) * (t_m * (sin(k_m) ** 2.0d0))) / cos(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5.2e-37) {
		tmp = Math.pow((l * ((Math.sqrt(2.0) / Math.pow(k_m, 2.0)) * Math.sqrt((1.0 / t_m)))), 2.0);
	} else {
		tmp = (l * l) * (2.0 / ((Math.pow(k_m, 2.0) * (t_m * Math.pow(Math.sin(k_m), 2.0))) / Math.cos(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 5.2e-37:
		tmp = math.pow((l * ((math.sqrt(2.0) / math.pow(k_m, 2.0)) * math.sqrt((1.0 / t_m)))), 2.0)
	else:
		tmp = (l * l) * (2.0 / ((math.pow(k_m, 2.0) * (t_m * math.pow(math.sin(k_m), 2.0))) / math.cos(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 5.2e-37)
		tmp = Float64(l * Float64(Float64(sqrt(2.0) / (k_m ^ 2.0)) * sqrt(Float64(1.0 / t_m)))) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t_m * (sin(k_m) ^ 2.0))) / cos(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 5.2e-37)
		tmp = (l * ((sqrt(2.0) / (k_m ^ 2.0)) * sqrt((1.0 / t_m)))) ^ 2.0;
	else
		tmp = (l * l) * (2.0 / (((k_m ^ 2.0) * (t_m * (sin(k_m) ^ 2.0))) / cos(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 5.2e-37], N[Power[N[(l * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.19999999999999959e-37

   1. Initial program 36.3%

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

   3. Simplified 41.2%

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

      1. add-sqr-sqrt 28.0%

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

      2. pow2 28.0%

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

   6. Applied egg-rr 24.8%

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

   7. Taylor expanded in k around 0 39.5%

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

   if 5.19999999999999959e-37 < k

   1. Initial program 25.7%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in t around 0 71.6%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-37}:\\ \;\;\;\;{\left(\ell \cdot \left(\frac{\sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 5.2 \cdot 10^{-37}:\\ \;\;\;\;{\left(\ell \cdot \left(\frac{\sqrt{2}}{{k\_m}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k\_m}^{2} \cdot \left(t\_m \cdot \frac{{\sin k\_m}^{2}}{\cos k\_m}\right)}\\ \end{array} \end{array} \]

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 5.2e-37)
    (pow (* l (* (/ (sqrt 2.0) (pow k_m 2.0)) (sqrt (/ 1.0 t_m)))) 2.0)
    (*
     (* l l)
     (/ 2.0 (* (pow k_m 2.0) (* t_m (/ (pow (sin k_m) 2.0) (cos k_m)))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5.2e-37) {
		tmp = pow((l * ((sqrt(2.0) / pow(k_m, 2.0)) * sqrt((1.0 / t_m)))), 2.0);
	} else {
		tmp = (l * l) * (2.0 / (pow(k_m, 2.0) * (t_m * (pow(sin(k_m), 2.0) / cos(k_m)))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5.2d-37) then
        tmp = (l * ((sqrt(2.0d0) / (k_m ** 2.0d0)) * sqrt((1.0d0 / t_m)))) ** 2.0d0
    else
        tmp = (l * l) * (2.0d0 / ((k_m ** 2.0d0) * (t_m * ((sin(k_m) ** 2.0d0) / cos(k_m)))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5.2e-37) {
		tmp = Math.pow((l * ((Math.sqrt(2.0) / Math.pow(k_m, 2.0)) * Math.sqrt((1.0 / t_m)))), 2.0);
	} else {
		tmp = (l * l) * (2.0 / (Math.pow(k_m, 2.0) * (t_m * (Math.pow(Math.sin(k_m), 2.0) / Math.cos(k_m)))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 5.2e-37:
		tmp = math.pow((l * ((math.sqrt(2.0) / math.pow(k_m, 2.0)) * math.sqrt((1.0 / t_m)))), 2.0)
	else:
		tmp = (l * l) * (2.0 / (math.pow(k_m, 2.0) * (t_m * (math.pow(math.sin(k_m), 2.0) / math.cos(k_m)))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 5.2e-37)
		tmp = Float64(l * Float64(Float64(sqrt(2.0) / (k_m ^ 2.0)) * sqrt(Float64(1.0 / t_m)))) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64((k_m ^ 2.0) * Float64(t_m * Float64((sin(k_m) ^ 2.0) / cos(k_m))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 5.2e-37)
		tmp = (l * ((sqrt(2.0) / (k_m ^ 2.0)) * sqrt((1.0 / t_m)))) ^ 2.0;
	else
		tmp = (l * l) * (2.0 / ((k_m ^ 2.0) * (t_m * ((sin(k_m) ^ 2.0) / cos(k_m)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 5.2e-37], N[Power[N[(l * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.19999999999999959e-37

   1. Initial program 36.3%

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

   3. Simplified 41.2%

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

      1. add-sqr-sqrt 28.0%

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

      2. pow2 28.0%

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

   6. Applied egg-rr 24.8%

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

   7. Taylor expanded in k around 0 39.5%

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

   if 5.19999999999999959e-37 < k

   1. Initial program 25.7%

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

   3. Simplified 44.0%

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

   5. Taylor expanded in t around 0 71.6%

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

      1. associate-/l* 71.6%

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

      2. associate-/l* 71.5%

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

   7. Simplified 71.5%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-37}:\\ \;\;\;\;{\left(\ell \cdot \left(\frac{\sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1300:\\ \;\;\;\;{\left(\ell \cdot \left(\frac{\sqrt{2}}{{k\_m}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k\_m}^{2}}\\ \end{array} \end{array} \]

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 1300.0)
    (pow (* l (* (/ (sqrt 2.0) (pow k_m 2.0)) (sqrt (/ 1.0 t_m)))) 2.0)
    (* -0.3333333333333333 (/ (pow l 2.0) (* 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) {
	double tmp;
	if (k_m <= 1300.0) {
		tmp = pow((l * ((sqrt(2.0) / pow(k_m, 2.0)) * sqrt((1.0 / t_m)))), 2.0);
	} else {
		tmp = -0.3333333333333333 * (pow(l, 2.0) / (t_m * pow(k_m, 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1300.0d0) then
        tmp = (l * ((sqrt(2.0d0) / (k_m ** 2.0d0)) * sqrt((1.0d0 / t_m)))) ** 2.0d0
    else
        tmp = (-0.3333333333333333d0) * ((l ** 2.0d0) / (t_m * (k_m ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1300.0) {
		tmp = Math.pow((l * ((Math.sqrt(2.0) / Math.pow(k_m, 2.0)) * Math.sqrt((1.0 / t_m)))), 2.0);
	} else {
		tmp = -0.3333333333333333 * (Math.pow(l, 2.0) / (t_m * Math.pow(k_m, 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1300.0:
		tmp = math.pow((l * ((math.sqrt(2.0) / math.pow(k_m, 2.0)) * math.sqrt((1.0 / t_m)))), 2.0)
	else:
		tmp = -0.3333333333333333 * (math.pow(l, 2.0) / (t_m * math.pow(k_m, 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1300.0)
		tmp = Float64(l * Float64(Float64(sqrt(2.0) / (k_m ^ 2.0)) * sqrt(Float64(1.0 / t_m)))) ^ 2.0;
	else
		tmp = Float64(-0.3333333333333333 * Float64((l ^ 2.0) / Float64(t_m * (k_m ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1300.0)
		tmp = (l * ((sqrt(2.0) / (k_m ^ 2.0)) * sqrt((1.0 / t_m)))) ^ 2.0;
	else
		tmp = -0.3333333333333333 * ((l ^ 2.0) / (t_m * (k_m ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1300.0], N[Power[N[(l * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(-0.3333333333333333 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1300

   1. Initial program 35.5%

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

   3. Simplified 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-sqr-sqrt 27.5%

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

      2. pow2 27.5%

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

   6. Applied egg-rr 25.5%

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

   7. Taylor expanded in k around 0 39.2%

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

   if 1300 < k

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

      2. associate-/r* 26.9%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in k around 0 12.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. +-commutative 12.0%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{{\ell}^{2}}{t} + -0.3333333333333333 \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}}{{k}^{4}} \]

      2. fma-define 12.0%

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

      3. *-commutative 12.0%

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

      4. associate-/l* 16.7%

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

      5. associate-*r* 16.7%

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

      6. *-commutative 16.7%

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

   7. Simplified 16.7%

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

   8. Taylor expanded in k around inf 52.3%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1300:\\ \;\;\;\;{\left(\ell \cdot \left(\frac{\sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{\frac{{\ell}^{2}}{t\_m} \cdot \left(-0.3333333333333333 + \frac{2}{{k\_m}^{2}}\right)}{{k\_m}^{2}} \end{array} \]

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
  (/
   (* (/ (pow l 2.0) t_m) (+ -0.3333333333333333 (/ 2.0 (pow k_m 2.0))))
   (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 * (((pow(l, 2.0) / t_m) * (-0.3333333333333333 + (2.0 / pow(k_m, 2.0)))) / 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 ** 2.0d0) / t_m) * ((-0.3333333333333333d0) + (2.0d0 / (k_m ** 2.0d0)))) / (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 * (((Math.pow(l, 2.0) / t_m) * (-0.3333333333333333 + (2.0 / Math.pow(k_m, 2.0)))) / 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 * (((math.pow(l, 2.0) / t_m) * (-0.3333333333333333 + (2.0 / math.pow(k_m, 2.0)))) / 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(Float64((l ^ 2.0) / t_m) * Float64(-0.3333333333333333 + Float64(2.0 / (k_m ^ 2.0)))) / (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 ^ 2.0) / t_m) * (-0.3333333333333333 + (2.0 / (k_m ^ 2.0)))) / (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[(N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(-0.3333333333333333 + N[(2.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.3%

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

   1. *-commutative 33.3%

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

   2. associate-/r* 33.3%

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

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. Taylor expanded in k around 0 27.3%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
6. Step-by-step derivation

   1. +-commutative 27.3%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{{\ell}^{2}}{t} + -0.3333333333333333 \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}}{{k}^{4}} \]

   2. fma-define 27.3%

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

   3. *-commutative 27.3%

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

   4. associate-/l* 27.4%

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

   5. associate-*r* 27.4%

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

   6. *-commutative 27.4%

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

7. Simplified 27.4%

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

8. Taylor expanded in k around inf 46.9%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}{{k}^{2}}} \]
9. Step-by-step derivation

   1. associate-*r/ 46.9%

      \[\leadsto \frac{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{t} + \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}}{{k}^{2}} \]

   2. times-frac 45.6%

      \[\leadsto \frac{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{t} + \color{blue}{\frac{2}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t}}}{{k}^{2}} \]

   3. distribute-rgt-out 61.1%

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

10. Simplified 61.1%

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

Alternative 13: 62.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1300:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t\_m \cdot {k\_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k\_m}^{2}}\\ \end{array} \end{array} \]

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 1300.0)
    (* (* l l) (/ 2.0 (* t_m (pow k_m 4.0))))
    (* -0.3333333333333333 (/ (pow l 2.0) (* 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) {
	double tmp;
	if (k_m <= 1300.0) {
		tmp = (l * l) * (2.0 / (t_m * pow(k_m, 4.0)));
	} else {
		tmp = -0.3333333333333333 * (pow(l, 2.0) / (t_m * pow(k_m, 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1300.0d0) then
        tmp = (l * l) * (2.0d0 / (t_m * (k_m ** 4.0d0)))
    else
        tmp = (-0.3333333333333333d0) * ((l ** 2.0d0) / (t_m * (k_m ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1300.0) {
		tmp = (l * l) * (2.0 / (t_m * Math.pow(k_m, 4.0)));
	} else {
		tmp = -0.3333333333333333 * (Math.pow(l, 2.0) / (t_m * Math.pow(k_m, 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1300.0:
		tmp = (l * l) * (2.0 / (t_m * math.pow(k_m, 4.0)))
	else:
		tmp = -0.3333333333333333 * (math.pow(l, 2.0) / (t_m * math.pow(k_m, 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1300.0)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t_m * (k_m ^ 4.0))));
	else
		tmp = Float64(-0.3333333333333333 * Float64((l ^ 2.0) / Float64(t_m * (k_m ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1300.0)
		tmp = (l * l) * (2.0 / (t_m * (k_m ^ 4.0)));
	else
		tmp = -0.3333333333333333 * ((l ^ 2.0) / (t_m * (k_m ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1300.0], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1300

   1. Initial program 35.5%

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

   3. Simplified 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. Taylor expanded in k around 0 62.2%

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

   if 1300 < k

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

      2. associate-/r* 26.9%

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

   3. Simplified 45.7%

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

   5. Taylor expanded in k around 0 12.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   6. Step-by-step derivation

      1. +-commutative 12.0%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{{\ell}^{2}}{t} + -0.3333333333333333 \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}}{{k}^{4}} \]

      2. fma-define 12.0%

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

      3. *-commutative 12.0%

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

      4. associate-/l* 16.7%

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

      5. associate-*r* 16.7%

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

      6. *-commutative 16.7%

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

   7. Simplified 16.7%

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

   8. Taylor expanded in k around inf 52.3%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1300:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.7% 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 33.3%

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

3. Simplified 42.0%

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

5. Taylor expanded in k around 0 57.0%

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

6. Final simplification 57.0%

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

Reproduce

herbie shell --seed 2024111 
(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))))