Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.3% → 85.8%

Time: 21.9s

Alternatives: 21

Speedup: 35.0×

• Could not converge on a ground truth

  1. reg0 = (bf "1.22985843363224137004609019086575266964e-89")
  2. reg1 = (bf "8.057241607364539934023368773979716205863e209")
  3. reg2 = (bf "5.372098602581293092520916290893934583009e219")

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.3% 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: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_3 := \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t_4 := \frac{\sqrt{2}}{k\_m}\\ t_5 := t\_m \cdot t\_4\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2000000000:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 9.2 \cdot 10^{+222}:\\ \;\;\;\;\left(t\_5 \cdot {\left(\left(t\_m \cdot t\_2\right) \cdot t\_3\right)}^{-2}\right) \cdot \frac{\frac{t\_5}{\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{{\ell}^{4} \cdot \frac{{\cos k\_m}^{2}}{{\sin k\_m}^{4}}} \cdot \left(t\_4 \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{4}}{t\_m}\right)\right) \cdot \frac{1}{\frac{t\_2}{t\_4}}}{t\_3}\\ \end{array} \end{array} \end{array} \]

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_3 (cbrt (* (sin k_m) (tan k_m))))
        (t_4 (/ (sqrt 2.0) k_m))
        (t_5 (* t_m t_4)))
   (*
    t_s
    (if (<= k_m 2000000000.0)
      (pow
       (/ (sqrt 2.0) (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m)))))
       2.0)
      (if (<= k_m 9.2e+222)
        (*
         (* t_5 (pow (* (* t_m t_2) t_3) -2.0))
         (/ (/ t_5 (/ t_m (pow (cbrt l) 2.0))) t_3))
        (/
         (*
          (*
           (cbrt (* (pow l 4.0) (/ (pow (cos k_m) 2.0) (pow (sin k_m) 4.0))))
           (* t_4 (/ (pow (cbrt -1.0) 4.0) t_m)))
          (/ 1.0 (/ t_2 t_4)))
         t_3))))))

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 t_3 = cbrt((sin(k_m) * tan(k_m)));
	double t_4 = sqrt(2.0) / k_m;
	double t_5 = t_m * t_4;
	double tmp;
	if (k_m <= 2000000000.0) {
		tmp = pow((sqrt(2.0) / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))), 2.0);
	} else if (k_m <= 9.2e+222) {
		tmp = (t_5 * pow(((t_m * t_2) * t_3), -2.0)) * ((t_5 / (t_m / pow(cbrt(l), 2.0))) / t_3);
	} else {
		tmp = ((cbrt((pow(l, 4.0) * (pow(cos(k_m), 2.0) / pow(sin(k_m), 4.0)))) * (t_4 * (pow(cbrt(-1.0), 4.0) / t_m))) * (1.0 / (t_2 / t_4))) / t_3;
	}
	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 t_3 = Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double t_4 = Math.sqrt(2.0) / k_m;
	double t_5 = t_m * t_4;
	double tmp;
	if (k_m <= 2000000000.0) {
		tmp = Math.pow((Math.sqrt(2.0) / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else if (k_m <= 9.2e+222) {
		tmp = (t_5 * Math.pow(((t_m * t_2) * t_3), -2.0)) * ((t_5 / (t_m / Math.pow(Math.cbrt(l), 2.0))) / t_3);
	} else {
		tmp = ((Math.cbrt((Math.pow(l, 4.0) * (Math.pow(Math.cos(k_m), 2.0) / Math.pow(Math.sin(k_m), 4.0)))) * (t_4 * (Math.pow(Math.cbrt(-1.0), 4.0) / t_m))) * (1.0 / (t_2 / t_4))) / t_3;
	}
	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
	t_3 = cbrt(Float64(sin(k_m) * tan(k_m)))
	t_4 = Float64(sqrt(2.0) / k_m)
	t_5 = Float64(t_m * t_4)
	tmp = 0.0
	if (k_m <= 2000000000.0)
		tmp = Float64(sqrt(2.0) / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0;
	elseif (k_m <= 9.2e+222)
		tmp = Float64(Float64(t_5 * (Float64(Float64(t_m * t_2) * t_3) ^ -2.0)) * Float64(Float64(t_5 / Float64(t_m / (cbrt(l) ^ 2.0))) / t_3));
	else
		tmp = Float64(Float64(Float64(cbrt(Float64((l ^ 4.0) * Float64((cos(k_m) ^ 2.0) / (sin(k_m) ^ 4.0)))) * Float64(t_4 * Float64((cbrt(-1.0) ^ 4.0) / t_m))) * Float64(1.0 / Float64(t_2 / t_4))) / t_3);
	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]}, Block[{t$95$3 = N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$m * t$95$4), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 2000000000.0], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 9.2e+222], N[(N[(t$95$5 * N[Power[N[(N[(t$95$m * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$5 / N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[(N[Power[l, 4.0], $MachinePrecision] * N[(N[Power[N[Cos[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$4 * N[(N[Power[N[Power[-1.0, 1/3], $MachinePrecision], 4.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t$95$2 / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2e9

   1. Initial program 39.1%

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

      1. *-commutative 39.1%

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

      2. associate-/r* 39.1%

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

   3. Simplified 43.6%

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

      1. add-sqr-sqrt 29.4%

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

   6. Applied egg-rr 33.2%

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

      1. unpow2 33.2%

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

      2. associate-/l/ 33.7%

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

      3. associate-*l/ 35.3%

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

      4. associate-*l/ 35.3%

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

      5. *-commutative 35.3%

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

   8. Simplified 35.3%

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

   9. Taylor expanded in k around inf 51.7%

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

       1. associate-/l* 52.6%

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

   11. Simplified 52.6%

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

   if 2e9 < k < 9.20000000000000043e222

   1. Initial program 28.0%

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

      1. *-commutative 28.0%

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

      2. associate-/r* 28.0%

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

   3. Simplified 48.2%

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

      1. add-sqr-sqrt 48.2%

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

      2. add-cube-cbrt 48.2%

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

      3. times-frac 48.2%

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

   6. Applied egg-rr 82.2%

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

      1. associate-/r/ 82.3%

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

      2. associate-/r* 82.3%

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

      3. associate-/r/ 82.3%

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

   8. Simplified 82.3%

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

      1. div-inv 82.3%

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

      2. pow-flip 84.3%

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

      3. div-inv 84.4%

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

      4. pow-flip 84.3%

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

      5. metadata-eval 84.3%

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

      6. metadata-eval 84.3%

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

   10. Applied egg-rr 84.3%

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

   if 9.20000000000000043e222 < k

   1. Initial program 16.7%

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

      1. *-commutative 16.7%

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

      2. associate-/r* 16.7%

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

   3. Simplified 27.8%

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

      1. add-sqr-sqrt 27.8%

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

      2. add-cube-cbrt 27.8%

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

      3. times-frac 27.8%

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

   6. Applied egg-rr 56.3%

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

      1. associate-/r/ 56.3%

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

      2. associate-/r* 56.3%

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

      3. associate-/r/ 56.3%

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

   8. Simplified 56.3%

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

      1. associate-*r/ 56.3%

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

   10. Applied egg-rr 56.3%

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

       1. clear-num 56.3%

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

       2. inv-pow 56.3%

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

   12. Applied egg-rr 56.3%

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

       1. unpow-1 56.3%

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

       2. *-commutative 56.3%

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

       3. times-frac 56.3%

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

       4. *-inverses 56.3%

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

   14. Simplified 56.3%

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

   15. Taylor expanded in l around -inf 72.6%

       \[\leadsto \frac{\color{blue}{\left(\frac{{\left(\sqrt[3]{-1}\right)}^{4} \cdot \sqrt{2}}{k \cdot t} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k}^{2}}{{\sin k}^{4}}}\right)} \cdot \frac{1}{1 \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{-2}}{\frac{\sqrt{2}}{k}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]
   16. Step-by-step derivation

       1. *-commutative 72.6%

          \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k}^{2}}{{\sin k}^{4}}} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{4} \cdot \sqrt{2}}{k \cdot t}\right)} \cdot \frac{1}{1 \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{-2}}{\frac{\sqrt{2}}{k}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

       2. associate-/l* 72.6%

          \[\leadsto \frac{\left(\sqrt[3]{\color{blue}{{\ell}^{4} \cdot \frac{{\cos k}^{2}}{{\sin k}^{4}}}} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{4} \cdot \sqrt{2}}{k \cdot t}\right) \cdot \frac{1}{1 \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{-2}}{\frac{\sqrt{2}}{k}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

       3. *-commutative 72.6%

          \[\leadsto \frac{\left(\sqrt[3]{{\ell}^{4} \cdot \frac{{\cos k}^{2}}{{\sin k}^{4}}} \cdot \frac{\color{blue}{\sqrt{2} \cdot {\left(\sqrt[3]{-1}\right)}^{4}}}{k \cdot t}\right) \cdot \frac{1}{1 \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{-2}}{\frac{\sqrt{2}}{k}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

       4. times-frac 72.7%

          \[\leadsto \frac{\left(\sqrt[3]{{\ell}^{4} \cdot \frac{{\cos k}^{2}}{{\sin k}^{4}}} \cdot \color{blue}{\left(\frac{\sqrt{2}}{k} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{4}}{t}\right)}\right) \cdot \frac{1}{1 \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{-2}}{\frac{\sqrt{2}}{k}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]

   17. Simplified 72.7%

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{\ell}^{4} \cdot \frac{{\cos k}^{2}}{{\sin k}^{4}}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{4}}{t}\right)\right)} \cdot \frac{1}{1 \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{-2}}{\frac{\sqrt{2}}{k}}}}{\sqrt[3]{\sin k \cdot \tan k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2000000000:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 9.2 \cdot 10^{+222}:\\ \;\;\;\;\left(\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{t \cdot \frac{\sqrt{2}}{k}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{{\ell}^{4} \cdot \frac{{\cos k}^{2}}{{\sin k}^{4}}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{4}}{t}\right)\right) \cdot \frac{1}{\frac{{\left(\sqrt[3]{\ell}\right)}^{-2}}{\frac{\sqrt{2}}{k}}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_3 := \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t_4 := \frac{\sqrt{2}}{k\_m}\\ t_5 := t\_m \cdot t\_4\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2000000000:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 1.35 \cdot 10^{+223}:\\ \;\;\;\;\left(t\_5 \cdot {\left(\left(t\_m \cdot t\_2\right) \cdot t\_3\right)}^{-2}\right) \cdot \frac{\frac{t\_5}{\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k\_m \cdot t\_m} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k\_m}^{2}}{{\sin k\_m}^{4}}}\right) \cdot \frac{\frac{t\_4}{t\_2}}{t\_3}\\ \end{array} \end{array} \end{array} \]

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_3 (cbrt (* (sin k_m) (tan k_m))))
        (t_4 (/ (sqrt 2.0) k_m))
        (t_5 (* t_m t_4)))
   (*
    t_s
    (if (<= k_m 2000000000.0)
      (pow
       (/ (sqrt 2.0) (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m)))))
       2.0)
      (if (<= k_m 1.35e+223)
        (*
         (* t_5 (pow (* (* t_m t_2) t_3) -2.0))
         (/ (/ t_5 (/ t_m (pow (cbrt l) 2.0))) t_3))
        (*
         (*
          (/ (sqrt 2.0) (* k_m t_m))
          (cbrt (/ (* (pow l 4.0) (pow (cos k_m) 2.0)) (pow (sin k_m) 4.0))))
         (/ (/ t_4 t_2) t_3)))))))

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 t_3 = cbrt((sin(k_m) * tan(k_m)));
	double t_4 = sqrt(2.0) / k_m;
	double t_5 = t_m * t_4;
	double tmp;
	if (k_m <= 2000000000.0) {
		tmp = pow((sqrt(2.0) / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))), 2.0);
	} else if (k_m <= 1.35e+223) {
		tmp = (t_5 * pow(((t_m * t_2) * t_3), -2.0)) * ((t_5 / (t_m / pow(cbrt(l), 2.0))) / t_3);
	} else {
		tmp = ((sqrt(2.0) / (k_m * t_m)) * cbrt(((pow(l, 4.0) * pow(cos(k_m), 2.0)) / pow(sin(k_m), 4.0)))) * ((t_4 / t_2) / t_3);
	}
	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 t_3 = Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double t_4 = Math.sqrt(2.0) / k_m;
	double t_5 = t_m * t_4;
	double tmp;
	if (k_m <= 2000000000.0) {
		tmp = Math.pow((Math.sqrt(2.0) / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else if (k_m <= 1.35e+223) {
		tmp = (t_5 * Math.pow(((t_m * t_2) * t_3), -2.0)) * ((t_5 / (t_m / Math.pow(Math.cbrt(l), 2.0))) / t_3);
	} else {
		tmp = ((Math.sqrt(2.0) / (k_m * t_m)) * Math.cbrt(((Math.pow(l, 4.0) * Math.pow(Math.cos(k_m), 2.0)) / Math.pow(Math.sin(k_m), 4.0)))) * ((t_4 / t_2) / t_3);
	}
	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
	t_3 = cbrt(Float64(sin(k_m) * tan(k_m)))
	t_4 = Float64(sqrt(2.0) / k_m)
	t_5 = Float64(t_m * t_4)
	tmp = 0.0
	if (k_m <= 2000000000.0)
		tmp = Float64(sqrt(2.0) / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0;
	elseif (k_m <= 1.35e+223)
		tmp = Float64(Float64(t_5 * (Float64(Float64(t_m * t_2) * t_3) ^ -2.0)) * Float64(Float64(t_5 / Float64(t_m / (cbrt(l) ^ 2.0))) / t_3));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / Float64(k_m * t_m)) * cbrt(Float64(Float64((l ^ 4.0) * (cos(k_m) ^ 2.0)) / (sin(k_m) ^ 4.0)))) * Float64(Float64(t_4 / t_2) / t_3));
	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]}, Block[{t$95$3 = N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$m * t$95$4), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 2000000000.0], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 1.35e+223], N[(N[(t$95$5 * N[Power[N[(N[(t$95$m * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$5 / N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[Power[l, 4.0], $MachinePrecision] * N[Power[N[Cos[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$4 / t$95$2), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2e9

   1. Initial program 39.1%

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

      1. *-commutative 39.1%

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

      2. associate-/r* 39.1%

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

   3. Simplified 43.6%

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

      1. add-sqr-sqrt 29.4%

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

   6. Applied egg-rr 33.2%

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

      1. unpow2 33.2%

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

      2. associate-/l/ 33.7%

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

      3. associate-*l/ 35.3%

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

      4. associate-*l/ 35.3%

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

      5. *-commutative 35.3%

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

   8. Simplified 35.3%

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

   9. Taylor expanded in k around inf 51.7%

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

       1. associate-/l* 52.6%

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

   11. Simplified 52.6%

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

   if 2e9 < k < 1.35e223

   1. Initial program 28.0%

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

      1. *-commutative 28.0%

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

      2. associate-/r* 28.0%

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

   3. Simplified 48.2%

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

      1. add-sqr-sqrt 48.2%

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

      2. add-cube-cbrt 48.2%

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

      3. times-frac 48.2%

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

   6. Applied egg-rr 82.2%

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

      1. associate-/r/ 82.3%

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

      2. associate-/r* 82.3%

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

      3. associate-/r/ 82.3%

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

   8. Simplified 82.3%

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

      1. div-inv 82.3%

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

      2. pow-flip 84.3%

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

      3. div-inv 84.4%

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

      4. pow-flip 84.3%

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

      5. metadata-eval 84.3%

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

      6. metadata-eval 84.3%

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

   10. Applied egg-rr 84.3%

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

   if 1.35e223 < k

   1. Initial program 16.7%

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

      1. *-commutative 16.7%

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

      2. associate-/r* 16.7%

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

   3. Simplified 27.8%

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

      1. add-sqr-sqrt 27.8%

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

      2. add-cube-cbrt 27.8%

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

      3. times-frac 27.8%

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

   6. Applied egg-rr 56.3%

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

      1. associate-/r/ 56.3%

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

      2. associate-/r* 56.3%

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

      3. associate-/r/ 56.3%

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

   8. Simplified 56.3%

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

      1. div-inv 56.3%

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

      2. div-inv 56.3%

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

      3. pow-flip 56.3%

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

      4. metadata-eval 56.3%

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

   10. Applied egg-rr 56.3%

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

       1. associate-*r/ 56.3%

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

       2. *-rgt-identity 56.3%

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

       3. associate-/r* 56.3%

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

       4. associate-/l* 56.3%

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

       5. *-inverses 56.3%

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

   12. Simplified 56.3%

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

   13. Taylor expanded in k around inf 72.7%

       \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k \cdot t} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k}^{2}}{{\sin k}^{4}}}\right)} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot 1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2000000000:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{+223}:\\ \;\;\;\;\left(\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \frac{\frac{t \cdot \frac{\sqrt{2}}{k}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k \cdot t} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k}^{2}}{{\sin k}^{4}}}\right) \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_3 := \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t_4 := \frac{\sqrt{2}}{k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2000000000:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 1.12 \cdot 10^{+223}:\\ \;\;\;\;\frac{\left(\left(t\_m \cdot t\_4\right) \cdot {\left(\left(t\_m \cdot t\_2\right) \cdot t\_3\right)}^{-2}\right) \cdot \left(t\_4 \cdot \frac{1}{t\_2}\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k\_m \cdot t\_m} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k\_m}^{2}}{{\sin k\_m}^{4}}}\right) \cdot \frac{\frac{t\_4}{t\_2}}{t\_3}\\ \end{array} \end{array} \end{array} \]

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_3 (cbrt (* (sin k_m) (tan k_m))))
        (t_4 (/ (sqrt 2.0) k_m)))
   (*
    t_s
    (if (<= k_m 2000000000.0)
      (pow
       (/ (sqrt 2.0) (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m)))))
       2.0)
      (if (<= k_m 1.12e+223)
        (/
         (* (* (* t_m t_4) (pow (* (* t_m t_2) t_3) -2.0)) (* t_4 (/ 1.0 t_2)))
         t_3)
        (*
         (*
          (/ (sqrt 2.0) (* k_m t_m))
          (cbrt (/ (* (pow l 4.0) (pow (cos k_m) 2.0)) (pow (sin k_m) 4.0))))
         (/ (/ t_4 t_2) t_3)))))))

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 t_3 = cbrt((sin(k_m) * tan(k_m)));
	double t_4 = sqrt(2.0) / k_m;
	double tmp;
	if (k_m <= 2000000000.0) {
		tmp = pow((sqrt(2.0) / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))), 2.0);
	} else if (k_m <= 1.12e+223) {
		tmp = (((t_m * t_4) * pow(((t_m * t_2) * t_3), -2.0)) * (t_4 * (1.0 / t_2))) / t_3;
	} else {
		tmp = ((sqrt(2.0) / (k_m * t_m)) * cbrt(((pow(l, 4.0) * pow(cos(k_m), 2.0)) / pow(sin(k_m), 4.0)))) * ((t_4 / t_2) / t_3);
	}
	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 t_3 = Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double t_4 = Math.sqrt(2.0) / k_m;
	double tmp;
	if (k_m <= 2000000000.0) {
		tmp = Math.pow((Math.sqrt(2.0) / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else if (k_m <= 1.12e+223) {
		tmp = (((t_m * t_4) * Math.pow(((t_m * t_2) * t_3), -2.0)) * (t_4 * (1.0 / t_2))) / t_3;
	} else {
		tmp = ((Math.sqrt(2.0) / (k_m * t_m)) * Math.cbrt(((Math.pow(l, 4.0) * Math.pow(Math.cos(k_m), 2.0)) / Math.pow(Math.sin(k_m), 4.0)))) * ((t_4 / t_2) / t_3);
	}
	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
	t_3 = cbrt(Float64(sin(k_m) * tan(k_m)))
	t_4 = Float64(sqrt(2.0) / k_m)
	tmp = 0.0
	if (k_m <= 2000000000.0)
		tmp = Float64(sqrt(2.0) / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0;
	elseif (k_m <= 1.12e+223)
		tmp = Float64(Float64(Float64(Float64(t_m * t_4) * (Float64(Float64(t_m * t_2) * t_3) ^ -2.0)) * Float64(t_4 * Float64(1.0 / t_2))) / t_3);
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / Float64(k_m * t_m)) * cbrt(Float64(Float64((l ^ 4.0) * (cos(k_m) ^ 2.0)) / (sin(k_m) ^ 4.0)))) * Float64(Float64(t_4 / t_2) / t_3));
	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]}, Block[{t$95$3 = N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 2000000000.0], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 1.12e+223], N[(N[(N[(N[(t$95$m * t$95$4), $MachinePrecision] * N[Power[N[(N[(t$95$m * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$4 * N[(1.0 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[Power[l, 4.0], $MachinePrecision] * N[Power[N[Cos[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$4 / t$95$2), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2e9

   1. Initial program 39.1%

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

      1. *-commutative 39.1%

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

      2. associate-/r* 39.1%

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

   3. Simplified 43.6%

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

      1. add-sqr-sqrt 29.4%

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

   6. Applied egg-rr 33.2%

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

      1. unpow2 33.2%

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

      2. associate-/l/ 33.7%

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

      3. associate-*l/ 35.3%

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

      4. associate-*l/ 35.3%

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

      5. *-commutative 35.3%

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

   8. Simplified 35.3%

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

   9. Taylor expanded in k around inf 51.7%

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

       1. associate-/l* 52.6%

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

   11. Simplified 52.6%

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

   if 2e9 < k < 1.1200000000000001e223

   1. Initial program 28.0%

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

      1. *-commutative 28.0%

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

      2. associate-/r* 28.0%

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

   3. Simplified 48.2%

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

      1. add-sqr-sqrt 48.2%

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

      2. add-cube-cbrt 48.2%

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

      3. times-frac 48.2%

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

   6. Applied egg-rr 82.2%

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

      1. associate-/r/ 82.3%

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

      2. associate-/r* 82.3%

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

      3. associate-/r/ 82.3%

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

   8. Simplified 82.3%

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

      1. associate-*r/ 82.2%

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

   10. Applied egg-rr 84.3%

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

       1. associate-/l* 84.3%

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

   12. Applied egg-rr 84.3%

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

       1. associate-/r* 84.3%

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

       2. *-inverses 84.3%

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

   14. Simplified 84.3%

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

   if 1.1200000000000001e223 < k

   1. Initial program 16.7%

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

      1. *-commutative 16.7%

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

      2. associate-/r* 16.7%

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

   3. Simplified 27.8%

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

      1. add-sqr-sqrt 27.8%

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

      2. add-cube-cbrt 27.8%

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

      3. times-frac 27.8%

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

   6. Applied egg-rr 56.3%

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

      1. associate-/r/ 56.3%

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

      2. associate-/r* 56.3%

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

      3. associate-/r/ 56.3%

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

   8. Simplified 56.3%

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

      1. div-inv 56.3%

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

      2. div-inv 56.3%

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

      3. pow-flip 56.3%

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

      4. metadata-eval 56.3%

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

   10. Applied egg-rr 56.3%

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

       1. associate-*r/ 56.3%

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

       2. *-rgt-identity 56.3%

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

       3. associate-/r* 56.3%

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

       4. associate-/l* 56.3%

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

       5. *-inverses 56.3%

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

   12. Simplified 56.3%

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

   13. Taylor expanded in k around inf 72.7%

       \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k \cdot t} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k}^{2}}{{\sin k}^{4}}}\right)} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot 1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2000000000:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.12 \cdot 10^{+223}:\\ \;\;\;\;\frac{\left(\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k \cdot t} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k}^{2}}{{\sin k}^{4}}}\right) \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t_3 := \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t_4 := \frac{\sqrt{2}}{k\_m}\\ t_5 := \frac{t\_4}{t\_2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2000000000:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 5 \cdot 10^{+222}:\\ \;\;\;\;\frac{\left(t\_m \cdot t\_4\right) \cdot \left({\left(\left(t\_m \cdot t\_2\right) \cdot t\_3\right)}^{-2} \cdot t\_5\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k\_m \cdot t\_m} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k\_m}^{2}}{{\sin k\_m}^{4}}}\right) \cdot \frac{t\_5}{t\_3}\\ \end{array} \end{array} \end{array} \]

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_3 (cbrt (* (sin k_m) (tan k_m))))
        (t_4 (/ (sqrt 2.0) k_m))
        (t_5 (/ t_4 t_2)))
   (*
    t_s
    (if (<= k_m 2000000000.0)
      (pow
       (/ (sqrt 2.0) (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m)))))
       2.0)
      (if (<= k_m 5e+222)
        (/ (* (* t_m t_4) (* (pow (* (* t_m t_2) t_3) -2.0) t_5)) t_3)
        (*
         (*
          (/ (sqrt 2.0) (* k_m t_m))
          (cbrt (/ (* (pow l 4.0) (pow (cos k_m) 2.0)) (pow (sin k_m) 4.0))))
         (/ t_5 t_3)))))))

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 t_3 = cbrt((sin(k_m) * tan(k_m)));
	double t_4 = sqrt(2.0) / k_m;
	double t_5 = t_4 / t_2;
	double tmp;
	if (k_m <= 2000000000.0) {
		tmp = pow((sqrt(2.0) / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))), 2.0);
	} else if (k_m <= 5e+222) {
		tmp = ((t_m * t_4) * (pow(((t_m * t_2) * t_3), -2.0) * t_5)) / t_3;
	} else {
		tmp = ((sqrt(2.0) / (k_m * t_m)) * cbrt(((pow(l, 4.0) * pow(cos(k_m), 2.0)) / pow(sin(k_m), 4.0)))) * (t_5 / t_3);
	}
	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 t_3 = Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double t_4 = Math.sqrt(2.0) / k_m;
	double t_5 = t_4 / t_2;
	double tmp;
	if (k_m <= 2000000000.0) {
		tmp = Math.pow((Math.sqrt(2.0) / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else if (k_m <= 5e+222) {
		tmp = ((t_m * t_4) * (Math.pow(((t_m * t_2) * t_3), -2.0) * t_5)) / t_3;
	} else {
		tmp = ((Math.sqrt(2.0) / (k_m * t_m)) * Math.cbrt(((Math.pow(l, 4.0) * Math.pow(Math.cos(k_m), 2.0)) / Math.pow(Math.sin(k_m), 4.0)))) * (t_5 / t_3);
	}
	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
	t_3 = cbrt(Float64(sin(k_m) * tan(k_m)))
	t_4 = Float64(sqrt(2.0) / k_m)
	t_5 = Float64(t_4 / t_2)
	tmp = 0.0
	if (k_m <= 2000000000.0)
		tmp = Float64(sqrt(2.0) / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0;
	elseif (k_m <= 5e+222)
		tmp = Float64(Float64(Float64(t_m * t_4) * Float64((Float64(Float64(t_m * t_2) * t_3) ^ -2.0) * t_5)) / t_3);
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / Float64(k_m * t_m)) * cbrt(Float64(Float64((l ^ 4.0) * (cos(k_m) ^ 2.0)) / (sin(k_m) ^ 4.0)))) * Float64(t_5 / t_3));
	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]}, Block[{t$95$3 = N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 / t$95$2), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 2000000000.0], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 5e+222], N[(N[(N[(t$95$m * t$95$4), $MachinePrecision] * N[(N[Power[N[(N[(t$95$m * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision], -2.0], $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[Power[l, 4.0], $MachinePrecision] * N[Power[N[Cos[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(t$95$5 / t$95$3), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2e9

   1. Initial program 39.1%

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

      1. *-commutative 39.1%

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

      2. associate-/r* 39.1%

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

   3. Simplified 43.6%

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

      1. add-sqr-sqrt 29.4%

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

   6. Applied egg-rr 33.2%

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

      1. unpow2 33.2%

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

      2. associate-/l/ 33.7%

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

      3. associate-*l/ 35.3%

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

      4. associate-*l/ 35.3%

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

      5. *-commutative 35.3%

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

   8. Simplified 35.3%

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

   9. Taylor expanded in k around inf 51.7%

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

       1. associate-/l* 52.6%

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

   11. Simplified 52.6%

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

   if 2e9 < k < 5.00000000000000023e222

   1. Initial program 28.0%

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

      1. *-commutative 28.0%

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

      2. associate-/r* 28.0%

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

   3. Simplified 48.2%

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

      1. add-sqr-sqrt 48.2%

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

      2. add-cube-cbrt 48.2%

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

      3. times-frac 48.2%

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

   6. Applied egg-rr 82.2%

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

      1. associate-/r/ 82.3%

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

      2. associate-/r* 82.3%

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

      3. associate-/r/ 82.3%

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

   8. Simplified 82.3%

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

      1. div-inv 82.3%

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

      2. div-inv 82.3%

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

      3. pow-flip 82.3%

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

      4. metadata-eval 82.3%

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

   10. Applied egg-rr 82.3%

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

       1. associate-*r/ 82.4%

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

       2. *-rgt-identity 82.4%

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

       3. associate-/r* 82.4%

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

       4. associate-/l* 82.4%

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

       5. *-inverses 82.4%

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

   12. Simplified 82.4%

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

       1. div-inv 82.4%

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

       2. pow-flip 84.4%

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

       3. div-inv 84.4%

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

       4. pow-flip 84.3%

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

       5. metadata-eval 84.3%

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

       6. metadata-eval 84.3%

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

   14. Applied egg-rr 84.3%

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

   if 5.00000000000000023e222 < k

   1. Initial program 16.7%

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

      1. *-commutative 16.7%

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

      2. associate-/r* 16.7%

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

   3. Simplified 27.8%

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

      1. add-sqr-sqrt 27.8%

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

      2. add-cube-cbrt 27.8%

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

      3. times-frac 27.8%

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

   6. Applied egg-rr 56.3%

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

      1. associate-/r/ 56.3%

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

      2. associate-/r* 56.3%

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

      3. associate-/r/ 56.3%

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

   8. Simplified 56.3%

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

      1. div-inv 56.3%

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

      2. div-inv 56.3%

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

      3. pow-flip 56.3%

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

      4. metadata-eval 56.3%

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

   10. Applied egg-rr 56.3%

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

       1. associate-*r/ 56.3%

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

       2. *-rgt-identity 56.3%

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

       3. associate-/r* 56.3%

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

       4. associate-/l* 56.3%

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

       5. *-inverses 56.3%

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

   12. Simplified 56.3%

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

   13. Taylor expanded in k around inf 72.7%

       \[\leadsto \color{blue}{\left(\frac{\sqrt{2}}{k \cdot t} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k}^{2}}{{\sin k}^{4}}}\right)} \cdot \frac{\frac{\frac{\sqrt{2}}{k} \cdot 1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2000000000:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+222}:\\ \;\;\;\;\frac{\left(t \cdot \frac{\sqrt{2}}{k}\right) \cdot \left({\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{-2} \cdot \frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}{\sqrt[3]{\sin k \cdot \tan k}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k \cdot t} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k}^{2}}{{\sin k}^{4}}}\right) \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sin k\_m \cdot \tan k\_m\\ t_3 := \sqrt[3]{t\_2}\\ t_4 := \frac{\sqrt{2}}{k\_m}\\ t_5 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2000000000:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 1.6 \cdot 10^{+131}:\\ \;\;\;\;\frac{{t\_4}^{2}}{t\_5 \cdot t\_3} \cdot \left(t\_m \cdot {\left(t\_5 \cdot \left(t\_m \cdot t\_3\right)\right)}^{-2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{t\_5}{t\_4}} \cdot \left(t\_4 \cdot \left(t\_m \cdot {\left(t\_m \cdot \sqrt[3]{t\_2 \cdot {\ell}^{-2}}\right)}^{-2}\right)\right)}{t\_3}\\ \end{array} \end{array} \end{array} \]

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 (* (sin k_m) (tan k_m)))
        (t_3 (cbrt t_2))
        (t_4 (/ (sqrt 2.0) k_m))
        (t_5 (pow (cbrt l) -2.0)))
   (*
    t_s
    (if (<= k_m 2000000000.0)
      (pow
       (/ (sqrt 2.0) (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m)))))
       2.0)
      (if (<= k_m 1.6e+131)
        (*
         (/ (pow t_4 2.0) (* t_5 t_3))
         (* t_m (pow (* t_5 (* t_m t_3)) -2.0)))
        (/
         (*
          (/ 1.0 (/ t_5 t_4))
          (* t_4 (* t_m (pow (* t_m (cbrt (* t_2 (pow l -2.0)))) -2.0))))
         t_3))))))

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 = sin(k_m) * tan(k_m);
	double t_3 = cbrt(t_2);
	double t_4 = sqrt(2.0) / k_m;
	double t_5 = pow(cbrt(l), -2.0);
	double tmp;
	if (k_m <= 2000000000.0) {
		tmp = pow((sqrt(2.0) / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))), 2.0);
	} else if (k_m <= 1.6e+131) {
		tmp = (pow(t_4, 2.0) / (t_5 * t_3)) * (t_m * pow((t_5 * (t_m * t_3)), -2.0));
	} else {
		tmp = ((1.0 / (t_5 / t_4)) * (t_4 * (t_m * pow((t_m * cbrt((t_2 * pow(l, -2.0)))), -2.0)))) / t_3;
	}
	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.sin(k_m) * Math.tan(k_m);
	double t_3 = Math.cbrt(t_2);
	double t_4 = Math.sqrt(2.0) / k_m;
	double t_5 = Math.pow(Math.cbrt(l), -2.0);
	double tmp;
	if (k_m <= 2000000000.0) {
		tmp = Math.pow((Math.sqrt(2.0) / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else if (k_m <= 1.6e+131) {
		tmp = (Math.pow(t_4, 2.0) / (t_5 * t_3)) * (t_m * Math.pow((t_5 * (t_m * t_3)), -2.0));
	} else {
		tmp = ((1.0 / (t_5 / t_4)) * (t_4 * (t_m * Math.pow((t_m * Math.cbrt((t_2 * Math.pow(l, -2.0)))), -2.0)))) / t_3;
	}
	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(sin(k_m) * tan(k_m))
	t_3 = cbrt(t_2)
	t_4 = Float64(sqrt(2.0) / k_m)
	t_5 = cbrt(l) ^ -2.0
	tmp = 0.0
	if (k_m <= 2000000000.0)
		tmp = Float64(sqrt(2.0) / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0;
	elseif (k_m <= 1.6e+131)
		tmp = Float64(Float64((t_4 ^ 2.0) / Float64(t_5 * t_3)) * Float64(t_m * (Float64(t_5 * Float64(t_m * t_3)) ^ -2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(t_5 / t_4)) * Float64(t_4 * Float64(t_m * (Float64(t_m * cbrt(Float64(t_2 * (l ^ -2.0)))) ^ -2.0)))) / t_3);
	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[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 1/3], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 2000000000.0], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 1.6e+131], N[(N[(N[Power[t$95$4, 2.0], $MachinePrecision] / N[(t$95$5 * t$95$3), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Power[N[(t$95$5 * N[(t$95$m * t$95$3), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(t$95$5 / t$95$4), $MachinePrecision]), $MachinePrecision] * N[(t$95$4 * N[(t$95$m * N[Power[N[(t$95$m * N[Power[N[(t$95$2 * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 2e9

   1. Initial program 39.1%

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

      1. *-commutative 39.1%

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

      2. associate-/r* 39.1%

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

   3. Simplified 43.6%

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

      1. add-sqr-sqrt 29.4%

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

   6. Applied egg-rr 33.2%

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

      1. unpow2 33.2%

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

      2. associate-/l/ 33.7%

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

      3. associate-*l/ 35.3%

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

      4. associate-*l/ 35.3%

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

      5. *-commutative 35.3%

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

   8. Simplified 35.3%

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

   9. Taylor expanded in k around inf 51.7%

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

       1. associate-/l* 52.6%

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

   11. Simplified 52.6%

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

   if 2e9 < k < 1.6000000000000001e131

   1. Initial program 28.7%

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

      1. *-commutative 28.7%

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

      2. associate-/r* 28.7%

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

   3. Simplified 46.5%

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

      1. add-sqr-sqrt 46.5%

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

      2. add-cube-cbrt 46.5%

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

      3. times-frac 46.4%

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

   6. Applied egg-rr 84.8%

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

      1. associate-/r/ 84.8%

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

      2. associate-/r* 84.8%

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

      3. associate-/r/ 84.9%

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

   8. Simplified 84.9%

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

      1. associate-*r/ 84.8%

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

   10. Applied egg-rr 88.0%

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

       1. clear-num 88.0%

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

       2. inv-pow 88.0%

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

   12. Applied egg-rr 88.0%

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

       1. unpow-1 88.0%

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

       2. *-commutative 88.0%

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

       3. times-frac 88.0%

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

       4. *-inverses 88.0%

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

   14. Simplified 88.0%

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

       1. associate-/l* 88.0%

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

       2. associate-*l* 87.9%

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

       3. *-commutative 87.9%

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

   16. Applied egg-rr 87.9%

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

       1. *-commutative 87.9%

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

       2. associate-*r* 87.9%

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

       3. associate-/l/ 87.9%

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

       4. associate-*l/ 87.9%

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

       5. unpow2 87.9%

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

   18. Simplified 87.8%

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

   if 1.6000000000000001e131 < k

   1. Initial program 21.2%

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

      1. *-commutative 21.2%

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

      2. associate-/r* 21.2%

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

   3. Simplified 38.5%

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

      1. add-sqr-sqrt 38.5%

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

      2. add-cube-cbrt 38.5%

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

      3. times-frac 38.5%

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

   6. Applied egg-rr 65.9%

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

      1. associate-/r/ 65.9%

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

      2. associate-/r* 65.9%

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

      3. associate-/r/ 65.9%

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

   8. Simplified 65.9%

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

      1. associate-*r/ 65.9%

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

   10. Applied egg-rr 65.9%

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

       1. clear-num 65.9%

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

       2. inv-pow 65.9%

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

   12. Applied egg-rr 65.9%

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

       1. unpow-1 65.9%

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

       2. *-commutative 65.9%

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

       3. times-frac 65.9%

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

       4. *-inverses 65.9%

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

   14. Simplified 65.9%

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

       1. pow1 65.9%

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

   16. Applied egg-rr 60.8%

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

       1. unpow1 60.8%

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

   18. Simplified 60.8%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2000000000:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+131}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt{2}}{k}\right)}^{2}}{{\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}} \cdot \left(t \cdot {\left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{\sin k \cdot \tan k}\right)\right)}^{-2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{{\left(\sqrt[3]{\ell}\right)}^{-2}}{\frac{\sqrt{2}}{k}}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot {\ell}^{-2}}\right)}^{-2}\right)\right)}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sqrt{2}}{k\_m}\\ t_3 := \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2000000000:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot t\_2}{{\left(t\_3 \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{\frac{t\_2}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{t\_3}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (sqrt 2.0) k_m)) (t_3 (cbrt (* (sin k_m) (tan k_m)))))
   (*
    t_s
    (if (<= k_m 2000000000.0)
      (pow
       (/ (sqrt 2.0) (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m)))))
       2.0)
      (*
       (/ (* t_m t_2) (pow (* t_3 (/ t_m (pow (cbrt l) 2.0))) 2.0))
       (/ (/ t_2 (pow (cbrt l) -2.0)) t_3))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = sqrt(2.0) / k_m;
	double t_3 = cbrt((sin(k_m) * tan(k_m)));
	double tmp;
	if (k_m <= 2000000000.0) {
		tmp = pow((sqrt(2.0) / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))), 2.0);
	} else {
		tmp = ((t_m * t_2) / pow((t_3 * (t_m / pow(cbrt(l), 2.0))), 2.0)) * ((t_2 / pow(cbrt(l), -2.0)) / t_3);
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.sqrt(2.0) / k_m;
	double t_3 = Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double tmp;
	if (k_m <= 2000000000.0) {
		tmp = Math.pow((Math.sqrt(2.0) / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else {
		tmp = ((t_m * t_2) / Math.pow((t_3 * (t_m / Math.pow(Math.cbrt(l), 2.0))), 2.0)) * ((t_2 / Math.pow(Math.cbrt(l), -2.0)) / t_3);
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(sqrt(2.0) / k_m)
	t_3 = cbrt(Float64(sin(k_m) * tan(k_m)))
	tmp = 0.0
	if (k_m <= 2000000000.0)
		tmp = Float64(sqrt(2.0) / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(t_m * t_2) / (Float64(t_3 * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 2.0)) * Float64(Float64(t_2 / (cbrt(l) ^ -2.0)) / t_3));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 2000000000.0], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(t$95$m * t$95$2), $MachinePrecision] / N[Power[N[(t$95$3 * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 / N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2e9

   1. Initial program 39.1%

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

      1. *-commutative 39.1%

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

      2. associate-/r* 39.1%

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

   3. Simplified 43.6%

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

      1. add-sqr-sqrt 29.4%

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

   6. Applied egg-rr 33.2%

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

      1. unpow2 33.2%

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

      2. associate-/l/ 33.7%

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

      3. associate-*l/ 35.3%

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

      4. associate-*l/ 35.3%

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

      5. *-commutative 35.3%

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

   8. Simplified 35.3%

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

   9. Taylor expanded in k around inf 51.7%

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

       1. associate-/l* 52.6%

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

   11. Simplified 52.6%

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

   if 2e9 < k

   1. Initial program 24.6%

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

      1. *-commutative 24.6%

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

      2. associate-/r* 24.6%

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

   3. Simplified 42.2%

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

      1. add-sqr-sqrt 42.2%

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

      2. add-cube-cbrt 42.2%

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

      3. times-frac 42.1%

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

   6. Applied egg-rr 74.6%

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

      1. associate-/r/ 74.6%

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

      2. associate-/r* 74.6%

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

      3. associate-/r/ 74.6%

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

   8. Simplified 74.6%

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

      1. div-inv 74.6%

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

      2. div-inv 74.6%

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

      3. pow-flip 74.6%

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

      4. metadata-eval 74.6%

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

   10. Applied egg-rr 74.6%

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

       1. associate-*r/ 74.7%

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

       2. *-rgt-identity 74.7%

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

       3. associate-/r* 74.7%

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

       4. associate-/l* 74.7%

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

       5. *-inverses 74.7%

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

   12. Simplified 74.7%

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

   13. Taylor expanded in k around 0 74.7%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2000000000:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sin k\_m \cdot \tan k\_m\\ t_3 := \frac{\sqrt{2}}{k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.48 \cdot 10^{-6}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 1.1 \cdot 10^{+146}:\\ \;\;\;\;\left(2 \cdot \left(\left(\cos k\_m \cdot {k\_m}^{-2}\right) \cdot \frac{1}{t\_m \cdot {\sin k\_m}^{2}}\right)\right) \cdot \left(\ell \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{{\left(\sqrt[3]{\ell}\right)}^{-2}}{t\_3}} \cdot \left(t\_3 \cdot \left(t\_m \cdot {\left(t\_m \cdot \sqrt[3]{t\_2 \cdot {\ell}^{-2}}\right)}^{-2}\right)\right)}{\sqrt[3]{t\_2}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* (sin k_m) (tan k_m))) (t_3 (/ (sqrt 2.0) k_m)))
   (*
    t_s
    (if (<= k_m 1.48e-6)
      (pow
       (/ (sqrt 2.0) (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m)))))
       2.0)
      (if (<= k_m 1.1e+146)
        (*
         (*
          2.0
          (* (* (cos k_m) (pow k_m -2.0)) (/ 1.0 (* t_m (pow (sin k_m) 2.0)))))
         (* l l))
        (/
         (*
          (/ 1.0 (/ (pow (cbrt l) -2.0) t_3))
          (* t_3 (* t_m (pow (* t_m (cbrt (* t_2 (pow l -2.0)))) -2.0))))
         (cbrt t_2)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = sin(k_m) * tan(k_m);
	double t_3 = sqrt(2.0) / k_m;
	double tmp;
	if (k_m <= 1.48e-6) {
		tmp = pow((sqrt(2.0) / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))), 2.0);
	} else if (k_m <= 1.1e+146) {
		tmp = (2.0 * ((cos(k_m) * pow(k_m, -2.0)) * (1.0 / (t_m * pow(sin(k_m), 2.0))))) * (l * l);
	} else {
		tmp = ((1.0 / (pow(cbrt(l), -2.0) / t_3)) * (t_3 * (t_m * pow((t_m * cbrt((t_2 * pow(l, -2.0)))), -2.0)))) / cbrt(t_2);
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.sin(k_m) * Math.tan(k_m);
	double t_3 = Math.sqrt(2.0) / k_m;
	double tmp;
	if (k_m <= 1.48e-6) {
		tmp = Math.pow((Math.sqrt(2.0) / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else if (k_m <= 1.1e+146) {
		tmp = (2.0 * ((Math.cos(k_m) * Math.pow(k_m, -2.0)) * (1.0 / (t_m * Math.pow(Math.sin(k_m), 2.0))))) * (l * l);
	} else {
		tmp = ((1.0 / (Math.pow(Math.cbrt(l), -2.0) / t_3)) * (t_3 * (t_m * Math.pow((t_m * Math.cbrt((t_2 * Math.pow(l, -2.0)))), -2.0)))) / Math.cbrt(t_2);
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(sin(k_m) * tan(k_m))
	t_3 = Float64(sqrt(2.0) / k_m)
	tmp = 0.0
	if (k_m <= 1.48e-6)
		tmp = Float64(sqrt(2.0) / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0;
	elseif (k_m <= 1.1e+146)
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k_m) * (k_m ^ -2.0)) * Float64(1.0 / Float64(t_m * (sin(k_m) ^ 2.0))))) * Float64(l * l));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64((cbrt(l) ^ -2.0) / t_3)) * Float64(t_3 * Float64(t_m * (Float64(t_m * cbrt(Float64(t_2 * (l ^ -2.0)))) ^ -2.0)))) / cbrt(t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 1.48e-6], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 1.1e+146], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[(t$95$m * N[Power[N[(t$95$m * N[Power[N[(t$95$2 * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$2, 1/3], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.48000000000000002e-6

   1. Initial program 39.0%

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

      1. *-commutative 39.0%

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

      2. associate-/r* 39.0%

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

   3. Simplified 43.6%

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

      1. add-sqr-sqrt 29.2%

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

   6. Applied egg-rr 33.0%

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

      1. unpow2 33.0%

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

      2. associate-/l/ 33.5%

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

      3. associate-*l/ 35.1%

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

      4. associate-*l/ 35.2%

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

      5. *-commutative 35.2%

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

   8. Simplified 35.2%

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

   9. Taylor expanded in k around inf 51.7%

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

       1. associate-/l* 52.6%

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

   11. Simplified 52.6%

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

   if 1.48000000000000002e-6 < k < 1.0999999999999999e146

   1. Initial program 28.2%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in t around 0 78.2%

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

      1. associate-/r* 78.3%

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

   7. Simplified 78.3%

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

      1. div-inv 78.4%

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

      2. div-inv 78.4%

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

      3. pow-flip 78.4%

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

      4. metadata-eval 78.4%

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

   9. Applied egg-rr 78.4%

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

   if 1.0999999999999999e146 < k

   1. Initial program 22.6%

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

      1. *-commutative 22.6%

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

      2. associate-/r* 22.6%

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

   3. Simplified 37.8%

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

      1. add-sqr-sqrt 37.8%

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

      2. add-cube-cbrt 37.8%

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

      3. times-frac 37.8%

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

   6. Applied egg-rr 66.9%

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

      1. associate-/r/ 66.9%

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

      2. associate-/r* 66.9%

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

      3. associate-/r/ 66.9%

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

   8. Simplified 66.9%

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

      1. associate-*r/ 66.9%

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

   10. Applied egg-rr 66.9%

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

       1. clear-num 66.9%

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

       2. inv-pow 66.9%

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

   12. Applied egg-rr 66.9%

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

       1. unpow-1 66.9%

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

       2. *-commutative 66.9%

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

       3. times-frac 66.9%

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

       4. *-inverses 66.9%

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

   14. Simplified 66.9%

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

       1. pow1 66.9%

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

   16. Applied egg-rr 61.3%

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

       1. unpow1 61.3%

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

   18. Simplified 61.3%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.48 \cdot 10^{-6}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{+146}:\\ \;\;\;\;\left(2 \cdot \left(\left(\cos k \cdot {k}^{-2}\right) \cdot \frac{1}{t \cdot {\sin k}^{2}}\right)\right) \cdot \left(\ell \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{{\left(\sqrt[3]{\ell}\right)}^{-2}}{\frac{\sqrt{2}}{k}}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot {\left(t \cdot \sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot {\ell}^{-2}}\right)}^{-2}\right)\right)}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k\_m \cdot \tan k\_m} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 1.15 \cdot 10^{+157}:\\ \;\;\;\;\left(2 \cdot \left(\left(\cos k\_m \cdot {k\_m}^{-2}\right) \cdot \frac{1}{t\_m \cdot {\sin k\_m}^{2}}\right)\right) \cdot \left(\ell \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{t\_2}^{2}} \cdot \frac{{\left(\frac{k\_m}{t\_m}\right)}^{-2}}{t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* (cbrt (* (sin k_m) (tan k_m))) (/ t_m (pow (cbrt l) 2.0)))))
   (*
    t_s
    (if (<= k_m 6.2e-6)
      (pow
       (/ (sqrt 2.0) (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m)))))
       2.0)
      (if (<= k_m 1.15e+157)
        (*
         (*
          2.0
          (* (* (cos k_m) (pow k_m -2.0)) (/ 1.0 (* t_m (pow (sin k_m) 2.0)))))
         (* l l))
        (* (/ 2.0 (pow t_2 2.0)) (/ (pow (/ k_m t_m) -2.0) t_2)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = cbrt((sin(k_m) * tan(k_m))) * (t_m / pow(cbrt(l), 2.0));
	double tmp;
	if (k_m <= 6.2e-6) {
		tmp = pow((sqrt(2.0) / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))), 2.0);
	} else if (k_m <= 1.15e+157) {
		tmp = (2.0 * ((cos(k_m) * pow(k_m, -2.0)) * (1.0 / (t_m * pow(sin(k_m), 2.0))))) * (l * l);
	} else {
		tmp = (2.0 / pow(t_2, 2.0)) * (pow((k_m / t_m), -2.0) / t_2);
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.cbrt((Math.sin(k_m) * Math.tan(k_m))) * (t_m / Math.pow(Math.cbrt(l), 2.0));
	double tmp;
	if (k_m <= 6.2e-6) {
		tmp = Math.pow((Math.sqrt(2.0) / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	} else if (k_m <= 1.15e+157) {
		tmp = (2.0 * ((Math.cos(k_m) * Math.pow(k_m, -2.0)) * (1.0 / (t_m * Math.pow(Math.sin(k_m), 2.0))))) * (l * l);
	} else {
		tmp = (2.0 / Math.pow(t_2, 2.0)) * (Math.pow((k_m / t_m), -2.0) / t_2);
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(cbrt(Float64(sin(k_m) * tan(k_m))) * Float64(t_m / (cbrt(l) ^ 2.0)))
	tmp = 0.0
	if (k_m <= 6.2e-6)
		tmp = Float64(sqrt(2.0) / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0;
	elseif (k_m <= 1.15e+157)
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k_m) * (k_m ^ -2.0)) * Float64(1.0 / Float64(t_m * (sin(k_m) ^ 2.0))))) * Float64(l * l));
	else
		tmp = Float64(Float64(2.0 / (t_2 ^ 2.0)) * Float64((Float64(k_m / t_m) ^ -2.0) / t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[(N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 6.2e-6], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 1.15e+157], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k$95$m / t$95$m), $MachinePrecision], -2.0], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 6.1999999999999999e-6

   1. Initial program 39.0%

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

      1. *-commutative 39.0%

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

      2. associate-/r* 39.0%

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

   3. Simplified 43.6%

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

      1. add-sqr-sqrt 29.2%

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

   6. Applied egg-rr 33.0%

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

      1. unpow2 33.0%

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

      2. associate-/l/ 33.5%

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

      3. associate-*l/ 35.1%

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

      4. associate-*l/ 35.2%

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

      5. *-commutative 35.2%

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

   8. Simplified 35.2%

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

   9. Taylor expanded in k around inf 51.7%

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

       1. associate-/l* 52.6%

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

   11. Simplified 52.6%

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

   if 6.1999999999999999e-6 < k < 1.15000000000000002e157

   1. Initial program 29.5%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in t around 0 78.7%

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

      1. associate-/r* 78.7%

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

   7. Simplified 78.7%

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

      1. div-inv 78.8%

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

      2. div-inv 78.8%

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

      3. pow-flip 78.8%

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

      4. metadata-eval 78.8%

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

   9. Applied egg-rr 78.8%

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

   if 1.15000000000000002e157 < k

   1. Initial program 20.7%

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

      1. *-commutative 20.7%

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

      2. associate-/r* 20.7%

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

   3. Simplified 34.5%

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

      1. add-cube-cbrt 34.5%

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

      2. div-inv 34.5%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. times-frac 34.5%

         \[\leadsto \color{blue}{\frac{2}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}} \]

   6. Applied egg-rr 59.3%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \sqrt{\frac{t}{\cos k}}}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{+157}:\\ \;\;\;\;\left(2 \cdot \left(\left(\cos k \cdot {k}^{-2}\right) \cdot \frac{1}{t \cdot {\sin k}^{2}}\right)\right) \cdot \left(\ell \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{{\left(\frac{k}{t}\right)}^{-2}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{\sqrt{2}}{\left(k\_m \cdot \frac{\sin k\_m}{\ell}\right) \cdot \sqrt{\frac{t\_m}{\cos k\_m}}}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 6.7 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;k\_m \leq 1.3 \cdot 10^{+178}:\\ \;\;\;\;\left(2 \cdot \left(\left(\cos k\_m \cdot {k\_m}^{-2}\right) \cdot \frac{1}{t\_m \cdot {\sin k\_m}^{2}}\right)\right) \cdot \left(\ell \cdot \ell\right)\\ \mathbf{elif}\;k\_m \leq 1.02 \cdot 10^{+231}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \end{array} \]

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
          (/ (sqrt 2.0) (* (* k_m (/ (sin k_m) l)) (sqrt (/ t_m (cos k_m)))))
          2.0)))
   (*
    t_s
    (if (<= k_m 6.7e-6)
      t_2
      (if (<= k_m 1.3e+178)
        (*
         (*
          2.0
          (* (* (cos k_m) (pow k_m -2.0)) (/ 1.0 (* t_m (pow (sin k_m) 2.0)))))
         (* l l))
        (if (<= k_m 1.02e+231) t_2 0.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = pow((sqrt(2.0) / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))), 2.0);
	double tmp;
	if (k_m <= 6.7e-6) {
		tmp = t_2;
	} else if (k_m <= 1.3e+178) {
		tmp = (2.0 * ((cos(k_m) * pow(k_m, -2.0)) * (1.0 / (t_m * pow(sin(k_m), 2.0))))) * (l * l);
	} else if (k_m <= 1.02e+231) {
		tmp = t_2;
	} else {
		tmp = 0.0;
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (sqrt(2.0d0) / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))) ** 2.0d0
    if (k_m <= 6.7d-6) then
        tmp = t_2
    else if (k_m <= 1.3d+178) then
        tmp = (2.0d0 * ((cos(k_m) * (k_m ** (-2.0d0))) * (1.0d0 / (t_m * (sin(k_m) ** 2.0d0))))) * (l * l)
    else if (k_m <= 1.02d+231) then
        tmp = t_2
    else
        tmp = 0.0d0
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.pow((Math.sqrt(2.0) / ((k_m * (Math.sin(k_m) / l)) * Math.sqrt((t_m / Math.cos(k_m))))), 2.0);
	double tmp;
	if (k_m <= 6.7e-6) {
		tmp = t_2;
	} else if (k_m <= 1.3e+178) {
		tmp = (2.0 * ((Math.cos(k_m) * Math.pow(k_m, -2.0)) * (1.0 / (t_m * Math.pow(Math.sin(k_m), 2.0))))) * (l * l);
	} else if (k_m <= 1.02e+231) {
		tmp = t_2;
	} else {
		tmp = 0.0;
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = math.pow((math.sqrt(2.0) / ((k_m * (math.sin(k_m) / l)) * math.sqrt((t_m / math.cos(k_m))))), 2.0)
	tmp = 0
	if k_m <= 6.7e-6:
		tmp = t_2
	elif k_m <= 1.3e+178:
		tmp = (2.0 * ((math.cos(k_m) * math.pow(k_m, -2.0)) * (1.0 / (t_m * math.pow(math.sin(k_m), 2.0))))) * (l * l)
	elif k_m <= 1.02e+231:
		tmp = t_2
	else:
		tmp = 0.0
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(sqrt(2.0) / Float64(Float64(k_m * Float64(sin(k_m) / l)) * sqrt(Float64(t_m / cos(k_m))))) ^ 2.0
	tmp = 0.0
	if (k_m <= 6.7e-6)
		tmp = t_2;
	elseif (k_m <= 1.3e+178)
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k_m) * (k_m ^ -2.0)) * Float64(1.0 / Float64(t_m * (sin(k_m) ^ 2.0))))) * Float64(l * l));
	elseif (k_m <= 1.02e+231)
		tmp = t_2;
	else
		tmp = 0.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = (sqrt(2.0) / ((k_m * (sin(k_m) / l)) * sqrt((t_m / cos(k_m))))) ^ 2.0;
	tmp = 0.0;
	if (k_m <= 6.7e-6)
		tmp = t_2;
	elseif (k_m <= 1.3e+178)
		tmp = (2.0 * ((cos(k_m) * (k_m ^ -2.0)) * (1.0 / (t_m * (sin(k_m) ^ 2.0))))) * (l * l);
	elseif (k_m <= 1.02e+231)
		tmp = t_2;
	else
		tmp = 0.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 6.7e-6], t$95$2, If[LessEqual[k$95$m, 1.3e+178], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.02e+231], t$95$2, 0.0]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 6.7e-6 or 1.3e178 < k < 1.02e231

   1. Initial program 38.3%

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

      1. *-commutative 38.3%

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

      2. associate-/r* 38.3%

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

   3. Simplified 43.6%

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

      1. add-sqr-sqrt 29.9%

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

   6. Applied egg-rr 32.1%

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

      1. unpow2 32.1%

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

      2. associate-/l/ 32.6%

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

      3. associate-*l/ 34.1%

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

      4. associate-*l/ 34.1%

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

      5. *-commutative 34.1%

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

   8. Simplified 34.1%

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

   9. Taylor expanded in k around inf 51.9%

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

       1. associate-/l* 52.7%

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

   11. Simplified 52.7%

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

   if 6.7e-6 < k < 1.3e178

   1. Initial program 31.6%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in t around 0 75.6%

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

      1. associate-/r* 75.7%

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

   7. Simplified 75.7%

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

      1. div-inv 75.8%

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

      2. div-inv 75.8%

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

      3. pow-flip 75.8%

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

      4. metadata-eval 75.8%

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

   9. Applied egg-rr 75.8%

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

   if 1.02e231 < k

   1. Initial program 12.5%

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

   3. Simplified 25.0%

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

      1. add-log-exp 25.0%

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

      2. exp-prod 26.5%

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

      3. associate-*r* 26.5%

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

      4. *-commutative 26.5%

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

      5. associate-*l* 26.5%

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

   6. Applied egg-rr 26.5%

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

   7. Taylor expanded in t around 0 2.0%

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

      1. add-cube-cbrt 2.0%

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

      2. pow2 2.0%

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

      3. clear-num 2.0%

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

      4. metadata-eval 2.0%

         \[\leadsto \left({\left(\sqrt[3]{\frac{1}{\frac{\color{blue}{0}}{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      5. metadata-eval 2.0%

         \[\leadsto \left({\left(\sqrt[3]{\frac{1}{\color{blue}{0}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      6. metadata-eval 2.0%

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

      7. cbrt-div 2.0%

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

      8. metadata-eval 2.0%

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

      9. metadata-eval 2.0%

         \[\leadsto \left({\left(\frac{1}{\sqrt[3]{\color{blue}{0}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      10. clear-num 2.0%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{1}{\frac{\log 1}{2}}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      11. metadata-eval 2.0%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\frac{\color{blue}{0}}{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      12. metadata-eval 2.0%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\color{blue}{0}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      13. metadata-eval 2.0%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\color{blue}{\log 1}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      14. cbrt-div 2.0%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\log 1}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      15. metadata-eval 2.0%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{\color{blue}{1}}{\sqrt[3]{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      16. metadata-eval 2.0%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{1}{\sqrt[3]{\color{blue}{0}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   9. Applied egg-rr 2.0%

      \[\leadsto \color{blue}{\left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{1}{\sqrt[3]{0}}\right)} \cdot \left(\ell \cdot \ell\right) \]
   10. Step-by-step derivation

       1. pow-plus 2.0%

          \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt[3]{0}}\right)}^{\left(2 + 1\right)}} \cdot \left(\ell \cdot \ell\right) \]

       2. metadata-eval 2.0%

          \[\leadsto {\left(\frac{1}{\sqrt[3]{0}}\right)}^{\color{blue}{3}} \cdot \left(\ell \cdot \ell\right) \]

       3. cube-div 2.0%

          \[\leadsto \color{blue}{\frac{{1}^{3}}{{\left(\sqrt[3]{0}\right)}^{3}}} \cdot \left(\ell \cdot \ell\right) \]

       4. metadata-eval 2.0%

          \[\leadsto \frac{\color{blue}{1}}{{\left(\sqrt[3]{0}\right)}^{3}} \cdot \left(\ell \cdot \ell\right) \]

       5. rem-cube-cbrt 2.0%

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

       6. unpow-1 2.0%

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

       7. pow-base-0 52.0%

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

   11. Simplified 52.0%

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

   12. Taylor expanded in l around 0 59.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 83.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{\sqrt{2} \cdot \ell}{k\_m \cdot \sin k\_m} \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;k\_m \leq 1.3 \cdot 10^{+178}:\\ \;\;\;\;\left(2 \cdot \left(\left(\cos k\_m \cdot {k\_m}^{-2}\right) \cdot \frac{1}{t\_m \cdot {\sin k\_m}^{2}}\right)\right) \cdot \left(\ell \cdot \ell\right)\\ \mathbf{elif}\;k\_m \leq 1.02 \cdot 10^{+231}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \end{array} \]

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
          (* (/ (* (sqrt 2.0) l) (* k_m (sin k_m))) (sqrt (/ (cos k_m) t_m)))
          2.0)))
   (*
    t_s
    (if (<= k_m 6.5e-6)
      t_2
      (if (<= k_m 1.3e+178)
        (*
         (*
          2.0
          (* (* (cos k_m) (pow k_m -2.0)) (/ 1.0 (* t_m (pow (sin k_m) 2.0)))))
         (* l l))
        (if (<= k_m 1.02e+231) t_2 0.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = pow((((sqrt(2.0) * l) / (k_m * sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	double tmp;
	if (k_m <= 6.5e-6) {
		tmp = t_2;
	} else if (k_m <= 1.3e+178) {
		tmp = (2.0 * ((cos(k_m) * pow(k_m, -2.0)) * (1.0 / (t_m * pow(sin(k_m), 2.0))))) * (l * l);
	} else if (k_m <= 1.02e+231) {
		tmp = t_2;
	} else {
		tmp = 0.0;
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (((sqrt(2.0d0) * l) / (k_m * sin(k_m))) * sqrt((cos(k_m) / t_m))) ** 2.0d0
    if (k_m <= 6.5d-6) then
        tmp = t_2
    else if (k_m <= 1.3d+178) then
        tmp = (2.0d0 * ((cos(k_m) * (k_m ** (-2.0d0))) * (1.0d0 / (t_m * (sin(k_m) ** 2.0d0))))) * (l * l)
    else if (k_m <= 1.02d+231) then
        tmp = t_2
    else
        tmp = 0.0d0
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = Math.pow((((Math.sqrt(2.0) * l) / (k_m * Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	double tmp;
	if (k_m <= 6.5e-6) {
		tmp = t_2;
	} else if (k_m <= 1.3e+178) {
		tmp = (2.0 * ((Math.cos(k_m) * Math.pow(k_m, -2.0)) * (1.0 / (t_m * Math.pow(Math.sin(k_m), 2.0))))) * (l * l);
	} else if (k_m <= 1.02e+231) {
		tmp = t_2;
	} else {
		tmp = 0.0;
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = math.pow((((math.sqrt(2.0) * l) / (k_m * math.sin(k_m))) * math.sqrt((math.cos(k_m) / t_m))), 2.0)
	tmp = 0
	if k_m <= 6.5e-6:
		tmp = t_2
	elif k_m <= 1.3e+178:
		tmp = (2.0 * ((math.cos(k_m) * math.pow(k_m, -2.0)) * (1.0 / (t_m * math.pow(math.sin(k_m), 2.0))))) * (l * l)
	elif k_m <= 1.02e+231:
		tmp = t_2
	else:
		tmp = 0.0
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(Float64(Float64(sqrt(2.0) * l) / Float64(k_m * sin(k_m))) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0
	tmp = 0.0
	if (k_m <= 6.5e-6)
		tmp = t_2;
	elseif (k_m <= 1.3e+178)
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k_m) * (k_m ^ -2.0)) * Float64(1.0 / Float64(t_m * (sin(k_m) ^ 2.0))))) * Float64(l * l));
	elseif (k_m <= 1.02e+231)
		tmp = t_2;
	else
		tmp = 0.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = (((sqrt(2.0) * l) / (k_m * sin(k_m))) * sqrt((cos(k_m) / t_m))) ^ 2.0;
	tmp = 0.0;
	if (k_m <= 6.5e-6)
		tmp = t_2;
	elseif (k_m <= 1.3e+178)
		tmp = (2.0 * ((cos(k_m) * (k_m ^ -2.0)) * (1.0 / (t_m * (sin(k_m) ^ 2.0))))) * (l * l);
	elseif (k_m <= 1.02e+231)
		tmp = t_2;
	else
		tmp = 0.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := Block[{t$95$2 = N[Power[N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * l), $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 6.5e-6], t$95$2, If[LessEqual[k$95$m, 1.3e+178], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.02e+231], t$95$2, 0.0]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 6.4999999999999996e-6 or 1.3e178 < k < 1.02e231

   1. Initial program 38.3%

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

      1. *-commutative 38.3%

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

      2. associate-/r* 38.3%

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

   3. Simplified 43.6%

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

      1. add-sqr-sqrt 29.9%

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

   6. Applied egg-rr 32.1%

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

      1. unpow2 32.1%

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

      2. associate-/l/ 32.6%

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

      3. associate-*l/ 34.1%

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

      4. associate-*l/ 34.1%

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

      5. *-commutative 34.1%

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

   8. Simplified 34.1%

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

   9. Taylor expanded in k around inf 51.9%

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

   if 6.4999999999999996e-6 < k < 1.3e178

   1. Initial program 31.6%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in t around 0 75.6%

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

      1. associate-/r* 75.7%

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

   7. Simplified 75.7%

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

      1. div-inv 75.8%

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

      2. div-inv 75.8%

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

      3. pow-flip 75.8%

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

      4. metadata-eval 75.8%

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

   9. Applied egg-rr 75.8%

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

   if 1.02e231 < k

   1. Initial program 12.5%

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

   3. Simplified 25.0%

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

      1. add-log-exp 25.0%

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

      2. exp-prod 26.5%

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

      3. associate-*r* 26.5%

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

      4. *-commutative 26.5%

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

      5. associate-*l* 26.5%

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

   6. Applied egg-rr 26.5%

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

   7. Taylor expanded in t around 0 2.0%

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

      1. add-cube-cbrt 2.0%

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

      2. pow2 2.0%

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

      3. clear-num 2.0%

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

      4. metadata-eval 2.0%

         \[\leadsto \left({\left(\sqrt[3]{\frac{1}{\frac{\color{blue}{0}}{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      5. metadata-eval 2.0%

         \[\leadsto \left({\left(\sqrt[3]{\frac{1}{\color{blue}{0}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      6. metadata-eval 2.0%

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

      7. cbrt-div 2.0%

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

      8. metadata-eval 2.0%

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

      9. metadata-eval 2.0%

         \[\leadsto \left({\left(\frac{1}{\sqrt[3]{\color{blue}{0}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      10. clear-num 2.0%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{1}{\frac{\log 1}{2}}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      11. metadata-eval 2.0%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\frac{\color{blue}{0}}{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      12. metadata-eval 2.0%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\color{blue}{0}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      13. metadata-eval 2.0%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\color{blue}{\log 1}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      14. cbrt-div 2.0%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\log 1}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      15. metadata-eval 2.0%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{\color{blue}{1}}{\sqrt[3]{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      16. metadata-eval 2.0%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{1}{\sqrt[3]{\color{blue}{0}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   9. Applied egg-rr 2.0%

      \[\leadsto \color{blue}{\left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{1}{\sqrt[3]{0}}\right)} \cdot \left(\ell \cdot \ell\right) \]
   10. Step-by-step derivation

       1. pow-plus 2.0%

          \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt[3]{0}}\right)}^{\left(2 + 1\right)}} \cdot \left(\ell \cdot \ell\right) \]

       2. metadata-eval 2.0%

          \[\leadsto {\left(\frac{1}{\sqrt[3]{0}}\right)}^{\color{blue}{3}} \cdot \left(\ell \cdot \ell\right) \]

       3. cube-div 2.0%

          \[\leadsto \color{blue}{\frac{{1}^{3}}{{\left(\sqrt[3]{0}\right)}^{3}}} \cdot \left(\ell \cdot \ell\right) \]

       4. metadata-eval 2.0%

          \[\leadsto \frac{\color{blue}{1}}{{\left(\sqrt[3]{0}\right)}^{3}} \cdot \left(\ell \cdot \ell\right) \]

       5. rem-cube-cbrt 2.0%

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

       6. unpow-1 2.0%

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

       7. pow-base-0 52.0%

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

   11. Simplified 52.0%

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

   12. Taylor expanded in l around 0 59.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;{\left(\frac{\sqrt{2} \cdot \ell}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{+178}:\\ \;\;\;\;\left(2 \cdot \left(\left(\cos k \cdot {k}^{-2}\right) \cdot \frac{1}{t \cdot {\sin k}^{2}}\right)\right) \cdot \left(\ell \cdot \ell\right)\\ \mathbf{elif}\;k \leq 1.02 \cdot 10^{+231}:\\ \;\;\;\;{\left(\frac{\sqrt{2} \cdot \ell}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 5.9 \cdot 10^{-6}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{k\_m}^{2}}{\ell} \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(\cos k\_m \cdot {k\_m}^{-2}\right) \cdot \frac{1}{t\_m \cdot {\sin k\_m}^{2}}\right)\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 5.9e-6)
    (pow (/ (sqrt 2.0) (* (/ (pow k_m 2.0) l) (sqrt t_m))) 2.0)
    (*
     (*
      2.0
      (* (* (cos k_m) (pow k_m -2.0)) (/ 1.0 (* t_m (pow (sin k_m) 2.0)))))
     (* l l)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5.9e-6) {
		tmp = pow((sqrt(2.0) / ((pow(k_m, 2.0) / l) * sqrt(t_m))), 2.0);
	} else {
		tmp = (2.0 * ((cos(k_m) * pow(k_m, -2.0)) * (1.0 / (t_m * pow(sin(k_m), 2.0))))) * (l * l);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5.9d-6) then
        tmp = (sqrt(2.0d0) / (((k_m ** 2.0d0) / l) * sqrt(t_m))) ** 2.0d0
    else
        tmp = (2.0d0 * ((cos(k_m) * (k_m ** (-2.0d0))) * (1.0d0 / (t_m * (sin(k_m) ** 2.0d0))))) * (l * l)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5.9e-6) {
		tmp = Math.pow((Math.sqrt(2.0) / ((Math.pow(k_m, 2.0) / l) * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = (2.0 * ((Math.cos(k_m) * Math.pow(k_m, -2.0)) * (1.0 / (t_m * Math.pow(Math.sin(k_m), 2.0))))) * (l * l);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 5.9e-6:
		tmp = math.pow((math.sqrt(2.0) / ((math.pow(k_m, 2.0) / l) * math.sqrt(t_m))), 2.0)
	else:
		tmp = (2.0 * ((math.cos(k_m) * math.pow(k_m, -2.0)) * (1.0 / (t_m * math.pow(math.sin(k_m), 2.0))))) * (l * l)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 5.9e-6)
		tmp = Float64(sqrt(2.0) / Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k_m) * (k_m ^ -2.0)) * Float64(1.0 / Float64(t_m * (sin(k_m) ^ 2.0))))) * Float64(l * l));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 5.9e-6)
		tmp = (sqrt(2.0) / (((k_m ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	else
		tmp = (2.0 * ((cos(k_m) * (k_m ^ -2.0)) * (1.0 / (t_m * (sin(k_m) ^ 2.0))))) * (l * l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 5.9e-6], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.90000000000000026e-6

   1. Initial program 39.0%

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

      1. *-commutative 39.0%

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

      2. associate-/r* 39.0%

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

   3. Simplified 43.6%

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

      1. add-sqr-sqrt 29.2%

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

   6. Applied egg-rr 33.0%

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

      1. unpow2 33.0%

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

      2. associate-/l/ 33.5%

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

      3. associate-*l/ 35.1%

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

      4. associate-*l/ 35.2%

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

      5. *-commutative 35.2%

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

   8. Simplified 35.2%

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

   9. Taylor expanded in k around 0 43.8%

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

   if 5.90000000000000026e-6 < k

   1. Initial program 25.4%

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

   3. Simplified 45.6%

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

   5. Taylor expanded in t around 0 66.9%

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

      1. associate-/r* 67.0%

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

   7. Simplified 67.0%

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

      1. div-inv 67.0%

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

      2. div-inv 67.0%

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

      3. pow-flip 67.0%

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

      4. metadata-eval 67.0%

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

   9. Applied egg-rr 67.0%

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

Alternative 12: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 7 \cdot 10^{-7}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{k\_m}^{2}}{\ell} \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k\_m}{{k\_m}^{2}}}{t\_m \cdot {\sin k\_m}^{2}}\right)\\ \end{array} \end{array} \]

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 7e-7)
    (pow (/ (sqrt 2.0) (* (/ (pow k_m 2.0) l) (sqrt t_m))) 2.0)
    (*
     (* l l)
     (* 2.0 (/ (/ (cos k_m) (pow k_m 2.0)) (* 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 <= 7e-7) {
		tmp = pow((sqrt(2.0) / ((pow(k_m, 2.0) / l) * sqrt(t_m))), 2.0);
	} else {
		tmp = (l * l) * (2.0 * ((cos(k_m) / pow(k_m, 2.0)) / (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 <= 7d-7) then
        tmp = (sqrt(2.0d0) / (((k_m ** 2.0d0) / l) * sqrt(t_m))) ** 2.0d0
    else
        tmp = (l * l) * (2.0d0 * ((cos(k_m) / (k_m ** 2.0d0)) / (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 <= 7e-7) {
		tmp = Math.pow((Math.sqrt(2.0) / ((Math.pow(k_m, 2.0) / l) * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = (l * l) * (2.0 * ((Math.cos(k_m) / Math.pow(k_m, 2.0)) / (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 <= 7e-7:
		tmp = math.pow((math.sqrt(2.0) / ((math.pow(k_m, 2.0) / l) * math.sqrt(t_m))), 2.0)
	else:
		tmp = (l * l) * (2.0 * ((math.cos(k_m) / math.pow(k_m, 2.0)) / (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 <= 7e-7)
		tmp = Float64(sqrt(2.0) / Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 * Float64(Float64(cos(k_m) / (k_m ^ 2.0)) / 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 <= 7e-7)
		tmp = (sqrt(2.0) / (((k_m ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	else
		tmp = (l * l) * (2.0 * ((cos(k_m) / (k_m ^ 2.0)) / (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, 7e-7], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 * N[(N[(N[Cos[k$95$m], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $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 < 6.99999999999999968e-7

   1. Initial program 39.0%

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

      1. *-commutative 39.0%

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

      2. associate-/r* 39.0%

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

   3. Simplified 43.6%

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

      1. add-sqr-sqrt 29.2%

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

   6. Applied egg-rr 33.0%

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

      1. unpow2 33.0%

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

      2. associate-/l/ 33.5%

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

      3. associate-*l/ 35.1%

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

      4. associate-*l/ 35.2%

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

      5. *-commutative 35.2%

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

   8. Simplified 35.2%

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

   9. Taylor expanded in k around 0 43.8%

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

   if 6.99999999999999968e-7 < k

   1. Initial program 25.4%

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

   3. Simplified 45.6%

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

   5. Taylor expanded in t around 0 66.9%

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

      1. associate-/r* 67.0%

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

   7. Simplified 67.0%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-7}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{k}^{2}}{\ell} \cdot \sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3.15 \cdot 10^{-6}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{k\_m}^{2}}{\ell} \cdot \sqrt{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k\_m}^{2}}}{t\_m \cdot \left(\sin k\_m \cdot \tan k\_m\right)}\\ \end{array} \end{array} \]

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 3.15e-6)
    (pow (/ (sqrt 2.0) (* (/ (pow k_m 2.0) l) (sqrt t_m))) 2.0)
    (* (* l l) (/ (/ 2.0 (pow k_m 2.0)) (* t_m (* (sin k_m) (tan k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 3.15e-6) {
		tmp = pow((sqrt(2.0) / ((pow(k_m, 2.0) / l) * sqrt(t_m))), 2.0);
	} else {
		tmp = (l * l) * ((2.0 / pow(k_m, 2.0)) / (t_m * (sin(k_m) * tan(k_m))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3.15d-6) then
        tmp = (sqrt(2.0d0) / (((k_m ** 2.0d0) / l) * sqrt(t_m))) ** 2.0d0
    else
        tmp = (l * l) * ((2.0d0 / (k_m ** 2.0d0)) / (t_m * (sin(k_m) * tan(k_m))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 3.15e-6) {
		tmp = Math.pow((Math.sqrt(2.0) / ((Math.pow(k_m, 2.0) / l) * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = (l * l) * ((2.0 / Math.pow(k_m, 2.0)) / (t_m * (Math.sin(k_m) * Math.tan(k_m))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 3.15e-6:
		tmp = math.pow((math.sqrt(2.0) / ((math.pow(k_m, 2.0) / l) * math.sqrt(t_m))), 2.0)
	else:
		tmp = (l * l) * ((2.0 / math.pow(k_m, 2.0)) / (t_m * (math.sin(k_m) * math.tan(k_m))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 3.15e-6)
		tmp = Float64(sqrt(2.0) / Float64(Float64((k_m ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 / (k_m ^ 2.0)) / Float64(t_m * Float64(sin(k_m) * tan(k_m)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 3.15e-6)
		tmp = (sqrt(2.0) / (((k_m ^ 2.0) / l) * sqrt(t_m))) ^ 2.0;
	else
		tmp = (l * l) * ((2.0 / (k_m ^ 2.0)) / (t_m * (sin(k_m) * tan(k_m))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 3.15e-6], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.14999999999999991e-6

   1. Initial program 39.0%

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

      1. *-commutative 39.0%

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

      2. associate-/r* 39.0%

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

   3. Simplified 43.6%

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

      1. add-sqr-sqrt 29.2%

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

   6. Applied egg-rr 33.0%

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

      1. unpow2 33.0%

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

      2. associate-/l/ 33.5%

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

      3. associate-*l/ 35.1%

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

      4. associate-*l/ 35.2%

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

      5. *-commutative 35.2%

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

   8. Simplified 35.2%

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

   9. Taylor expanded in k around 0 43.8%

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

   if 3.14999999999999991e-6 < k

   1. Initial program 25.4%

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

   3. Simplified 45.6%

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

      1. add-log-exp 31.8%

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

      2. exp-prod 34.1%

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

      3. associate-*r* 34.1%

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

      4. *-commutative 34.1%

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

      5. associate-*l* 34.1%

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

   6. Applied egg-rr 34.1%

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

   7. Taylor expanded in k around inf 66.8%

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

      1. associate-/r* 66.9%

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

   9. Simplified 66.9%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.15 \cdot 10^{-6}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{{k}^{2}}{\ell} \cdot \sqrt{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k}^{2}}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 78.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-313}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{\frac{k\_m}{t\_m} \cdot \left(k\_m \cdot {t\_m}^{1.5}\right)}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k\_m}^{2}}}{t\_m \cdot \left(\sin k\_m \cdot \tan k\_m\right)}\\ \end{array} \end{array} \]

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 (<= (* l l) 4e-313)
    (pow (/ (sqrt 2.0) (/ (* (/ k_m t_m) (* k_m (pow t_m 1.5))) l)) 2.0)
    (* (* l l) (/ (/ 2.0 (pow k_m 2.0)) (* t_m (* (sin k_m) (tan k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 4e-313) {
		tmp = pow((sqrt(2.0) / (((k_m / t_m) * (k_m * pow(t_m, 1.5))) / l)), 2.0);
	} else {
		tmp = (l * l) * ((2.0 / pow(k_m, 2.0)) / (t_m * (sin(k_m) * tan(k_m))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 4d-313) then
        tmp = (sqrt(2.0d0) / (((k_m / t_m) * (k_m * (t_m ** 1.5d0))) / l)) ** 2.0d0
    else
        tmp = (l * l) * ((2.0d0 / (k_m ** 2.0d0)) / (t_m * (sin(k_m) * tan(k_m))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 4e-313) {
		tmp = Math.pow((Math.sqrt(2.0) / (((k_m / t_m) * (k_m * Math.pow(t_m, 1.5))) / l)), 2.0);
	} else {
		tmp = (l * l) * ((2.0 / Math.pow(k_m, 2.0)) / (t_m * (Math.sin(k_m) * Math.tan(k_m))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 4e-313:
		tmp = math.pow((math.sqrt(2.0) / (((k_m / t_m) * (k_m * math.pow(t_m, 1.5))) / l)), 2.0)
	else:
		tmp = (l * l) * ((2.0 / math.pow(k_m, 2.0)) / (t_m * (math.sin(k_m) * math.tan(k_m))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 4e-313)
		tmp = Float64(sqrt(2.0) / Float64(Float64(Float64(k_m / t_m) * Float64(k_m * (t_m ^ 1.5))) / l)) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 / (k_m ^ 2.0)) / Float64(t_m * Float64(sin(k_m) * tan(k_m)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 4e-313)
		tmp = (sqrt(2.0) / (((k_m / t_m) * (k_m * (t_m ^ 1.5))) / l)) ^ 2.0;
	else
		tmp = (l * l) * ((2.0 / (k_m ^ 2.0)) / (t_m * (sin(k_m) * tan(k_m))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 4e-313], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.0000000000037e-313

   1. Initial program 16.3%

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

      1. *-commutative 16.3%

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

      2. associate-/r* 16.3%

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

   3. Simplified 24.2%

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

      1. add-sqr-sqrt 23.0%

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

   6. Applied egg-rr 14.4%

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

      1. unpow2 14.4%

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

      2. associate-/l/ 15.9%

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

      3. associate-*l/ 19.4%

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

      4. associate-*l/ 19.4%

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

      5. *-commutative 19.4%

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

   8. Simplified 19.4%

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

   9. Taylor expanded in k around 0 33.8%

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

   if 4.0000000000037e-313 < (*.f64 l l)

   1. Initial program 41.9%

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

   3. Simplified 50.6%

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

      1. add-log-exp 31.7%

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

      2. exp-prod 39.9%

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

      3. associate-*r* 39.9%

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

      4. *-commutative 39.9%

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

      5. associate-*l* 39.9%

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

   6. Applied egg-rr 39.9%

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

   7. Taylor expanded in k around inf 79.7%

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

      1. associate-/r* 80.2%

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

   9. Simplified 80.2%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-313}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{\frac{k}{t} \cdot \left(k \cdot {t}^{1.5}\right)}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k}^{2}}}{t \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 78.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-313}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{\frac{k\_m}{t\_m} \cdot \left(k\_m \cdot {t\_m}^{1.5}\right)}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k\_m}^{2} \cdot \left(t\_m \cdot \left(\sin k\_m \cdot \tan k\_m\right)\right)}\\ \end{array} \end{array} \]

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 (<= (* l l) 4e-313)
    (pow (/ (sqrt 2.0) (/ (* (/ k_m t_m) (* k_m (pow t_m 1.5))) l)) 2.0)
    (* (* l l) (/ 2.0 (* (pow k_m 2.0) (* t_m (* (sin k_m) (tan k_m)))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 4e-313) {
		tmp = pow((sqrt(2.0) / (((k_m / t_m) * (k_m * pow(t_m, 1.5))) / l)), 2.0);
	} else {
		tmp = (l * l) * (2.0 / (pow(k_m, 2.0) * (t_m * (sin(k_m) * tan(k_m)))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if ((l * l) <= 4d-313) then
        tmp = (sqrt(2.0d0) / (((k_m / t_m) * (k_m * (t_m ** 1.5d0))) / l)) ** 2.0d0
    else
        tmp = (l * l) * (2.0d0 / ((k_m ** 2.0d0) * (t_m * (sin(k_m) * tan(k_m)))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if ((l * l) <= 4e-313) {
		tmp = Math.pow((Math.sqrt(2.0) / (((k_m / t_m) * (k_m * Math.pow(t_m, 1.5))) / l)), 2.0);
	} else {
		tmp = (l * l) * (2.0 / (Math.pow(k_m, 2.0) * (t_m * (Math.sin(k_m) * Math.tan(k_m)))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if (l * l) <= 4e-313:
		tmp = math.pow((math.sqrt(2.0) / (((k_m / t_m) * (k_m * math.pow(t_m, 1.5))) / l)), 2.0)
	else:
		tmp = (l * l) * (2.0 / (math.pow(k_m, 2.0) * (t_m * (math.sin(k_m) * math.tan(k_m)))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (Float64(l * l) <= 4e-313)
		tmp = Float64(sqrt(2.0) / Float64(Float64(Float64(k_m / t_m) * Float64(k_m * (t_m ^ 1.5))) / l)) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64((k_m ^ 2.0) * Float64(t_m * Float64(sin(k_m) * tan(k_m))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if ((l * l) <= 4e-313)
		tmp = (sqrt(2.0) / (((k_m / t_m) * (k_m * (t_m ^ 1.5))) / l)) ^ 2.0;
	else
		tmp = (l * l) * (2.0 / ((k_m ^ 2.0) * (t_m * (sin(k_m) * tan(k_m)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 4e-313], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $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[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.0000000000037e-313

   1. Initial program 16.3%

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

      1. *-commutative 16.3%

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

      2. associate-/r* 16.3%

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

   3. Simplified 24.2%

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

      1. add-sqr-sqrt 23.0%

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

   6. Applied egg-rr 14.4%

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

      1. unpow2 14.4%

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

      2. associate-/l/ 15.9%

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

      3. associate-*l/ 19.4%

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

      4. associate-*l/ 19.4%

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

      5. *-commutative 19.4%

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

   8. Simplified 19.4%

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

   9. Taylor expanded in k around 0 33.8%

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

   if 4.0000000000037e-313 < (*.f64 l l)

   1. Initial program 41.9%

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

   3. Simplified 50.6%

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

      1. add-log-exp 31.7%

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

      2. exp-prod 39.9%

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

      3. associate-*r* 39.9%

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

      4. *-commutative 39.9%

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

      5. associate-*l* 39.9%

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

   6. Applied egg-rr 39.9%

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

   7. Taylor expanded in k around inf 79.7%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-313}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{\frac{k}{t} \cdot \left(k \cdot {t}^{1.5}\right)}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{+206}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{\frac{k\_m}{t\_m} \cdot \left(k\_m \cdot {t\_m}^{1.5}\right)}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t\_m \cdot e^{4 \cdot \log k\_m}}\\ \end{array} \end{array} \]

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 (<= t_m 1.1e+206)
    (pow (/ (sqrt 2.0) (/ (* (/ k_m t_m) (* k_m (pow t_m 1.5))) l)) 2.0)
    (* (* l l) (/ 2.0 (* t_m (exp (* 4.0 (log 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 (t_m <= 1.1e+206) {
		tmp = pow((sqrt(2.0) / (((k_m / t_m) * (k_m * pow(t_m, 1.5))) / l)), 2.0);
	} else {
		tmp = (l * l) * (2.0 / (t_m * exp((4.0 * log(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 (t_m <= 1.1d+206) then
        tmp = (sqrt(2.0d0) / (((k_m / t_m) * (k_m * (t_m ** 1.5d0))) / l)) ** 2.0d0
    else
        tmp = (l * l) * (2.0d0 / (t_m * exp((4.0d0 * log(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 (t_m <= 1.1e+206) {
		tmp = Math.pow((Math.sqrt(2.0) / (((k_m / t_m) * (k_m * Math.pow(t_m, 1.5))) / l)), 2.0);
	} else {
		tmp = (l * l) * (2.0 / (t_m * Math.exp((4.0 * Math.log(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 t_m <= 1.1e+206:
		tmp = math.pow((math.sqrt(2.0) / (((k_m / t_m) * (k_m * math.pow(t_m, 1.5))) / l)), 2.0)
	else:
		tmp = (l * l) * (2.0 / (t_m * math.exp((4.0 * math.log(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 (t_m <= 1.1e+206)
		tmp = Float64(sqrt(2.0) / Float64(Float64(Float64(k_m / t_m) * Float64(k_m * (t_m ^ 1.5))) / l)) ^ 2.0;
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t_m * exp(Float64(4.0 * log(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 (t_m <= 1.1e+206)
		tmp = (sqrt(2.0) / (((k_m / t_m) * (k_m * (t_m ^ 1.5))) / l)) ^ 2.0;
	else
		tmp = (l * l) * (2.0 / (t_m * exp((4.0 * log(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[t$95$m, 1.1e+206], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Exp[N[(4.0 * N[Log[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.10000000000000001e206

   1. Initial program 39.3%

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

      1. *-commutative 39.3%

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

      2. associate-/r* 39.3%

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

   3. Simplified 45.2%

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

      1. add-sqr-sqrt 31.3%

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

   6. Applied egg-rr 30.8%

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

      1. unpow2 30.8%

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

      2. associate-/l/ 30.8%

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

      3. associate-*l/ 32.2%

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

      4. associate-*l/ 32.4%

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

      5. *-commutative 32.4%

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

   8. Simplified 32.4%

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

   9. Taylor expanded in k around 0 33.5%

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

   if 1.10000000000000001e206 < t

   1. Initial program 3.6%

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

   3. Simplified 30.6%

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

   5. Taylor expanded in k around 0 66.2%

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

      1. add-exp-log 66.2%

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

      2. log-pow 39.3%

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

   7. Applied egg-rr 39.3%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{+206}:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\frac{\frac{k}{t} \cdot \left(k \cdot {t}^{1.5}\right)}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot e^{4 \cdot \log k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 62.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{+63}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 \cdot \frac{{k\_m}^{2}}{t\_m} + 2 \cdot \frac{1}{t\_m}}{{k\_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.4e+63)
    (*
     (* l l)
     (/
      (+ (* -0.3333333333333333 (/ (pow k_m 2.0) t_m)) (* 2.0 (/ 1.0 t_m)))
      (pow k_m 4.0)))
    0.0)))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e+63) {
		tmp = (l * l) * (((-0.3333333333333333 * (pow(k_m, 2.0) / t_m)) + (2.0 * (1.0 / t_m))) / pow(k_m, 4.0));
	} else {
		tmp = 0.0;
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.4d+63) then
        tmp = (l * l) * ((((-0.3333333333333333d0) * ((k_m ** 2.0d0) / t_m)) + (2.0d0 * (1.0d0 / t_m))) / (k_m ** 4.0d0))
    else
        tmp = 0.0d0
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e+63) {
		tmp = (l * l) * (((-0.3333333333333333 * (Math.pow(k_m, 2.0) / t_m)) + (2.0 * (1.0 / t_m))) / Math.pow(k_m, 4.0));
	} else {
		tmp = 0.0;
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.4e+63:
		tmp = (l * l) * (((-0.3333333333333333 * (math.pow(k_m, 2.0) / t_m)) + (2.0 * (1.0 / t_m))) / math.pow(k_m, 4.0))
	else:
		tmp = 0.0
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.4e+63)
		tmp = Float64(Float64(l * l) * Float64(Float64(Float64(-0.3333333333333333 * Float64((k_m ^ 2.0) / t_m)) + Float64(2.0 * Float64(1.0 / t_m))) / (k_m ^ 4.0)));
	else
		tmp = 0.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.4e+63)
		tmp = (l * l) * (((-0.3333333333333333 * ((k_m ^ 2.0) / t_m)) + (2.0 * (1.0 / t_m))) / (k_m ^ 4.0));
	else
		tmp = 0.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.4e+63], N[(N[(l * l), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(N[Power[k$95$m, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.4e63

   1. Initial program 39.0%

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

   3. Simplified 43.8%

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

   5. Taylor expanded in k around 0 52.5%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   if 2.4e63 < k

   1. Initial program 20.0%

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

   3. Simplified 43.8%

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

      1. add-log-exp 37.6%

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

      2. exp-prod 40.2%

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

      3. associate-*r* 40.2%

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

      4. *-commutative 40.2%

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

      5. associate-*l* 40.2%

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

   6. Applied egg-rr 40.2%

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

   7. Taylor expanded in t around 0 3.8%

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

      1. add-cube-cbrt 3.8%

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

      2. pow2 3.8%

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

      3. clear-num 3.8%

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

      4. metadata-eval 3.8%

         \[\leadsto \left({\left(\sqrt[3]{\frac{1}{\frac{\color{blue}{0}}{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      5. metadata-eval 3.8%

         \[\leadsto \left({\left(\sqrt[3]{\frac{1}{\color{blue}{0}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      6. metadata-eval 3.8%

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

      7. cbrt-div 3.8%

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

      8. metadata-eval 3.8%

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

      9. metadata-eval 3.8%

         \[\leadsto \left({\left(\frac{1}{\sqrt[3]{\color{blue}{0}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      10. clear-num 3.8%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{1}{\frac{\log 1}{2}}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      11. metadata-eval 3.8%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\frac{\color{blue}{0}}{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      12. metadata-eval 3.8%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\color{blue}{0}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      13. metadata-eval 3.8%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\color{blue}{\log 1}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      14. cbrt-div 3.8%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\log 1}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      15. metadata-eval 3.8%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{\color{blue}{1}}{\sqrt[3]{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      16. metadata-eval 3.8%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{1}{\sqrt[3]{\color{blue}{0}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   9. Applied egg-rr 3.8%

      \[\leadsto \color{blue}{\left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{1}{\sqrt[3]{0}}\right)} \cdot \left(\ell \cdot \ell\right) \]
   10. Step-by-step derivation

       1. pow-plus 3.8%

          \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt[3]{0}}\right)}^{\left(2 + 1\right)}} \cdot \left(\ell \cdot \ell\right) \]

       2. metadata-eval 3.8%

          \[\leadsto {\left(\frac{1}{\sqrt[3]{0}}\right)}^{\color{blue}{3}} \cdot \left(\ell \cdot \ell\right) \]

       3. cube-div 3.8%

          \[\leadsto \color{blue}{\frac{{1}^{3}}{{\left(\sqrt[3]{0}\right)}^{3}}} \cdot \left(\ell \cdot \ell\right) \]

       4. metadata-eval 3.8%

          \[\leadsto \frac{\color{blue}{1}}{{\left(\sqrt[3]{0}\right)}^{3}} \cdot \left(\ell \cdot \ell\right) \]

       5. rem-cube-cbrt 3.8%

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

       6. unpow-1 3.8%

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

       7. pow-base-0 51.8%

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

   11. Simplified 51.8%

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

   12. Taylor expanded in l around 0 54.9%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{+63}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 \cdot \frac{{k}^{2}}{t} + 2 \cdot \frac{1}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 62.1% accurate, 3.8× speedup

Math

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

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(Float64(2.0 / 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[(N[(2.0 / t$95$m), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.7%

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

3. Simplified 43.8%

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

5. Taylor expanded in k around 0 57.6%

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

   1. *-commutative 57.6%

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

   2. associate-/r* 57.6%

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

7. Simplified 57.6%

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

8. Final simplification 57.6%

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

Alternative 19: 62.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.7%

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

3. Simplified 43.8%

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

5. Taylor expanded in k around 0 57.6%

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

6. Final simplification 57.6%

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

Alternative 20: 56.2% accurate, 35.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1900:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{4}{0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (if (<= k_m 1900.0) (* (* l l) (/ 4.0 0.0)) 0.0)))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1900.0) {
		tmp = (l * l) * (4.0 / 0.0);
	} else {
		tmp = 0.0;
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1900.0d0) then
        tmp = (l * l) * (4.0d0 / 0.0d0)
    else
        tmp = 0.0d0
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 1900.0) {
		tmp = (l * l) * (4.0 / 0.0);
	} else {
		tmp = 0.0;
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 1900.0:
		tmp = (l * l) * (4.0 / 0.0)
	else:
		tmp = 0.0
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 1900.0)
		tmp = Float64(Float64(l * l) * Float64(4.0 / 0.0));
	else
		tmp = 0.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 1900.0)
		tmp = (l * l) * (4.0 / 0.0);
	else
		tmp = 0.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1900.0], N[(N[(l * l), $MachinePrecision] * N[(4.0 / 0.0), $MachinePrecision]), $MachinePrecision], 0.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1900

   1. Initial program 39.3%

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

   3. Simplified 43.5%

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

      1. add-log-exp 28.6%

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

      2. exp-prod 35.1%

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

      3. associate-*r* 35.1%

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

      4. *-commutative 35.1%

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

      5. associate-*l* 35.1%

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

   6. Applied egg-rr 35.1%

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

   7. Taylor expanded in t around 0 29.9%

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

      1. div-inv 29.9%

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

      2. metadata-eval 29.9%

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

      3. metadata-eval 29.9%

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

      4. metadata-eval 29.9%

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

      5. clear-num 29.9%

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

      6. metadata-eval 29.9%

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

   9. Applied egg-rr 29.9%

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

       1. associate-*r/ 29.9%

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

       2. metadata-eval 29.9%

          \[\leadsto \frac{\color{blue}{4}}{0} \cdot \left(\ell \cdot \ell\right) \]

   11. Simplified 29.9%

       \[\leadsto \color{blue}{\frac{4}{0}} \cdot \left(\ell \cdot \ell\right) \]

   if 1900 < k

   1. Initial program 24.2%

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

   3. Simplified 44.7%

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

      1. add-log-exp 30.7%

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

      2. exp-prod 33.0%

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

      3. associate-*r* 33.0%

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

      4. *-commutative 33.0%

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

      5. associate-*l* 33.0%

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

   6. Applied egg-rr 33.0%

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

   7. Taylor expanded in t around 0 3.5%

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

      1. add-cube-cbrt 3.5%

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

      2. pow2 3.5%

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

      3. clear-num 3.5%

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

      4. metadata-eval 3.5%

         \[\leadsto \left({\left(\sqrt[3]{\frac{1}{\frac{\color{blue}{0}}{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      5. metadata-eval 3.5%

         \[\leadsto \left({\left(\sqrt[3]{\frac{1}{\color{blue}{0}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      6. metadata-eval 3.5%

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

      7. cbrt-div 3.5%

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

      8. metadata-eval 3.5%

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

      9. metadata-eval 3.5%

         \[\leadsto \left({\left(\frac{1}{\sqrt[3]{\color{blue}{0}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      10. clear-num 3.5%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{1}{\frac{\log 1}{2}}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      11. metadata-eval 3.5%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\frac{\color{blue}{0}}{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      12. metadata-eval 3.5%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\color{blue}{0}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      13. metadata-eval 3.5%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\color{blue}{\log 1}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      14. cbrt-div 3.5%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\log 1}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      15. metadata-eval 3.5%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{\color{blue}{1}}{\sqrt[3]{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      16. metadata-eval 3.5%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{1}{\sqrt[3]{\color{blue}{0}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   9. Applied egg-rr 3.5%

      \[\leadsto \color{blue}{\left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{1}{\sqrt[3]{0}}\right)} \cdot \left(\ell \cdot \ell\right) \]
   10. Step-by-step derivation

       1. pow-plus 3.5%

          \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt[3]{0}}\right)}^{\left(2 + 1\right)}} \cdot \left(\ell \cdot \ell\right) \]

       2. metadata-eval 3.5%

          \[\leadsto {\left(\frac{1}{\sqrt[3]{0}}\right)}^{\color{blue}{3}} \cdot \left(\ell \cdot \ell\right) \]

       3. cube-div 3.5%

          \[\leadsto \color{blue}{\frac{{1}^{3}}{{\left(\sqrt[3]{0}\right)}^{3}}} \cdot \left(\ell \cdot \ell\right) \]

       4. metadata-eval 3.5%

          \[\leadsto \frac{\color{blue}{1}}{{\left(\sqrt[3]{0}\right)}^{3}} \cdot \left(\ell \cdot \ell\right) \]

       5. rem-cube-cbrt 3.5%

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

       6. unpow-1 3.5%

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

       7. pow-base-0 42.8%

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

   11. Simplified 42.8%

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

   12. Taylor expanded in l around 0 45.2%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1900:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{4}{0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 28.3% accurate, 421.0× speedup

Math

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

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 0.0))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * 0.0;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * 0.0d0
end function

Java

k_m = Math.abs(k);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k_m) {
	return t_s * 0.0;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * 0.0

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * 0.0)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * 0.0;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k$95$m_] := N[(t$95$s * 0.0), $MachinePrecision]

Derivation

1. Initial program 35.7%

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

3. Simplified 43.8%

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

   1. add-log-exp 29.1%

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

   2. exp-prod 34.6%

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

   3. associate-*r* 34.6%

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

   4. *-commutative 34.6%

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

   5. associate-*l* 34.6%

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

6. Applied egg-rr 34.6%

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

7. Taylor expanded in t around 0 23.5%

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

   1. add-cube-cbrt 23.5%

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

   2. pow2 23.5%

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

   3. clear-num 23.5%

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

   4. metadata-eval 23.5%

      \[\leadsto \left({\left(\sqrt[3]{\frac{1}{\frac{\color{blue}{0}}{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

   5. metadata-eval 23.5%

      \[\leadsto \left({\left(\sqrt[3]{\frac{1}{\color{blue}{0}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

   6. metadata-eval 23.5%

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

   7. cbrt-div 23.5%

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

   8. metadata-eval 23.5%

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

   9. metadata-eval 23.5%

      \[\leadsto \left({\left(\frac{1}{\sqrt[3]{\color{blue}{0}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

   10. clear-num 23.5%

       \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{1}{\frac{\log 1}{2}}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   11. metadata-eval 23.5%

       \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\frac{\color{blue}{0}}{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   12. metadata-eval 23.5%

       \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\color{blue}{0}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   13. metadata-eval 23.5%

       \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\color{blue}{\log 1}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   14. cbrt-div 23.5%

       \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\log 1}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   15. metadata-eval 23.5%

       \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{\color{blue}{1}}{\sqrt[3]{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

   16. metadata-eval 23.5%

       \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{1}{\sqrt[3]{\color{blue}{0}}}\right) \cdot \left(\ell \cdot \ell\right) \]

9. Applied egg-rr 23.5%

   \[\leadsto \color{blue}{\left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{1}{\sqrt[3]{0}}\right)} \cdot \left(\ell \cdot \ell\right) \]
10. Step-by-step derivation

    1. pow-plus 23.5%

       \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt[3]{0}}\right)}^{\left(2 + 1\right)}} \cdot \left(\ell \cdot \ell\right) \]

    2. metadata-eval 23.5%

       \[\leadsto {\left(\frac{1}{\sqrt[3]{0}}\right)}^{\color{blue}{3}} \cdot \left(\ell \cdot \ell\right) \]

    3. cube-div 23.5%

       \[\leadsto \color{blue}{\frac{{1}^{3}}{{\left(\sqrt[3]{0}\right)}^{3}}} \cdot \left(\ell \cdot \ell\right) \]

    4. metadata-eval 23.5%

       \[\leadsto \frac{\color{blue}{1}}{{\left(\sqrt[3]{0}\right)}^{3}} \cdot \left(\ell \cdot \ell\right) \]

    5. rem-cube-cbrt 23.5%

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

    6. unpow-1 23.5%

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

    7. pow-base-0 21.0%

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

11. Simplified 21.0%

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

12. Taylor expanded in l around 0 22.3%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(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))))