Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.6% → 82.8%

Time: 19.5s

Alternatives: 16

Speedup: 1.9×

• 27.6% 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: 54.6% 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: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ \mathbf{if}\;k_m \leq 1.4 \cdot 10^{-78}:\\ \;\;\;\;{\left(\frac{\frac{t_1}{{\left(\sqrt[3]{k_m}\right)}^{2}}}{t}\right)}^{3}\\ \mathbf{elif}\;k_m \leq 3.2 \cdot 10^{-7} \lor \neg \left(k_m \leq 2.45 \cdot 10^{+129}\right):\\ \;\;\;\;\frac{2}{{\left(\frac{t}{t_1} \cdot \sqrt[3]{\sin k_m \cdot \left(\tan k_m \cdot \left(2 + {\left(\frac{k_m}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{\frac{{k_m}^{2} \cdot \left(t \cdot {\sin k_m}^{2}\right)}{\cos k_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (let* ((t_1 (pow (cbrt l) 2.0)))
   (if (<= k_m 1.4e-78)
     (pow (/ (/ t_1 (pow (cbrt k_m) 2.0)) t) 3.0)
     (if (or (<= k_m 3.2e-7) (not (<= k_m 2.45e+129)))
       (/
        2.0
        (pow
         (*
          (/ t t_1)
          (cbrt (* (sin k_m) (* (tan k_m) (+ 2.0 (pow (/ k_m t) 2.0))))))
         3.0))
       (*
        2.0
        (/
         (pow l 2.0)
         (/ (* (pow k_m 2.0) (* t (pow (sin k_m) 2.0))) (cos k_m))))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double t_1 = pow(cbrt(l), 2.0);
	double tmp;
	if (k_m <= 1.4e-78) {
		tmp = pow(((t_1 / pow(cbrt(k_m), 2.0)) / t), 3.0);
	} else if ((k_m <= 3.2e-7) || !(k_m <= 2.45e+129)) {
		tmp = 2.0 / pow(((t / t_1) * cbrt((sin(k_m) * (tan(k_m) * (2.0 + pow((k_m / t), 2.0)))))), 3.0);
	} else {
		tmp = 2.0 * (pow(l, 2.0) / ((pow(k_m, 2.0) * (t * pow(sin(k_m), 2.0))) / cos(k_m)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double t_1 = Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if (k_m <= 1.4e-78) {
		tmp = Math.pow(((t_1 / Math.pow(Math.cbrt(k_m), 2.0)) / t), 3.0);
	} else if ((k_m <= 3.2e-7) || !(k_m <= 2.45e+129)) {
		tmp = 2.0 / Math.pow(((t / t_1) * Math.cbrt((Math.sin(k_m) * (Math.tan(k_m) * (2.0 + Math.pow((k_m / t), 2.0)))))), 3.0);
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / ((Math.pow(k_m, 2.0) * (t * Math.pow(Math.sin(k_m), 2.0))) / Math.cos(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	t_1 = cbrt(l) ^ 2.0
	tmp = 0.0
	if (k_m <= 1.4e-78)
		tmp = Float64(Float64(t_1 / (cbrt(k_m) ^ 2.0)) / t) ^ 3.0;
	elseif ((k_m <= 3.2e-7) || !(k_m <= 2.45e+129))
		tmp = Float64(2.0 / (Float64(Float64(t / t_1) * cbrt(Float64(sin(k_m) * Float64(tan(k_m) * Float64(2.0 + (Float64(k_m / t) ^ 2.0)))))) ^ 3.0));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(Float64((k_m ^ 2.0) * Float64(t * (sin(k_m) ^ 2.0))) / cos(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := Block[{t$95$1 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k$95$m, 1.4e-78], N[Power[N[(N[(t$95$1 / N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision], If[Or[LessEqual[k$95$m, 3.2e-7], N[Not[LessEqual[k$95$m, 2.45e+129]], $MachinePrecision]], N[(2.0 / N[Power[N[(N[(t / t$95$1), $MachinePrecision] * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.40000000000000012e-78

   1. Initial program 57.7%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in k around 0 54.4%

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

      1. add-cube-cbrt 54.4%

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

      2. pow3 54.4%

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

      3. associate-/r* 55.0%

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

      4. cbrt-div 55.0%

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

      5. rem-cbrt-cube 61.9%

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

   7. Applied egg-rr 61.9%

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

      1. cbrt-div 62.4%

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

      2. unpow2 62.4%

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

      3. cbrt-prod 68.5%

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

      4. unpow2 68.5%

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

      5. unpow2 68.5%

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

      6. cbrt-prod 79.9%

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

      7. pow2 79.9%

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

   9. Applied egg-rr 79.9%

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

   if 1.40000000000000012e-78 < k < 3.2000000000000001e-7 or 2.45e129 < k

   1. Initial program 38.0%

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

      1. associate-*l* 38.0%

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

      2. *-commutative 38.0%

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

      3. *-commutative 38.0%

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

      4. associate-/r* 43.1%

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

      5. distribute-rgt-in 43.1%

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

      6. unpow2 43.1%

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

      7. times-frac 33.8%

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

      8. sqr-neg 33.8%

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

      9. times-frac 43.1%

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

      10. unpow2 43.1%

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

      11. distribute-rgt-in 43.1%

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

      12. +-commutative 43.1%

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

   3. Simplified 43.1%

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

      1. add-cube-cbrt 43.0%

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

      2. pow3 43.0%

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

      3. associate-*l* 43.0%

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

      4. cbrt-prod 42.9%

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

      5. associate-/l/ 37.8%

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

      6. cbrt-div 38.0%

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

      7. rem-cbrt-cube 62.1%

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

      8. cbrt-prod 76.0%

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

      9. pow2 76.0%

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

   6. Applied egg-rr 76.0%

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

   if 3.2000000000000001e-7 < k < 2.45e129

   1. Initial program 37.0%

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

   3. Simplified 37.0%

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

      1. add-cube-cbrt 36.9%

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

      2. pow3 36.9%

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

      3. *-commutative 36.9%

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

      4. cbrt-prod 36.9%

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

      5. cbrt-prod 36.9%

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

      6. rem-cbrt-cube 36.9%

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

   6. Applied egg-rr 36.9%

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

   7. Taylor expanded in t around 0 79.3%

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

      1. associate-/l* 79.4%

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

   9. Simplified 79.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-78}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t}\right)}^{3}\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{-7} \lor \neg \left(k \leq 2.45 \cdot 10^{+129}\right):\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k_m \leq 1.7 \cdot 10^{-77}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k_m}\right)}^{2}}}{t}\right)}^{3}\\ \mathbf{elif}\;k_m \leq 6.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\left(2 + {\left(\frac{k_m}{t}\right)}^{2}\right) \cdot \left(\sin k_m \cdot \tan k_m\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sqrt[3]{{\left(k_m \cdot \left(\sin k_m \cdot \sqrt{t}\right)\right)}^{2}}\right)}^{3}}{{\ell}^{2} \cdot \cos k_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.7e-77)
   (pow (/ (/ (pow (cbrt l) 2.0) (pow (cbrt k_m) 2.0)) t) 3.0)
   (if (<= k_m 6.5e-8)
     (/
      2.0
      (pow
       (*
        (/ (pow t 1.5) l)
        (sqrt (* (+ 2.0 (pow (/ k_m t) 2.0)) (* (sin k_m) (tan k_m)))))
       2.0))
     (/
      2.0
      (/
       (pow (cbrt (pow (* k_m (* (sin k_m) (sqrt t))) 2.0)) 3.0)
       (* (pow l 2.0) (cos k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.7e-77) {
		tmp = pow(((pow(cbrt(l), 2.0) / pow(cbrt(k_m), 2.0)) / t), 3.0);
	} else if (k_m <= 6.5e-8) {
		tmp = 2.0 / pow(((pow(t, 1.5) / l) * sqrt(((2.0 + pow((k_m / t), 2.0)) * (sin(k_m) * tan(k_m))))), 2.0);
	} else {
		tmp = 2.0 / (pow(cbrt(pow((k_m * (sin(k_m) * sqrt(t))), 2.0)), 3.0) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.7e-77) {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / Math.pow(Math.cbrt(k_m), 2.0)) / t), 3.0);
	} else if (k_m <= 6.5e-8) {
		tmp = 2.0 / Math.pow(((Math.pow(t, 1.5) / l) * Math.sqrt(((2.0 + Math.pow((k_m / t), 2.0)) * (Math.sin(k_m) * Math.tan(k_m))))), 2.0);
	} else {
		tmp = 2.0 / (Math.pow(Math.cbrt(Math.pow((k_m * (Math.sin(k_m) * Math.sqrt(t))), 2.0)), 3.0) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.7e-77)
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / (cbrt(k_m) ^ 2.0)) / t) ^ 3.0;
	elseif (k_m <= 6.5e-8)
		tmp = Float64(2.0 / (Float64(Float64((t ^ 1.5) / l) * sqrt(Float64(Float64(2.0 + (Float64(k_m / t) ^ 2.0)) * Float64(sin(k_m) * tan(k_m))))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((cbrt((Float64(k_m * Float64(sin(k_m) * sqrt(t))) ^ 2.0)) ^ 3.0) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.7e-77], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[k$95$m, 6.5e-8], N[(2.0 / N[Power[N[(N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[Power[N[Power[N[(k$95$m * N[(N[Sin[k$95$m], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.69999999999999991e-77

   1. Initial program 57.7%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in k around 0 54.4%

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

      1. add-cube-cbrt 54.4%

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

      2. pow3 54.4%

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

      3. associate-/r* 55.0%

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

      4. cbrt-div 55.0%

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

      5. rem-cbrt-cube 61.9%

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

   7. Applied egg-rr 61.9%

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

      1. cbrt-div 62.4%

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

      2. unpow2 62.4%

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

      3. cbrt-prod 68.5%

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

      4. unpow2 68.5%

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

      5. unpow2 68.5%

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

      6. cbrt-prod 79.9%

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

      7. pow2 79.9%

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

   9. Applied egg-rr 79.9%

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

   if 1.69999999999999991e-77 < k < 6.49999999999999997e-8

   1. Initial program 77.3%

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

      1. associate-*l* 77.3%

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

      2. *-commutative 77.3%

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

      3. *-commutative 77.3%

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

      4. associate-/r* 77.7%

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

      5. distribute-rgt-in 77.7%

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

      6. unpow2 77.7%

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

      7. times-frac 77.7%

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

      8. sqr-neg 77.7%

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

      9. times-frac 77.7%

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

      10. unpow2 77.7%

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

      11. distribute-rgt-in 77.7%

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

      12. +-commutative 77.7%

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

   3. Simplified 77.7%

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

      1. add-sqr-sqrt 15.4%

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

      2. pow2 15.4%

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

      3. associate-*l* 15.4%

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

      4. sqrt-prod 15.4%

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

      5. associate-/l/ 15.4%

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

      6. sqrt-div 15.4%

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

      7. sqrt-pow1 15.3%

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

      8. metadata-eval 15.3%

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

      9. sqrt-prod 7.7%

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

      10. add-sqr-sqrt 15.3%

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

   6. Applied egg-rr 15.3%

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

      1. associate-*r* 15.3%

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

   8. Simplified 15.3%

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

   if 6.49999999999999997e-8 < k

   1. Initial program 28.6%

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

      1. associate-*l* 28.6%

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

      2. *-commutative 28.6%

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

      3. *-commutative 28.6%

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

      4. associate-/r* 34.4%

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

      5. distribute-rgt-in 34.4%

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

      6. unpow2 34.4%

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

      7. times-frac 27.1%

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

      8. sqr-neg 27.1%

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

      9. times-frac 34.4%

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

      10. unpow2 34.4%

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

      11. distribute-rgt-in 34.4%

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

      12. +-commutative 34.4%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in t around 0 63.5%

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

      1. add-cube-cbrt 63.4%

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

      2. pow3 63.4%

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

      3. add-sqr-sqrt 27.9%

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

      4. pow2 27.9%

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

      5. sqrt-prod 27.9%

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

      6. unpow2 27.9%

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

      7. sqrt-prod 29.5%

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

      8. add-sqr-sqrt 29.5%

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

      9. *-commutative 29.5%

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

      10. sqrt-prod 29.5%

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

      11. unpow2 29.5%

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

      12. sqrt-prod 15.9%

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

      13. add-sqr-sqrt 29.5%

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

   7. Applied egg-rr 29.5%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-77}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t}\right)}^{3}\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sqrt[3]{{\left(k \cdot \left(\sin k \cdot \sqrt{t}\right)\right)}^{2}}\right)}^{3}}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k_m \leq 7.5 \cdot 10^{-77}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k_m}\right)}^{2}}}{t}\right)}^{3}\\ \mathbf{elif}\;k_m \leq 5.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\left(2 + {\left(\frac{k_m}{t}\right)}^{2}\right) \cdot \left(\sin k_m \cdot \tan k_m\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sqrt{t} \cdot \left(k_m \cdot \sin k_m\right)\right)}^{2}}{{\ell}^{2} \cdot \cos k_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7.5e-77)
   (pow (/ (/ (pow (cbrt l) 2.0) (pow (cbrt k_m) 2.0)) t) 3.0)
   (if (<= k_m 5.1e-5)
     (/
      2.0
      (pow
       (*
        (/ (pow t 1.5) l)
        (sqrt (* (+ 2.0 (pow (/ k_m t) 2.0)) (* (sin k_m) (tan k_m)))))
       2.0))
     (/
      2.0
      (/
       (pow (* (sqrt t) (* k_m (sin k_m))) 2.0)
       (* (pow l 2.0) (cos k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.5e-77) {
		tmp = pow(((pow(cbrt(l), 2.0) / pow(cbrt(k_m), 2.0)) / t), 3.0);
	} else if (k_m <= 5.1e-5) {
		tmp = 2.0 / pow(((pow(t, 1.5) / l) * sqrt(((2.0 + pow((k_m / t), 2.0)) * (sin(k_m) * tan(k_m))))), 2.0);
	} else {
		tmp = 2.0 / (pow((sqrt(t) * (k_m * sin(k_m))), 2.0) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.5e-77) {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / Math.pow(Math.cbrt(k_m), 2.0)) / t), 3.0);
	} else if (k_m <= 5.1e-5) {
		tmp = 2.0 / Math.pow(((Math.pow(t, 1.5) / l) * Math.sqrt(((2.0 + Math.pow((k_m / t), 2.0)) * (Math.sin(k_m) * Math.tan(k_m))))), 2.0);
	} else {
		tmp = 2.0 / (Math.pow((Math.sqrt(t) * (k_m * Math.sin(k_m))), 2.0) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7.5e-77)
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / (cbrt(k_m) ^ 2.0)) / t) ^ 3.0;
	elseif (k_m <= 5.1e-5)
		tmp = Float64(2.0 / (Float64(Float64((t ^ 1.5) / l) * sqrt(Float64(Float64(2.0 + (Float64(k_m / t) ^ 2.0)) * Float64(sin(k_m) * tan(k_m))))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((Float64(sqrt(t) * Float64(k_m * sin(k_m))) ^ 2.0) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 7.5e-77], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[k$95$m, 5.1e-5], N[(2.0 / N[Power[N[(N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Sqrt[t], $MachinePrecision] * N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 7.5000000000000006e-77

   1. Initial program 57.7%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in k around 0 54.4%

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

      1. add-cube-cbrt 54.4%

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

      2. pow3 54.4%

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

      3. associate-/r* 55.0%

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

      4. cbrt-div 55.0%

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

      5. rem-cbrt-cube 61.9%

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

   7. Applied egg-rr 61.9%

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

      1. cbrt-div 62.4%

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

      2. unpow2 62.4%

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

      3. cbrt-prod 68.5%

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

      4. unpow2 68.5%

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

      5. unpow2 68.5%

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

      6. cbrt-prod 79.9%

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

      7. pow2 79.9%

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

   9. Applied egg-rr 79.9%

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

   if 7.5000000000000006e-77 < k < 5.09999999999999996e-5

   1. Initial program 77.3%

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

      1. associate-*l* 77.3%

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

      2. *-commutative 77.3%

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

      3. *-commutative 77.3%

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

      4. associate-/r* 77.7%

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

      5. distribute-rgt-in 77.7%

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

      6. unpow2 77.7%

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

      7. times-frac 77.7%

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

      8. sqr-neg 77.7%

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

      9. times-frac 77.7%

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

      10. unpow2 77.7%

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

      11. distribute-rgt-in 77.7%

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

      12. +-commutative 77.7%

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

   3. Simplified 77.7%

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

      1. add-sqr-sqrt 15.4%

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

      2. pow2 15.4%

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

      3. associate-*l* 15.4%

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

      4. sqrt-prod 15.4%

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

      5. associate-/l/ 15.4%

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

      6. sqrt-div 15.4%

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

      7. sqrt-pow1 15.3%

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

      8. metadata-eval 15.3%

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

      9. sqrt-prod 7.7%

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

      10. add-sqr-sqrt 15.3%

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

   6. Applied egg-rr 15.3%

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

      1. associate-*r* 15.3%

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

   8. Simplified 15.3%

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

   if 5.09999999999999996e-5 < k

   1. Initial program 28.6%

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

      1. associate-*l* 28.6%

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

      2. *-commutative 28.6%

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

      3. *-commutative 28.6%

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

      4. associate-/r* 34.4%

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

      5. distribute-rgt-in 34.4%

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

      6. unpow2 34.4%

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

      7. times-frac 27.1%

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

      8. sqr-neg 27.1%

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

      9. times-frac 34.4%

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

      10. unpow2 34.4%

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

      11. distribute-rgt-in 34.4%

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

      12. +-commutative 34.4%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in t around 0 63.5%

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

      1. expm1-log1p-u 47.7%

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

      2. expm1-udef 33.8%

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

   7. Applied egg-rr 10.7%

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

      1. expm1-def 17.7%

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

      2. expm1-log1p 29.5%

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

      3. associate-*r* 29.5%

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

   9. Simplified 29.5%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{-77}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t}\right)}^{3}\\ \mathbf{elif}\;k \leq 5.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sqrt{t} \cdot \left(k \cdot \sin k\right)\right)}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k_m \leq 7.2 \cdot 10^{-8}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k_m}\right)}^{2}}}{t}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sqrt{t} \cdot \left(k_m \cdot \sin k_m\right)\right)}^{2}}{{\ell}^{2} \cdot \cos k_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7.2e-8)
   (pow (/ (/ (pow (cbrt l) 2.0) (pow (cbrt k_m) 2.0)) t) 3.0)
   (/
    2.0
    (/ (pow (* (sqrt t) (* k_m (sin k_m))) 2.0) (* (pow l 2.0) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.2e-8) {
		tmp = pow(((pow(cbrt(l), 2.0) / pow(cbrt(k_m), 2.0)) / t), 3.0);
	} else {
		tmp = 2.0 / (pow((sqrt(t) * (k_m * sin(k_m))), 2.0) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.2e-8) {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / Math.pow(Math.cbrt(k_m), 2.0)) / t), 3.0);
	} else {
		tmp = 2.0 / (Math.pow((Math.sqrt(t) * (k_m * Math.sin(k_m))), 2.0) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7.2e-8)
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / (cbrt(k_m) ^ 2.0)) / t) ^ 3.0;
	else
		tmp = Float64(2.0 / Float64((Float64(sqrt(t) * Float64(k_m * sin(k_m))) ^ 2.0) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 7.2e-8], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Sqrt[t], $MachinePrecision] * N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.19999999999999962e-8

   1. Initial program 58.9%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in k around 0 55.9%

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

      1. add-cube-cbrt 55.9%

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

      2. pow3 55.9%

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

      3. associate-/r* 55.9%

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

      4. cbrt-div 56.0%

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

      5. rem-cbrt-cube 62.4%

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

   7. Applied egg-rr 62.4%

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

      1. cbrt-div 63.3%

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

      2. unpow2 63.3%

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

      3. cbrt-prod 69.5%

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

      4. unpow2 69.5%

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

      5. unpow2 69.5%

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

      6. cbrt-prod 80.1%

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

      7. pow2 80.1%

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

   9. Applied egg-rr 80.1%

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

   if 7.19999999999999962e-8 < k

   1. Initial program 28.6%

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

      1. associate-*l* 28.6%

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

      2. *-commutative 28.6%

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

      3. *-commutative 28.6%

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

      4. associate-/r* 34.4%

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

      5. distribute-rgt-in 34.4%

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

      6. unpow2 34.4%

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

      7. times-frac 27.1%

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

      8. sqr-neg 27.1%

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

      9. times-frac 34.4%

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

      10. unpow2 34.4%

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

      11. distribute-rgt-in 34.4%

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

      12. +-commutative 34.4%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in t around 0 63.5%

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

      1. expm1-log1p-u 47.7%

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

      2. expm1-udef 33.8%

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

   7. Applied egg-rr 10.7%

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

      1. expm1-def 17.7%

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

      2. expm1-log1p 29.5%

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

      3. associate-*r* 29.5%

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

   9. Simplified 29.5%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{-8}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sqrt{t} \cdot \left(k \cdot \sin k\right)\right)}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k_m \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{2}{\left(k_m \cdot 2\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k_m}\right)}^{2}}\\ \mathbf{elif}\;k_m \leq 8 \cdot 10^{-6}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{{k_m}^{-2}}}{t}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k_m \cdot 2\right)}{2}\right)\right)}{{\ell}^{2} \cdot \cos k_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7.5e-155)
   (/ 2.0 (* (* k_m 2.0) (pow (* (/ (pow t 1.5) l) (sqrt (sin k_m))) 2.0)))
   (if (<= k_m 8e-6)
     (pow (/ (* (pow (cbrt l) 2.0) (cbrt (pow k_m -2.0))) t) 3.0)
     (/
      2.0
      (/
       (* (pow k_m 2.0) (* t (- 0.5 (/ (cos (* k_m 2.0)) 2.0))))
       (* (pow l 2.0) (cos k_m)))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.5e-155) {
		tmp = 2.0 / ((k_m * 2.0) * pow(((pow(t, 1.5) / l) * sqrt(sin(k_m))), 2.0));
	} else if (k_m <= 8e-6) {
		tmp = pow(((pow(cbrt(l), 2.0) * cbrt(pow(k_m, -2.0))) / t), 3.0);
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t * (0.5 - (cos((k_m * 2.0)) / 2.0)))) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.5e-155) {
		tmp = 2.0 / ((k_m * 2.0) * Math.pow(((Math.pow(t, 1.5) / l) * Math.sqrt(Math.sin(k_m))), 2.0));
	} else if (k_m <= 8e-6) {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) * Math.cbrt(Math.pow(k_m, -2.0))) / t), 3.0);
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t * (0.5 - (Math.cos((k_m * 2.0)) / 2.0)))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7.5e-155)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * (Float64(Float64((t ^ 1.5) / l) * sqrt(sin(k_m))) ^ 2.0)));
	elseif (k_m <= 8e-6)
		tmp = Float64(Float64((cbrt(l) ^ 2.0) * cbrt((k_m ^ -2.0))) / t) ^ 3.0;
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0)))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 7.5e-155], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[Power[N[(N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 8e-6], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Power[k$95$m, -2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 7.5000000000000006e-155

   1. Initial program 57.4%

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

      1. associate-*l* 57.4%

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

      2. *-commutative 57.4%

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

      3. *-commutative 57.4%

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

      4. associate-/r* 65.5%

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

      5. distribute-rgt-in 65.5%

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

      6. unpow2 65.5%

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

      7. times-frac 48.9%

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

      8. sqr-neg 48.9%

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

      9. times-frac 65.5%

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

      10. unpow2 65.5%

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

      11. distribute-rgt-in 65.5%

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

      12. +-commutative 65.5%

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

   3. Simplified 65.5%

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

      1. add-sqr-sqrt 33.1%

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

      2. pow2 33.1%

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

      3. sqrt-prod 14.9%

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

      4. associate-/l/ 11.3%

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

      5. sqrt-div 11.3%

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

      6. sqrt-pow1 14.4%

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

      7. metadata-eval 14.4%

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

      8. sqrt-prod 10.7%

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

      9. add-sqr-sqrt 19.6%

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

   6. Applied egg-rr 19.6%

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

   7. Taylor expanded in k around 0 15.6%

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

      1. *-commutative 64.4%

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

   9. Simplified 15.6%

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

   if 7.5000000000000006e-155 < k < 7.99999999999999964e-6

   1. Initial program 67.5%

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

   3. Simplified 70.6%

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

   5. Taylor expanded in k around 0 73.9%

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

      1. add-cube-cbrt 73.7%

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

      2. pow3 73.7%

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

      3. associate-/r* 70.6%

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

      4. cbrt-div 70.5%

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

      5. rem-cbrt-cube 77.1%

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

   7. Applied egg-rr 77.1%

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

      1. pow1/3 75.8%

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

      2. div-inv 75.8%

         \[\leadsto {\left(\frac{{\color{blue}{\left({\ell}^{2} \cdot \frac{1}{{k}^{2}}\right)}}^{0.3333333333333333}}{t}\right)}^{3} \]

      3. unpow-prod-down 78.6%

         \[\leadsto {\left(\frac{\color{blue}{{\left({\ell}^{2}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{{k}^{2}}\right)}^{0.3333333333333333}}}{t}\right)}^{3} \]

      4. pow1/3 78.5%

         \[\leadsto {\left(\frac{\color{blue}{\sqrt[3]{{\ell}^{2}}} \cdot {\left(\frac{1}{{k}^{2}}\right)}^{0.3333333333333333}}{t}\right)}^{3} \]

      5. unpow2 78.5%

         \[\leadsto {\left(\frac{\sqrt[3]{\color{blue}{\ell \cdot \ell}} \cdot {\left(\frac{1}{{k}^{2}}\right)}^{0.3333333333333333}}{t}\right)}^{3} \]

      6. cbrt-prod 84.4%

         \[\leadsto {\left(\frac{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)} \cdot {\left(\frac{1}{{k}^{2}}\right)}^{0.3333333333333333}}{t}\right)}^{3} \]

      7. unpow2 84.4%

         \[\leadsto {\left(\frac{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\left(\frac{1}{{k}^{2}}\right)}^{0.3333333333333333}}{t}\right)}^{3} \]

      8. pow-flip 84.4%

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

      9. metadata-eval 84.4%

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

   9. Applied egg-rr 84.4%

      \[\leadsto {\left(\frac{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot {\left({k}^{-2}\right)}^{0.3333333333333333}}}{t}\right)}^{3} \]
   10. Step-by-step derivation

       1. unpow1/3 86.4%

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

   11. Simplified 86.4%

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

   if 7.99999999999999964e-6 < k

   1. Initial program 28.6%

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

      1. associate-*l* 28.6%

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

      2. *-commutative 28.6%

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

      3. *-commutative 28.6%

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

      4. associate-/r* 34.4%

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

      5. distribute-rgt-in 34.4%

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

      6. unpow2 34.4%

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

      7. times-frac 27.1%

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

      8. sqr-neg 27.1%

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

      9. times-frac 34.4%

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

      10. unpow2 34.4%

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

      11. distribute-rgt-in 34.4%

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

      12. +-commutative 34.4%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in t around 0 63.5%

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

      1. unpow2 63.5%

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

      2. sin-mult 63.2%

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

   7. Applied egg-rr 63.2%

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

      1. div-sub 63.2%

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

      2. +-inverses 63.2%

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

      3. cos-0 63.2%

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

      4. metadata-eval 63.2%

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

      5. count-2 63.2%

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

      6. *-commutative 63.2%

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

   9. Simplified 63.2%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}\right)}{{\ell}^{2} \cdot \cos k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{elif}\;k \leq 8 \cdot 10^{-6}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2} \cdot \sqrt[3]{{k}^{-2}}}{t}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k_m \leq 3.5 \cdot 10^{-7}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k_m}\right)}^{2}}}{t}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k_m \cdot 2\right)}{2}\right)\right)}{{\ell}^{2} \cdot \cos k_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 3.5e-7)
   (pow (/ (/ (pow (cbrt l) 2.0) (pow (cbrt k_m) 2.0)) t) 3.0)
   (/
    2.0
    (/
     (* (pow k_m 2.0) (* t (- 0.5 (/ (cos (* k_m 2.0)) 2.0))))
     (* (pow l 2.0) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.5e-7) {
		tmp = pow(((pow(cbrt(l), 2.0) / pow(cbrt(k_m), 2.0)) / t), 3.0);
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t * (0.5 - (cos((k_m * 2.0)) / 2.0)))) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 3.5e-7) {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / Math.pow(Math.cbrt(k_m), 2.0)) / t), 3.0);
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t * (0.5 - (Math.cos((k_m * 2.0)) / 2.0)))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 3.5e-7)
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / (cbrt(k_m) ^ 2.0)) / t) ^ 3.0;
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0)))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 3.5e-7], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], 3.0], $MachinePrecision], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.49999999999999984e-7

   1. Initial program 58.9%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in k around 0 55.9%

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

      1. add-cube-cbrt 55.9%

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

      2. pow3 55.9%

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

      3. associate-/r* 55.9%

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

      4. cbrt-div 56.0%

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

      5. rem-cbrt-cube 62.4%

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

   7. Applied egg-rr 62.4%

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

      1. cbrt-div 63.3%

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

      2. unpow2 63.3%

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

      3. cbrt-prod 69.5%

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

      4. unpow2 69.5%

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

      5. unpow2 69.5%

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

      6. cbrt-prod 80.1%

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

      7. pow2 80.1%

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

   9. Applied egg-rr 80.1%

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

   if 3.49999999999999984e-7 < k

   1. Initial program 28.6%

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

      1. associate-*l* 28.6%

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

      2. *-commutative 28.6%

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

      3. *-commutative 28.6%

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

      4. associate-/r* 34.4%

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

      5. distribute-rgt-in 34.4%

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

      6. unpow2 34.4%

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

      7. times-frac 27.1%

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

      8. sqr-neg 27.1%

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

      9. times-frac 34.4%

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

      10. unpow2 34.4%

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

      11. distribute-rgt-in 34.4%

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

      12. +-commutative 34.4%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in t around 0 63.5%

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

      1. unpow2 63.5%

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

      2. sin-mult 63.2%

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

   7. Applied egg-rr 63.2%

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

      1. div-sub 63.2%

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

      2. +-inverses 63.2%

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

      3. cos-0 63.2%

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

      4. metadata-eval 63.2%

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

      5. count-2 63.2%

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

      6. *-commutative 63.2%

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

   9. Simplified 63.2%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}\right)}{{\ell}^{2} \cdot \cos k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-7}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k_m \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(k_m \cdot 2\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k_m \cdot 2\right)}{2}\right)\right)}{{\ell}^{2} \cdot \cos k_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.9e-5)
   (/ 2.0 (* (* k_m 2.0) (pow (* (/ (pow t 1.5) l) (sqrt (sin k_m))) 2.0)))
   (/
    2.0
    (/
     (* (pow k_m 2.0) (* t (- 0.5 (/ (cos (* k_m 2.0)) 2.0))))
     (* (pow l 2.0) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.9e-5) {
		tmp = 2.0 / ((k_m * 2.0) * pow(((pow(t, 1.5) / l) * sqrt(sin(k_m))), 2.0));
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t * (0.5 - (cos((k_m * 2.0)) / 2.0)))) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.9d-5) then
        tmp = 2.0d0 / ((k_m * 2.0d0) * ((((t ** 1.5d0) / l) * sqrt(sin(k_m))) ** 2.0d0))
    else
        tmp = 2.0d0 / (((k_m ** 2.0d0) * (t * (0.5d0 - (cos((k_m * 2.0d0)) / 2.0d0)))) / ((l ** 2.0d0) * cos(k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.9e-5) {
		tmp = 2.0 / ((k_m * 2.0) * Math.pow(((Math.pow(t, 1.5) / l) * Math.sqrt(Math.sin(k_m))), 2.0));
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t * (0.5 - (Math.cos((k_m * 2.0)) / 2.0)))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.9e-5:
		tmp = 2.0 / ((k_m * 2.0) * math.pow(((math.pow(t, 1.5) / l) * math.sqrt(math.sin(k_m))), 2.0))
	else:
		tmp = 2.0 / ((math.pow(k_m, 2.0) * (t * (0.5 - (math.cos((k_m * 2.0)) / 2.0)))) / (math.pow(l, 2.0) * math.cos(k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.9e-5)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * (Float64(Float64((t ^ 1.5) / l) * sqrt(sin(k_m))) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t * Float64(0.5 - Float64(cos(Float64(k_m * 2.0)) / 2.0)))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.9e-5)
		tmp = 2.0 / ((k_m * 2.0) * ((((t ^ 1.5) / l) * sqrt(sin(k_m))) ^ 2.0));
	else
		tmp = 2.0 / (((k_m ^ 2.0) * (t * (0.5 - (cos((k_m * 2.0)) / 2.0)))) / ((l ^ 2.0) * cos(k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.9e-5], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[Power[N[(N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[(0.5 - N[(N[Cos[N[(k$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.9000000000000001e-5

   1. Initial program 58.9%

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

      1. associate-*l* 59.0%

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

      2. *-commutative 59.0%

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

      3. *-commutative 59.0%

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

      4. associate-/r* 65.9%

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

      5. distribute-rgt-in 65.9%

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

      6. unpow2 65.9%

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

      7. times-frac 50.2%

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

      8. sqr-neg 50.2%

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

      9. times-frac 65.9%

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

      10. unpow2 65.9%

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

      11. distribute-rgt-in 65.9%

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

      12. +-commutative 65.9%

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

   3. Simplified 65.9%

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

      1. add-sqr-sqrt 31.7%

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

      2. pow2 31.7%

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

      3. sqrt-prod 16.3%

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

      4. associate-/l/ 13.2%

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

      5. sqrt-div 13.2%

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

      6. sqrt-pow1 16.3%

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

      7. metadata-eval 16.3%

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

      8. sqrt-prod 12.6%

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

      9. add-sqr-sqrt 20.7%

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

   6. Applied egg-rr 20.7%

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

   7. Taylor expanded in k around 0 17.3%

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

      1. *-commutative 65.5%

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

   9. Simplified 17.3%

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

   if 1.9000000000000001e-5 < k

   1. Initial program 28.6%

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

      1. associate-*l* 28.6%

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

      2. *-commutative 28.6%

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

      3. *-commutative 28.6%

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

      4. associate-/r* 34.4%

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

      5. distribute-rgt-in 34.4%

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

      6. unpow2 34.4%

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

      7. times-frac 27.1%

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

      8. sqr-neg 27.1%

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

      9. times-frac 34.4%

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

      10. unpow2 34.4%

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

      11. distribute-rgt-in 34.4%

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

      12. +-commutative 34.4%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in t around 0 63.5%

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

      1. unpow2 63.5%

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

      2. sin-mult 63.2%

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

   7. Applied egg-rr 63.2%

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

      1. div-sub 63.2%

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

      2. +-inverses 63.2%

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

      3. cos-0 63.2%

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

      4. metadata-eval 63.2%

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

      5. count-2 63.2%

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

      6. *-commutative 63.2%

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

   9. Simplified 63.2%

      \[\leadsto \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}\right)}{{\ell}^{2} \cdot \cos k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k_m \leq 7.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{\left(k_m \cdot 2\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k_m}^{2} \cdot \left(t \cdot {k_m}^{2}\right)}{{\ell}^{2} \cdot \cos k_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 7.5e-10)
   (/ 2.0 (* (* k_m 2.0) (pow (* (/ (pow t 1.5) l) (sqrt (sin k_m))) 2.0)))
   (/
    2.0
    (/ (* (pow k_m 2.0) (* t (pow k_m 2.0))) (* (pow l 2.0) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.5e-10) {
		tmp = 2.0 / ((k_m * 2.0) * pow(((pow(t, 1.5) / l) * sqrt(sin(k_m))), 2.0));
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t * pow(k_m, 2.0))) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 7.5d-10) then
        tmp = 2.0d0 / ((k_m * 2.0d0) * ((((t ** 1.5d0) / l) * sqrt(sin(k_m))) ** 2.0d0))
    else
        tmp = 2.0d0 / (((k_m ** 2.0d0) * (t * (k_m ** 2.0d0))) / ((l ** 2.0d0) * cos(k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 7.5e-10) {
		tmp = 2.0 / ((k_m * 2.0) * Math.pow(((Math.pow(t, 1.5) / l) * Math.sqrt(Math.sin(k_m))), 2.0));
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t * Math.pow(k_m, 2.0))) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 7.5e-10:
		tmp = 2.0 / ((k_m * 2.0) * math.pow(((math.pow(t, 1.5) / l) * math.sqrt(math.sin(k_m))), 2.0))
	else:
		tmp = 2.0 / ((math.pow(k_m, 2.0) * (t * math.pow(k_m, 2.0))) / (math.pow(l, 2.0) * math.cos(k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 7.5e-10)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * (Float64(Float64((t ^ 1.5) / l) * sqrt(sin(k_m))) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t * (k_m ^ 2.0))) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 7.5e-10)
		tmp = 2.0 / ((k_m * 2.0) * ((((t ^ 1.5) / l) * sqrt(sin(k_m))) ^ 2.0));
	else
		tmp = 2.0 / (((k_m ^ 2.0) * (t * (k_m ^ 2.0))) / ((l ^ 2.0) * cos(k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 7.5e-10], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[Power[N[(N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.49999999999999995e-10

   1. Initial program 58.9%

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

      1. associate-*l* 59.0%

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

      2. *-commutative 59.0%

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

      3. *-commutative 59.0%

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

      4. associate-/r* 65.9%

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

      5. distribute-rgt-in 65.9%

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

      6. unpow2 65.9%

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

      7. times-frac 50.2%

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

      8. sqr-neg 50.2%

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

      9. times-frac 65.9%

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

      10. unpow2 65.9%

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

      11. distribute-rgt-in 65.9%

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

      12. +-commutative 65.9%

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

   3. Simplified 65.9%

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

      1. add-sqr-sqrt 31.7%

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

      2. pow2 31.7%

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

      3. sqrt-prod 16.3%

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

      4. associate-/l/ 13.2%

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

      5. sqrt-div 13.2%

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

      6. sqrt-pow1 16.3%

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

      7. metadata-eval 16.3%

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

      8. sqrt-prod 12.6%

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

      9. add-sqr-sqrt 20.7%

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

   6. Applied egg-rr 20.7%

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

   7. Taylor expanded in k around 0 17.3%

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

      1. *-commutative 65.5%

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

   9. Simplified 17.3%

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

   if 7.49999999999999995e-10 < k

   1. Initial program 28.6%

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

      1. associate-*l* 28.6%

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

      2. *-commutative 28.6%

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

      3. *-commutative 28.6%

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

      4. associate-/r* 34.4%

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

      5. distribute-rgt-in 34.4%

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

      6. unpow2 34.4%

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

      7. times-frac 27.1%

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

      8. sqr-neg 27.1%

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

      9. times-frac 34.4%

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

      10. unpow2 34.4%

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

      11. distribute-rgt-in 34.4%

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

      12. +-commutative 34.4%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in t around 0 63.5%

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

   6. Taylor expanded in k around 0 55.3%

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

4. Final simplification 25.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k_m \leq 4.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\left(k_m \cdot 2\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k_m}^{4}}{{\ell}^{2} \cdot \cos k_m}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 4.2e-12)
   (/ 2.0 (* (* k_m 2.0) (pow (* (/ (pow t 1.5) l) (sqrt (sin k_m))) 2.0)))
   (/ 2.0 (/ (* t (pow k_m 4.0)) (* (pow l 2.0) (cos k_m))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4.2e-12) {
		tmp = 2.0 / ((k_m * 2.0) * pow(((pow(t, 1.5) / l) * sqrt(sin(k_m))), 2.0));
	} else {
		tmp = 2.0 / ((t * pow(k_m, 4.0)) / (pow(l, 2.0) * cos(k_m)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4.2d-12) then
        tmp = 2.0d0 / ((k_m * 2.0d0) * ((((t ** 1.5d0) / l) * sqrt(sin(k_m))) ** 2.0d0))
    else
        tmp = 2.0d0 / ((t * (k_m ** 4.0d0)) / ((l ** 2.0d0) * cos(k_m)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 4.2e-12) {
		tmp = 2.0 / ((k_m * 2.0) * Math.pow(((Math.pow(t, 1.5) / l) * Math.sqrt(Math.sin(k_m))), 2.0));
	} else {
		tmp = 2.0 / ((t * Math.pow(k_m, 4.0)) / (Math.pow(l, 2.0) * Math.cos(k_m)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 4.2e-12:
		tmp = 2.0 / ((k_m * 2.0) * math.pow(((math.pow(t, 1.5) / l) * math.sqrt(math.sin(k_m))), 2.0))
	else:
		tmp = 2.0 / ((t * math.pow(k_m, 4.0)) / (math.pow(l, 2.0) * math.cos(k_m)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 4.2e-12)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * (Float64(Float64((t ^ 1.5) / l) * sqrt(sin(k_m))) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(t * (k_m ^ 4.0)) / Float64((l ^ 2.0) * cos(k_m))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 4.2e-12)
		tmp = 2.0 / ((k_m * 2.0) * ((((t ^ 1.5) / l) * sqrt(sin(k_m))) ^ 2.0));
	else
		tmp = 2.0 / ((t * (k_m ^ 4.0)) / ((l ^ 2.0) * cos(k_m)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 4.2e-12], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[Power[N[(N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.19999999999999988e-12

   1. Initial program 58.9%

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

      1. associate-*l* 59.0%

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

      2. *-commutative 59.0%

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

      3. *-commutative 59.0%

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

      4. associate-/r* 65.9%

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

      5. distribute-rgt-in 65.9%

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

      6. unpow2 65.9%

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

      7. times-frac 50.2%

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

      8. sqr-neg 50.2%

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

      9. times-frac 65.9%

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

      10. unpow2 65.9%

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

      11. distribute-rgt-in 65.9%

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

      12. +-commutative 65.9%

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

   3. Simplified 65.9%

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

      1. add-sqr-sqrt 31.7%

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

      2. pow2 31.7%

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

      3. sqrt-prod 16.3%

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

      4. associate-/l/ 13.2%

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

      5. sqrt-div 13.2%

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

      6. sqrt-pow1 16.3%

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

      7. metadata-eval 16.3%

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

      8. sqrt-prod 12.6%

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

      9. add-sqr-sqrt 20.7%

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

   6. Applied egg-rr 20.7%

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

   7. Taylor expanded in k around 0 17.3%

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

      1. *-commutative 65.5%

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

   9. Simplified 17.3%

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

   if 4.19999999999999988e-12 < k

   1. Initial program 28.6%

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

      1. associate-*l* 28.6%

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

      2. *-commutative 28.6%

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

      3. *-commutative 28.6%

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

      4. associate-/r* 34.4%

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

      5. distribute-rgt-in 34.4%

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

      6. unpow2 34.4%

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

      7. times-frac 27.1%

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

      8. sqr-neg 27.1%

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

      9. times-frac 34.4%

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

      10. unpow2 34.4%

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

      11. distribute-rgt-in 34.4%

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

      12. +-commutative 34.4%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in t around 0 63.5%

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

   6. Taylor expanded in k around 0 51.8%

      \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{4} \cdot t}}{{\ell}^{2} \cdot \cos k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{4}}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-92}:\\ \;\;\;\;{\left(\ell \cdot \sqrt{\frac{2}{t \cdot {k_m}^{4}}}\right)}^{2}\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{+59}:\\ \;\;\;\;\frac{2}{\left(\tan k_m \cdot \left(2 + {\left(\frac{k_m}{t}\right)}^{2}\right)\right) \cdot \frac{\frac{k_m}{\frac{\ell}{{t}^{3}}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{\left({\left(\sqrt[3]{k_m}\right)}^{2} \cdot t\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 3.3e-92)
   (pow (* l (sqrt (/ 2.0 (* t (pow k_m 4.0))))) 2.0)
   (if (<= t 2.45e+59)
     (/
      2.0
      (*
       (* (tan k_m) (+ 2.0 (pow (/ k_m t) 2.0)))
       (/ (/ k_m (/ l (pow t 3.0))) l)))
     (/ (pow l 2.0) (pow (* (pow (cbrt k_m) 2.0) t) 3.0)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3.3e-92) {
		tmp = pow((l * sqrt((2.0 / (t * pow(k_m, 4.0))))), 2.0);
	} else if (t <= 2.45e+59) {
		tmp = 2.0 / ((tan(k_m) * (2.0 + pow((k_m / t), 2.0))) * ((k_m / (l / pow(t, 3.0))) / l));
	} else {
		tmp = pow(l, 2.0) / pow((pow(cbrt(k_m), 2.0) * t), 3.0);
	}
	return tmp;
}

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3.3e-92) {
		tmp = Math.pow((l * Math.sqrt((2.0 / (t * Math.pow(k_m, 4.0))))), 2.0);
	} else if (t <= 2.45e+59) {
		tmp = 2.0 / ((Math.tan(k_m) * (2.0 + Math.pow((k_m / t), 2.0))) * ((k_m / (l / Math.pow(t, 3.0))) / l));
	} else {
		tmp = Math.pow(l, 2.0) / Math.pow((Math.pow(Math.cbrt(k_m), 2.0) * t), 3.0);
	}
	return tmp;
}

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 3.3e-92)
		tmp = Float64(l * sqrt(Float64(2.0 / Float64(t * (k_m ^ 4.0))))) ^ 2.0;
	elseif (t <= 2.45e+59)
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * Float64(2.0 + (Float64(k_m / t) ^ 2.0))) * Float64(Float64(k_m / Float64(l / (t ^ 3.0))) / l)));
	else
		tmp = Float64((l ^ 2.0) / (Float64((cbrt(k_m) ^ 2.0) * t) ^ 3.0));
	end
	return tmp
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 3.3e-92], N[Power[N[(l * N[Sqrt[N[(2.0 / N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t, 2.45e+59], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[(N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.29999999999999998e-92

   1. Initial program 48.8%

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

      1. associate-*l* 48.8%

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

      2. *-commutative 48.8%

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

      3. *-commutative 48.8%

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

      4. associate-/r* 55.9%

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

      5. distribute-rgt-in 55.9%

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

      6. unpow2 55.9%

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

      7. times-frac 39.7%

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

      8. sqr-neg 39.7%

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

      9. times-frac 55.9%

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

      10. unpow2 55.9%

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

      11. distribute-rgt-in 55.9%

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

      12. +-commutative 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in t around 0 63.6%

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

   6. Taylor expanded in k around 0 53.1%

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

      1. add-sqr-sqrt 30.7%

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

      2. pow2 30.7%

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

      3. associate-/r/ 30.7%

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

      4. sqrt-prod 26.2%

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

      5. *-commutative 26.2%

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

      6. unpow2 26.2%

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

      7. sqrt-prod 16.3%

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

      8. add-sqr-sqrt 29.8%

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

   8. Applied egg-rr 29.8%

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

   if 3.29999999999999998e-92 < t < 2.45000000000000004e59

   1. Initial program 88.3%

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

      1. associate-*l* 88.4%

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

      2. *-commutative 88.4%

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

      3. *-commutative 88.4%

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

      4. associate-/r* 88.4%

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

      5. distribute-rgt-in 88.4%

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

      6. unpow2 88.4%

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

      7. times-frac 88.5%

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

      8. sqr-neg 88.5%

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

      9. times-frac 88.4%

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

      10. unpow2 88.4%

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

      11. distribute-rgt-in 88.4%

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

      12. +-commutative 88.4%

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

   3. Simplified 88.4%

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

      1. associate-*l/ 94.1%

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

      2. clear-num 94.1%

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

      3. associate-*l/ 94.1%

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

      4. *-un-lft-identity 94.1%

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

   6. Applied egg-rr 94.1%

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

   7. Taylor expanded in k around 0 77.3%

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

      1. associate-/l* 77.2%

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

   9. Simplified 77.2%

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

   if 2.45000000000000004e59 < t

   1. Initial program 51.9%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in k around 0 46.1%

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

      1. add-cube-cbrt 46.1%

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

      2. pow3 46.1%

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

      3. *-commutative 46.1%

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

      4. cbrt-prod 46.1%

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

      5. rem-cbrt-cube 54.9%

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

      6. unpow2 54.9%

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

      7. cbrt-prod 72.7%

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

      8. pow2 72.7%

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

   7. Applied egg-rr 72.7%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-92}:\\ \;\;\;\;{\left(\ell \cdot \sqrt{\frac{2}{t \cdot {k}^{4}}}\right)}^{2}\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{+59}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{\frac{k}{\frac{\ell}{{t}^{3}}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{\left({\left(\sqrt[3]{k}\right)}^{2} \cdot t\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-92}:\\ \;\;\;\;{\left(\ell \cdot \sqrt{\frac{2}{t \cdot {k_m}^{4}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k_m \cdot \left(2 + {\left(\frac{k_m}{t}\right)}^{2}\right)\right) \cdot \frac{\frac{k_m}{\frac{\ell}{{t}^{3}}}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 3.3e-92)
   (pow (* l (sqrt (/ 2.0 (* t (pow k_m 4.0))))) 2.0)
   (/
    2.0
    (*
     (* (tan k_m) (+ 2.0 (pow (/ k_m t) 2.0)))
     (/ (/ k_m (/ l (pow t 3.0))) l)))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3.3e-92) {
		tmp = pow((l * sqrt((2.0 / (t * pow(k_m, 4.0))))), 2.0);
	} else {
		tmp = 2.0 / ((tan(k_m) * (2.0 + pow((k_m / t), 2.0))) * ((k_m / (l / pow(t, 3.0))) / l));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 3.3d-92) then
        tmp = (l * sqrt((2.0d0 / (t * (k_m ** 4.0d0))))) ** 2.0d0
    else
        tmp = 2.0d0 / ((tan(k_m) * (2.0d0 + ((k_m / t) ** 2.0d0))) * ((k_m / (l / (t ** 3.0d0))) / l))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 3.3e-92) {
		tmp = Math.pow((l * Math.sqrt((2.0 / (t * Math.pow(k_m, 4.0))))), 2.0);
	} else {
		tmp = 2.0 / ((Math.tan(k_m) * (2.0 + Math.pow((k_m / t), 2.0))) * ((k_m / (l / Math.pow(t, 3.0))) / l));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 3.3e-92:
		tmp = math.pow((l * math.sqrt((2.0 / (t * math.pow(k_m, 4.0))))), 2.0)
	else:
		tmp = 2.0 / ((math.tan(k_m) * (2.0 + math.pow((k_m / t), 2.0))) * ((k_m / (l / math.pow(t, 3.0))) / l))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 3.3e-92)
		tmp = Float64(l * sqrt(Float64(2.0 / Float64(t * (k_m ^ 4.0))))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * Float64(2.0 + (Float64(k_m / t) ^ 2.0))) * Float64(Float64(k_m / Float64(l / (t ^ 3.0))) / l)));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 3.3e-92)
		tmp = (l * sqrt((2.0 / (t * (k_m ^ 4.0))))) ^ 2.0;
	else
		tmp = 2.0 / ((tan(k_m) * (2.0 + ((k_m / t) ^ 2.0))) * ((k_m / (l / (t ^ 3.0))) / l));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 3.3e-92], N[Power[N[(l * N[Sqrt[N[(2.0 / N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.29999999999999998e-92

   1. Initial program 48.8%

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

      1. associate-*l* 48.8%

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

      2. *-commutative 48.8%

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

      3. *-commutative 48.8%

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

      4. associate-/r* 55.9%

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

      5. distribute-rgt-in 55.9%

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

      6. unpow2 55.9%

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

      7. times-frac 39.7%

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

      8. sqr-neg 39.7%

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

      9. times-frac 55.9%

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

      10. unpow2 55.9%

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

      11. distribute-rgt-in 55.9%

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

      12. +-commutative 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in t around 0 63.6%

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

   6. Taylor expanded in k around 0 53.1%

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

      1. add-sqr-sqrt 30.7%

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

      2. pow2 30.7%

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

      3. associate-/r/ 30.7%

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

      4. sqrt-prod 26.2%

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

      5. *-commutative 26.2%

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

      6. unpow2 26.2%

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

      7. sqrt-prod 16.3%

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

      8. add-sqr-sqrt 29.8%

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

   8. Applied egg-rr 29.8%

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

   if 3.29999999999999998e-92 < t

   1. Initial program 60.1%

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

      1. associate-*l* 60.1%

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

      2. *-commutative 60.1%

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

      3. *-commutative 60.1%

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

      4. associate-/r* 65.9%

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

      5. distribute-rgt-in 65.9%

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

      6. unpow2 65.9%

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

      7. times-frac 57.8%

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

      8. sqr-neg 57.8%

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

      9. times-frac 65.9%

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

      10. unpow2 65.9%

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

      11. distribute-rgt-in 65.9%

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

      12. +-commutative 65.9%

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

   3. Simplified 65.9%

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

      1. associate-*l/ 67.2%

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

      2. clear-num 67.1%

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

      3. associate-*l/ 69.4%

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

      4. *-un-lft-identity 69.4%

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

   6. Applied egg-rr 69.4%

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

   7. Taylor expanded in k around 0 63.4%

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

      1. associate-/l* 63.2%

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

   9. Simplified 63.2%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{-92}:\\ \;\;\;\;{\left(\ell \cdot \sqrt{\frac{2}{t \cdot {k}^{4}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{\frac{k}{\frac{\ell}{{t}^{3}}}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;t \leq 2.45 \cdot 10^{-25}:\\ \;\;\;\;{\left(\ell \cdot \sqrt{\frac{2}{t \cdot {k_m}^{4}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k_m \cdot 2\right) \cdot \left(\sin k_m \cdot \frac{t \cdot \left({t}^{2} \cdot \frac{1}{\ell}\right)}{\ell}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= t 2.45e-25)
   (pow (* l (sqrt (/ 2.0 (* t (pow k_m 4.0))))) 2.0)
   (/
    2.0
    (* (* k_m 2.0) (* (sin k_m) (/ (* t (* (pow t 2.0) (/ 1.0 l))) l))))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.45e-25) {
		tmp = pow((l * sqrt((2.0 / (t * pow(k_m, 4.0))))), 2.0);
	} else {
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * ((t * (pow(t, 2.0) * (1.0 / l))) / l)));
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t <= 2.45d-25) then
        tmp = (l * sqrt((2.0d0 / (t * (k_m ** 4.0d0))))) ** 2.0d0
    else
        tmp = 2.0d0 / ((k_m * 2.0d0) * (sin(k_m) * ((t * ((t ** 2.0d0) * (1.0d0 / l))) / l)))
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (t <= 2.45e-25) {
		tmp = Math.pow((l * Math.sqrt((2.0 / (t * Math.pow(k_m, 4.0))))), 2.0);
	} else {
		tmp = 2.0 / ((k_m * 2.0) * (Math.sin(k_m) * ((t * (Math.pow(t, 2.0) * (1.0 / l))) / l)));
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if t <= 2.45e-25:
		tmp = math.pow((l * math.sqrt((2.0 / (t * math.pow(k_m, 4.0))))), 2.0)
	else:
		tmp = 2.0 / ((k_m * 2.0) * (math.sin(k_m) * ((t * (math.pow(t, 2.0) * (1.0 / l))) / l)))
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (t <= 2.45e-25)
		tmp = Float64(l * sqrt(Float64(2.0 / Float64(t * (k_m ^ 4.0))))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * Float64(sin(k_m) * Float64(Float64(t * Float64((t ^ 2.0) * Float64(1.0 / l))) / l))));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (t <= 2.45e-25)
		tmp = (l * sqrt((2.0 / (t * (k_m ^ 4.0))))) ^ 2.0;
	else
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * ((t * ((t ^ 2.0) * (1.0 / l))) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[t, 2.45e-25], N[Power[N[(l * N[Sqrt[N[(2.0 / N[(t * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(t * N[(N[Power[t, 2.0], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.44999999999999995e-25

   1. Initial program 49.9%

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

      1. associate-*l* 49.9%

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

      2. *-commutative 49.9%

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

      3. *-commutative 49.9%

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

      4. associate-/r* 56.8%

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

      5. distribute-rgt-in 56.8%

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

      6. unpow2 56.8%

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

      7. times-frac 41.0%

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

      8. sqr-neg 41.0%

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

      9. times-frac 56.8%

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

      10. unpow2 56.8%

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

      11. distribute-rgt-in 56.8%

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

      12. +-commutative 56.8%

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

   3. Simplified 56.8%

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

   5. Taylor expanded in t around 0 63.8%

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

   6. Taylor expanded in k around 0 53.2%

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

      1. add-sqr-sqrt 31.2%

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

      2. pow2 31.2%

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

      3. associate-/r/ 31.2%

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

      4. sqrt-prod 26.8%

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

      5. *-commutative 26.8%

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

      6. unpow2 26.8%

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

      7. sqrt-prod 17.1%

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

      8. add-sqr-sqrt 30.4%

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

   8. Applied egg-rr 30.4%

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

   if 2.44999999999999995e-25 < t

   1. Initial program 57.9%

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

      1. associate-*l* 57.9%

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

      2. *-commutative 57.9%

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

      3. *-commutative 57.9%

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

      4. associate-/r* 64.0%

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

      5. distribute-rgt-in 64.0%

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

      6. unpow2 64.0%

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

      7. times-frac 55.5%

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

      8. sqr-neg 55.5%

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

      9. times-frac 64.0%

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

      10. unpow2 64.0%

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

      11. distribute-rgt-in 64.0%

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

      12. +-commutative 64.0%

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

   3. Simplified 64.0%

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

      1. div-inv 64.0%

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

      2. cube-mult 64.0%

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

      3. associate-*l* 72.1%

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

      4. pow2 72.1%

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

   6. Applied egg-rr 72.1%

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

   7. Taylor expanded in k around 0 64.3%

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

      1. *-commutative 62.5%

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

   9. Simplified 64.3%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.45 \cdot 10^{-25}:\\ \;\;\;\;{\left(\ell \cdot \sqrt{\frac{2}{t \cdot {k}^{4}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{t \cdot \left({t}^{2} \cdot \frac{1}{\ell}\right)}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k_m \leq 5.6 \cdot 10^{+128}:\\ \;\;\;\;\frac{2}{\left(k_m \cdot 2\right) \cdot \left(\sin k_m \cdot \frac{t \cdot \left({t}^{2} \cdot \frac{1}{\ell}\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k_m}^{4}}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 5.6e+128)
   (/ 2.0 (* (* k_m 2.0) (* (sin k_m) (/ (* t (* (pow t 2.0) (/ 1.0 l))) l))))
   (* 2.0 (/ (/ (pow l 2.0) (pow k_m 4.0)) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.6e+128) {
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * ((t * (pow(t, 2.0) * (1.0 / l))) / l)));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 4.0)) / t);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 5.6d+128) then
        tmp = 2.0d0 / ((k_m * 2.0d0) * (sin(k_m) * ((t * ((t ** 2.0d0) * (1.0d0 / l))) / l)))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k_m ** 4.0d0)) / t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 5.6e+128) {
		tmp = 2.0 / ((k_m * 2.0) * (Math.sin(k_m) * ((t * (Math.pow(t, 2.0) * (1.0 / l))) / l)));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 4.0)) / t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 5.6e+128:
		tmp = 2.0 / ((k_m * 2.0) * (math.sin(k_m) * ((t * (math.pow(t, 2.0) * (1.0 / l))) / l)))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 4.0)) / t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 5.6e+128)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * Float64(sin(k_m) * Float64(Float64(t * Float64((t ^ 2.0) * Float64(1.0 / l))) / l))));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 4.0)) / t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 5.6e+128)
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * ((t * ((t ^ 2.0) * (1.0 / l))) / l)));
	else
		tmp = 2.0 * (((l ^ 2.0) / (k_m ^ 4.0)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 5.6e+128], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(t * N[(N[Power[t, 2.0], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.59999999999999965e128

   1. Initial program 56.5%

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

      1. associate-*l* 56.5%

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

      2. *-commutative 56.5%

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

      3. *-commutative 56.5%

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

      4. associate-/r* 63.1%

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

      5. distribute-rgt-in 63.1%

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

      6. unpow2 63.1%

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

      7. times-frac 49.3%

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

      8. sqr-neg 49.3%

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

      9. times-frac 63.1%

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

      10. unpow2 63.1%

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

      11. distribute-rgt-in 63.1%

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

      12. +-commutative 63.1%

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

   3. Simplified 63.1%

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

      1. div-inv 63.1%

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

      2. cube-mult 63.1%

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

      3. associate-*l* 69.0%

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

      4. pow2 69.0%

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

   6. Applied egg-rr 69.0%

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

   7. Taylor expanded in k around 0 63.8%

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

      1. *-commutative 63.6%

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

   9. Simplified 63.8%

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

   if 5.59999999999999965e128 < k

   1. Initial program 22.0%

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

      1. associate-*l* 22.0%

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

      2. *-commutative 22.0%

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

      3. *-commutative 22.0%

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

      4. associate-/r* 29.0%

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

      5. distribute-rgt-in 29.0%

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

      6. unpow2 29.0%

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

      7. times-frac 15.9%

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

      8. sqr-neg 15.9%

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

      9. times-frac 29.0%

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

      10. unpow2 29.0%

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

      11. distribute-rgt-in 29.0%

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

      12. +-commutative 29.0%

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

   3. Simplified 29.0%

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

   5. Taylor expanded in t around 0 51.2%

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

   6. Taylor expanded in k around 0 51.2%

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

   7. Taylor expanded in k around 0 51.2%

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

      1. associate-/r* 51.2%

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

   9. Simplified 51.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.6 \cdot 10^{+128}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{t \cdot \left({t}^{2} \cdot \frac{1}{\ell}\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \begin{array}{l} \mathbf{if}\;k_m \leq 1.02 \cdot 10^{+129}:\\ \;\;\;\;\frac{2}{\left(k_m \cdot 2\right) \cdot \frac{\frac{\sin k_m}{\frac{\ell}{{t}^{3}}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k_m}^{4}}}{t}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (if (<= k_m 1.02e+129)
   (/ 2.0 (* (* k_m 2.0) (/ (/ (sin k_m) (/ l (pow t 3.0))) l)))
   (* 2.0 (/ (/ (pow l 2.0) (pow k_m 4.0)) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.02e+129) {
		tmp = 2.0 / ((k_m * 2.0) * ((sin(k_m) / (l / pow(t, 3.0))) / l));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 4.0)) / t);
	}
	return tmp;
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.02d+129) then
        tmp = 2.0d0 / ((k_m * 2.0d0) * ((sin(k_m) / (l / (t ** 3.0d0))) / l))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k_m ** 4.0d0)) / t)
    end if
    code = tmp
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	double tmp;
	if (k_m <= 1.02e+129) {
		tmp = 2.0 / ((k_m * 2.0) * ((Math.sin(k_m) / (l / Math.pow(t, 3.0))) / l));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 4.0)) / t);
	}
	return tmp;
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	tmp = 0
	if k_m <= 1.02e+129:
		tmp = 2.0 / ((k_m * 2.0) * ((math.sin(k_m) / (l / math.pow(t, 3.0))) / l))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 4.0)) / t)
	return tmp

Julia

k_m = abs(k)
function code(t, l, k_m)
	tmp = 0.0
	if (k_m <= 1.02e+129)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * Float64(Float64(sin(k_m) / Float64(l / (t ^ 3.0))) / l)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 4.0)) / t));
	end
	return tmp
end

MATLAB

k_m = abs(k);
function tmp_2 = code(t, l, k_m)
	tmp = 0.0;
	if (k_m <= 1.02e+129)
		tmp = 2.0 / ((k_m * 2.0) * ((sin(k_m) / (l / (t ^ 3.0))) / l));
	else
		tmp = 2.0 * (((l ^ 2.0) / (k_m ^ 4.0)) / t);
	end
	tmp_2 = tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := If[LessEqual[k$95$m, 1.02e+129], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[(N[(N[Sin[k$95$m], $MachinePrecision] / N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.01999999999999996e129

   1. Initial program 56.5%

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

      1. associate-*l* 56.5%

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

      2. *-commutative 56.5%

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

      3. *-commutative 56.5%

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

      4. associate-/r* 63.1%

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

      5. distribute-rgt-in 63.1%

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

      6. unpow2 63.1%

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

      7. times-frac 49.3%

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

      8. sqr-neg 49.3%

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

      9. times-frac 63.1%

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

      10. unpow2 63.1%

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

      11. distribute-rgt-in 63.1%

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

      12. +-commutative 63.1%

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

   3. Simplified 63.1%

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

      1. associate-*l/ 64.0%

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

      2. clear-num 64.0%

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

      3. associate-*l/ 64.8%

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

      4. *-un-lft-identity 64.8%

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

   6. Applied egg-rr 64.8%

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

   7. Taylor expanded in k around 0 63.6%

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

      1. *-commutative 63.6%

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

   9. Simplified 63.6%

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

   if 1.01999999999999996e129 < k

   1. Initial program 22.0%

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

      1. associate-*l* 22.0%

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

      2. *-commutative 22.0%

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

      3. *-commutative 22.0%

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

      4. associate-/r* 29.0%

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

      5. distribute-rgt-in 29.0%

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

      6. unpow2 29.0%

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

      7. times-frac 15.9%

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

      8. sqr-neg 15.9%

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

      9. times-frac 29.0%

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

      10. unpow2 29.0%

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

      11. distribute-rgt-in 29.0%

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

      12. +-commutative 29.0%

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

   3. Simplified 29.0%

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

   5. Taylor expanded in t around 0 51.2%

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

   6. Taylor expanded in k around 0 51.2%

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

   7. Taylor expanded in k around 0 51.2%

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

      1. associate-/r* 51.2%

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

   9. Simplified 51.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{+129}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \frac{\frac{\sin k}{\frac{\ell}{{t}^{3}}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ 2 \cdot \frac{\frac{{\ell}^{2}}{{k_m}^{4}}}{t} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (* 2.0 (/ (/ (pow l 2.0) (pow k_m 4.0)) t)))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 * ((pow(l, 2.0) / pow(k_m, 4.0)) / t);
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = 2.0d0 * (((l ** 2.0d0) / (k_m ** 4.0d0)) / t)
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 4.0)) / t);
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 4.0)) / t)

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 4.0)) / t))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 * (((l ^ 2.0) / (k_m ^ 4.0)) / t);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.2%

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

   1. associate-*l* 52.2%

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

   2. *-commutative 52.2%

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

   3. *-commutative 52.2%

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

   4. associate-/r* 58.9%

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

   5. distribute-rgt-in 58.9%

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

   6. unpow2 58.9%

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

   7. times-frac 45.1%

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

   8. sqr-neg 45.1%

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

   9. times-frac 58.9%

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

   10. unpow2 58.9%

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

   11. distribute-rgt-in 58.9%

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

   12. +-commutative 58.9%

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

3. Simplified 58.9%

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

5. Taylor expanded in t around 0 58.7%

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

6. Taylor expanded in k around 0 49.6%

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

7. Taylor expanded in k around 0 49.6%

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

   1. associate-/r* 50.5%

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

9. Simplified 50.5%

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

10. Final simplification 50.5%

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

Alternative 16: 52.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ \frac{2}{\frac{{k_m}^{4}}{\frac{{\ell}^{2}}{t}}} \end{array} \]

FPCore

k_m = (fabs.f64 k)
(FPCore (t l k_m)
 :precision binary64
 (/ 2.0 (/ (pow k_m 4.0) (/ (pow l 2.0) t))))

C

k_m = fabs(k);
double code(double t, double l, double k_m) {
	return 2.0 / (pow(k_m, 4.0) / (pow(l, 2.0) / t));
}

Fortran

k_m = abs(k)
real(8) function code(t, l, k_m)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = 2.0d0 / ((k_m ** 4.0d0) / ((l ** 2.0d0) / t))
end function

Java

k_m = Math.abs(k);
public static double code(double t, double l, double k_m) {
	return 2.0 / (Math.pow(k_m, 4.0) / (Math.pow(l, 2.0) / t));
}

Python

k_m = math.fabs(k)
def code(t, l, k_m):
	return 2.0 / (math.pow(k_m, 4.0) / (math.pow(l, 2.0) / t))

Julia

k_m = abs(k)
function code(t, l, k_m)
	return Float64(2.0 / Float64((k_m ^ 4.0) / Float64((l ^ 2.0) / t)))
end

MATLAB

k_m = abs(k);
function tmp = code(t, l, k_m)
	tmp = 2.0 / ((k_m ^ 4.0) / ((l ^ 2.0) / t));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
code[t_, l_, k$95$m_] := N[(2.0 / N[(N[Power[k$95$m, 4.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.2%

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

   1. associate-*l* 52.2%

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

   2. *-commutative 52.2%

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

   3. *-commutative 52.2%

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

   4. associate-/r* 58.9%

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

   5. distribute-rgt-in 58.9%

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

   6. unpow2 58.9%

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

   7. times-frac 45.1%

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

   8. sqr-neg 45.1%

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

   9. times-frac 58.9%

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

   10. unpow2 58.9%

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

   11. distribute-rgt-in 58.9%

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

   12. +-commutative 58.9%

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

3. Simplified 58.9%

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

5. Taylor expanded in t around 0 58.7%

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

6. Taylor expanded in k around 0 49.6%

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

   1. associate-/l* 50.8%

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

8. Simplified 50.8%

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

9. Final simplification 50.8%

   \[\leadsto \frac{2}{\frac{{k}^{4}}{\frac{{\ell}^{2}}{t}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024012 
(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))))