Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.7% → 83.1%

Time: 20.4s

Alternatives: 13

Speedup: 2.0×

• 27.7% 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: 54.7% 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: 83.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 6 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t_m \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t_m \leq 6 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \frac{{t_m}^{3}}{\ell}}{\frac{\ell}{t_2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))
   (*
    t_s
    (if (<= t_m 6e-106)
      (/
       2.0
       (/ (* (pow k 2.0) (* t_m (pow (sin k) 2.0))) (* (pow l 2.0) (cos k))))
      (if (<= t_m 6e+101)
        (/ 2.0 (/ (* (sin k) (/ (pow t_m 3.0) l)) (/ l t_2)))
        (/
         2.0
         (* t_2 (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = tan(k) * (2.0 + pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 6e-106) {
		tmp = 2.0 / ((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (pow(l, 2.0) * cos(k)));
	} else if (t_m <= 6e+101) {
		tmp = 2.0 / ((sin(k) * (pow(t_m, 3.0) / l)) / (l / t_2));
	} else {
		tmp = 2.0 / (t_2 * pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0));
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 6e-106) {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))) / (Math.pow(l, 2.0) * Math.cos(k)));
	} else if (t_m <= 6e+101) {
		tmp = 2.0 / ((Math.sin(k) * (Math.pow(t_m, 3.0) / l)) / (l / t_2));
	} else {
		tmp = 2.0 / (t_2 * Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))
	tmp = 0.0
	if (t_m <= 6e-106)
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64((l ^ 2.0) * cos(k))));
	elseif (t_m <= 6e+101)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l)) / Float64(l / t_2)));
	else
		tmp = Float64(2.0 / Float64(t_2 * (Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6e-106], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e+101], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(l / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 6.00000000000000037e-106

   1. Initial program 49.3%

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

      1. associate-*l* 49.3%

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

      2. sqr-neg 49.3%

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

      3. sqr-neg 49.3%

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

      4. associate-/r* 54.3%

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

      5. distribute-rgt-in 54.3%

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

      6. unpow2 54.3%

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

      7. times-frac 41.6%

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

      8. sqr-neg 41.6%

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

      9. times-frac 54.3%

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

      10. unpow2 54.3%

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

      11. distribute-rgt-in 54.3%

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

   3. Simplified 54.3%

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

      1. add-cube-cbrt 54.3%

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

      2. pow3 54.3%

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

      3. *-commutative 54.3%

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

      4. cbrt-prod 54.2%

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

      5. associate-/l/ 49.3%

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

      6. cbrt-div 49.8%

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

      7. rem-cbrt-cube 61.5%

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

      8. cbrt-unprod 71.7%

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

      9. pow2 71.7%

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

   6. Applied egg-rr 71.7%

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

   7. Taylor expanded in k around inf 65.3%

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

   if 6.00000000000000037e-106 < t < 5.99999999999999986e101

   1. Initial program 71.2%

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

      1. associate-*l* 71.3%

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

      2. sqr-neg 71.3%

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

      3. sqr-neg 71.3%

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

      4. associate-/r* 82.7%

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

      5. distribute-rgt-in 82.7%

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

      6. unpow2 82.7%

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

      7. times-frac 82.7%

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

      8. sqr-neg 82.7%

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

      9. times-frac 82.7%

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

      10. unpow2 82.7%

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

      11. distribute-rgt-in 82.7%

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

   3. Simplified 82.7%

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

      1. associate-*l/ 82.6%

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

   6. Applied egg-rr 82.6%

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

      1. associate-*l/ 88.1%

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

      2. *-commutative 88.1%

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

   8. Applied egg-rr 88.1%

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

      1. associate-/l* 88.1%

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

   10. Simplified 88.1%

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

   if 5.99999999999999986e101 < t

   1. Initial program 68.5%

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

      1. associate-*l* 68.5%

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

      2. sqr-neg 68.5%

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

      3. sqr-neg 68.5%

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

      4. associate-/r* 71.9%

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

      5. distribute-rgt-in 71.9%

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

      6. unpow2 71.9%

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

      7. times-frac 51.3%

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

      8. sqr-neg 51.3%

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

      9. times-frac 71.9%

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

      10. unpow2 71.9%

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

      11. distribute-rgt-in 71.9%

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

   3. Simplified 71.9%

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

      1. add-cube-cbrt 71.9%

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

      2. pow3 71.9%

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

      3. *-commutative 71.9%

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

      4. cbrt-prod 71.9%

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

      5. associate-/l/ 68.5%

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

      6. cbrt-div 68.5%

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

      7. rem-cbrt-cube 91.2%

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

      8. cbrt-unprod 96.7%

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

      9. pow2 96.7%

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

   6. Applied egg-rr 96.7%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4.6 \cdot 10^{-106}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t_m \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \frac{{t_m}^{3}}{\ell}}{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t_m}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.6e-106)
    (*
     2.0
     (/ (* (pow l 2.0) (cos k)) (* (pow k 2.0) (* t_m (pow (sin k) 2.0)))))
    (if (<= t_m 5.6e+102)
      (/
       2.0
       (/
        (* (sin k) (/ (pow t_m 3.0) l))
        (/ l (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))))
      (pow (/ (/ (pow (cbrt l) 2.0) (pow (cbrt k) 2.0)) t_m) 3.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.6e-106) {
		tmp = 2.0 * ((pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * (t_m * pow(sin(k), 2.0))));
	} else if (t_m <= 5.6e+102) {
		tmp = 2.0 / ((sin(k) * (pow(t_m, 3.0) / l)) / (l / (tan(k) * (2.0 + pow((k / t_m), 2.0)))));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) / pow(cbrt(k), 2.0)) / t_m), 3.0);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.6e-106) {
		tmp = 2.0 * ((Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))));
	} else if (t_m <= 5.6e+102) {
		tmp = 2.0 / ((Math.sin(k) * (Math.pow(t_m, 3.0) / l)) / (l / (Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / Math.pow(Math.cbrt(k), 2.0)) / t_m), 3.0);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.6e-106)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0)))));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l)) / Float64(l / Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))))));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / (cbrt(k) ^ 2.0)) / t_m) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.6e-106], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(l / N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.6000000000000002e-106

   1. Initial program 49.3%

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

      1. associate-/r* 49.7%

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

      2. sqr-neg 49.7%

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

      3. associate-*l* 46.9%

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

      4. sqr-neg 46.9%

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

      5. associate-/r* 51.9%

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

      6. associate-+l+ 51.9%

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

      7. unpow2 51.9%

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

      8. times-frac 39.2%

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

      9. sqr-neg 39.2%

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

      10. times-frac 51.9%

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

      11. unpow2 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in t around 0 65.6%

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

   if 4.6000000000000002e-106 < t < 5.60000000000000037e102

   1. Initial program 71.2%

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

      1. associate-*l* 71.3%

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

      2. sqr-neg 71.3%

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

      3. sqr-neg 71.3%

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

      4. associate-/r* 82.7%

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

      5. distribute-rgt-in 82.7%

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

      6. unpow2 82.7%

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

      7. times-frac 82.7%

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

      8. sqr-neg 82.7%

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

      9. times-frac 82.7%

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

      10. unpow2 82.7%

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

      11. distribute-rgt-in 82.7%

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

   3. Simplified 82.7%

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

      1. associate-*l/ 82.6%

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

   6. Applied egg-rr 82.6%

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

      1. associate-*l/ 88.1%

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

      2. *-commutative 88.1%

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

   8. Applied egg-rr 88.1%

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

      1. associate-/l* 88.1%

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

   10. Simplified 88.1%

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

   if 5.60000000000000037e102 < t

   1. Initial program 68.5%

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

      1. associate-/r* 68.5%

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

      2. sqr-neg 68.5%

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

      3. associate-*l* 56.6%

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

      4. sqr-neg 56.6%

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

      5. associate-/r* 59.7%

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

      6. associate-+l+ 59.7%

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

      7. unpow2 59.7%

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

      8. times-frac 39.1%

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

      9. sqr-neg 39.1%

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

      10. times-frac 59.7%

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

      11. unpow2 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in k around 0 56.6%

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

      1. add-cube-cbrt 56.6%

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

      2. pow3 56.6%

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

      3. associate-/r* 53.7%

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

      4. cbrt-div 53.7%

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

      5. rem-cbrt-cube 71.4%

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

   7. Applied egg-rr 71.4%

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

      1. cbrt-div 74.3%

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

      2. unpow2 74.3%

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

      3. cbrt-prod 80.0%

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

      4. pow2 80.0%

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

      5. unpow2 80.0%

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

      6. cbrt-prod 94.2%

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

      7. pow2 94.2%

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

   9. Applied egg-rr 94.2%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-106}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 5.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t_m \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \frac{{t_m}^{3}}{\ell}}{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t_m}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.5e-106)
    (/
     2.0
     (/ (* (pow k 2.0) (* t_m (pow (sin k) 2.0))) (* (pow l 2.0) (cos k))))
    (if (<= t_m 5.6e+102)
      (/
       2.0
       (/
        (* (sin k) (/ (pow t_m 3.0) l))
        (/ l (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))))
      (pow (/ (/ (pow (cbrt l) 2.0) (pow (cbrt k) 2.0)) t_m) 3.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.5e-106) {
		tmp = 2.0 / ((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (pow(l, 2.0) * cos(k)));
	} else if (t_m <= 5.6e+102) {
		tmp = 2.0 / ((sin(k) * (pow(t_m, 3.0) / l)) / (l / (tan(k) * (2.0 + pow((k / t_m), 2.0)))));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) / pow(cbrt(k), 2.0)) / t_m), 3.0);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.5e-106) {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))) / (Math.pow(l, 2.0) * Math.cos(k)));
	} else if (t_m <= 5.6e+102) {
		tmp = 2.0 / ((Math.sin(k) * (Math.pow(t_m, 3.0) / l)) / (l / (Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / Math.pow(Math.cbrt(k), 2.0)) / t_m), 3.0);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.5e-106)
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64((l ^ 2.0) * cos(k))));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l)) / Float64(l / Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))))));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / (cbrt(k) ^ 2.0)) / t_m) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.5e-106], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(l / N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.5000000000000001e-106

   1. Initial program 49.3%

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

      1. associate-*l* 49.3%

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

      2. sqr-neg 49.3%

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

      3. sqr-neg 49.3%

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

      4. associate-/r* 54.3%

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

      5. distribute-rgt-in 54.3%

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

      6. unpow2 54.3%

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

      7. times-frac 41.6%

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

      8. sqr-neg 41.6%

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

      9. times-frac 54.3%

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

      10. unpow2 54.3%

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

      11. distribute-rgt-in 54.3%

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

   3. Simplified 54.3%

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

      1. add-cube-cbrt 54.3%

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

      2. pow3 54.3%

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

      3. *-commutative 54.3%

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

      4. cbrt-prod 54.2%

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

      5. associate-/l/ 49.3%

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

      6. cbrt-div 49.8%

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

      7. rem-cbrt-cube 61.5%

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

      8. cbrt-unprod 71.7%

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

      9. pow2 71.7%

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

   6. Applied egg-rr 71.7%

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

   7. Taylor expanded in k around inf 65.3%

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

   if 5.5000000000000001e-106 < t < 5.60000000000000037e102

   1. Initial program 71.2%

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

      1. associate-*l* 71.3%

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

      2. sqr-neg 71.3%

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

      3. sqr-neg 71.3%

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

      4. associate-/r* 82.7%

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

      5. distribute-rgt-in 82.7%

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

      6. unpow2 82.7%

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

      7. times-frac 82.7%

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

      8. sqr-neg 82.7%

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

      9. times-frac 82.7%

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

      10. unpow2 82.7%

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

      11. distribute-rgt-in 82.7%

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

   3. Simplified 82.7%

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

      1. associate-*l/ 82.6%

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

   6. Applied egg-rr 82.6%

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

      1. associate-*l/ 88.1%

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

      2. *-commutative 88.1%

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

   8. Applied egg-rr 88.1%

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

      1. associate-/l* 88.1%

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

   10. Simplified 88.1%

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

   if 5.60000000000000037e102 < t

   1. Initial program 68.5%

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

      1. associate-/r* 68.5%

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

      2. sqr-neg 68.5%

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

      3. associate-*l* 56.6%

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

      4. sqr-neg 56.6%

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

      5. associate-/r* 59.7%

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

      6. associate-+l+ 59.7%

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

      7. unpow2 59.7%

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

      8. times-frac 39.1%

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

      9. sqr-neg 39.1%

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

      10. times-frac 59.7%

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

      11. unpow2 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in k around 0 56.6%

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

      1. add-cube-cbrt 56.6%

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

      2. pow3 56.6%

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

      3. associate-/r* 53.7%

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

      4. cbrt-div 53.7%

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

      5. rem-cbrt-cube 71.4%

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

   7. Applied egg-rr 71.4%

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

      1. cbrt-div 74.3%

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

      2. unpow2 74.3%

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

      3. cbrt-prod 80.0%

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

      4. pow2 80.0%

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

      5. unpow2 80.0%

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

      6. cbrt-prod 94.2%

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

      7. pow2 94.2%

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

   9. Applied egg-rr 94.2%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3.6 \cdot 10^{-106}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t_m \cdot {\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}^{2}\right)}\\ \mathbf{elif}\;t_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \frac{{t_m}^{3}}{\ell}}{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t_m}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.6e-106)
    (*
     2.0
     (/
      (pow l 2.0)
      (*
       (pow k 2.0)
       (* t_m (pow (+ k (* 0.08333333333333333 (pow k 3.0))) 2.0)))))
    (if (<= t_m 5.6e+102)
      (/
       2.0
       (/
        (* (sin k) (/ (pow t_m 3.0) l))
        (/ l (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))))
      (pow (/ (/ (pow (cbrt l) 2.0) (pow (cbrt k) 2.0)) t_m) 3.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.6e-106) {
		tmp = 2.0 * (pow(l, 2.0) / (pow(k, 2.0) * (t_m * pow((k + (0.08333333333333333 * pow(k, 3.0))), 2.0))));
	} else if (t_m <= 5.6e+102) {
		tmp = 2.0 / ((sin(k) * (pow(t_m, 3.0) / l)) / (l / (tan(k) * (2.0 + pow((k / t_m), 2.0)))));
	} else {
		tmp = pow(((pow(cbrt(l), 2.0) / pow(cbrt(k), 2.0)) / t_m), 3.0);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.6e-106) {
		tmp = 2.0 * (Math.pow(l, 2.0) / (Math.pow(k, 2.0) * (t_m * Math.pow((k + (0.08333333333333333 * Math.pow(k, 3.0))), 2.0))));
	} else if (t_m <= 5.6e+102) {
		tmp = 2.0 / ((Math.sin(k) * (Math.pow(t_m, 3.0) / l)) / (l / (Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))));
	} else {
		tmp = Math.pow(((Math.pow(Math.cbrt(l), 2.0) / Math.pow(Math.cbrt(k), 2.0)) / t_m), 3.0);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.6e-106)
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64((k ^ 2.0) * Float64(t_m * (Float64(k + Float64(0.08333333333333333 * (k ^ 3.0))) ^ 2.0)))));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l)) / Float64(l / Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))))));
	else
		tmp = Float64(Float64((cbrt(l) ^ 2.0) / (cbrt(k) ^ 2.0)) / t_m) ^ 3.0;
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.6e-106], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[(k + N[(0.08333333333333333 * N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(l / N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.60000000000000013e-106

   1. Initial program 49.3%

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

      1. associate-/r* 49.7%

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

      2. sqr-neg 49.7%

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

      3. associate-*l* 46.9%

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

      4. sqr-neg 46.9%

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

      5. associate-/r* 51.9%

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

      6. associate-+l+ 51.9%

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

      7. unpow2 51.9%

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

      8. times-frac 39.2%

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

      9. sqr-neg 39.2%

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

      10. times-frac 51.9%

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

      11. unpow2 51.9%

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

   3. Simplified 51.9%

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

      1. add-sqr-sqrt 13.5%

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

      2. pow2 13.5%

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

      3. sqrt-prod 4.3%

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

      4. associate-/l/ 3.8%

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

      5. sqrt-div 3.8%

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

      6. sqrt-pow1 5.3%

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

      7. metadata-eval 5.3%

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

      8. sqrt-unprod 2.1%

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

      9. add-sqr-sqrt 5.8%

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

   6. Applied egg-rr 5.8%

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

   7. Taylor expanded in k around 0 6.4%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-commutative 6.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k + \color{blue}{{k}^{3} \cdot 0.08333333333333333}\right)\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Simplified 6.4%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + {k}^{3} \cdot 0.08333333333333333\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in t around 0 54.9%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}^{2}\right)}} \]

   if 3.60000000000000013e-106 < t < 5.60000000000000037e102

   1. Initial program 71.2%

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

      1. associate-*l* 71.3%

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

      2. sqr-neg 71.3%

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

      3. sqr-neg 71.3%

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

      4. associate-/r* 82.7%

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

      5. distribute-rgt-in 82.7%

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

      6. unpow2 82.7%

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

      7. times-frac 82.7%

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

      8. sqr-neg 82.7%

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

      9. times-frac 82.7%

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

      10. unpow2 82.7%

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

      11. distribute-rgt-in 82.7%

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

   3. Simplified 82.7%

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

      1. associate-*l/ 82.6%

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

   6. Applied egg-rr 82.6%

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

      1. associate-*l/ 88.1%

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

      2. *-commutative 88.1%

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

   8. Applied egg-rr 88.1%

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

      1. associate-/l* 88.1%

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

   10. Simplified 88.1%

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

   if 5.60000000000000037e102 < t

   1. Initial program 68.5%

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

      1. associate-/r* 68.5%

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

      2. sqr-neg 68.5%

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

      3. associate-*l* 56.6%

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

      4. sqr-neg 56.6%

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

      5. associate-/r* 59.7%

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

      6. associate-+l+ 59.7%

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

      7. unpow2 59.7%

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

      8. times-frac 39.1%

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

      9. sqr-neg 39.1%

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

      10. times-frac 59.7%

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

      11. unpow2 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in k around 0 56.6%

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

      1. add-cube-cbrt 56.6%

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

      2. pow3 56.6%

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

      3. associate-/r* 53.7%

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

      4. cbrt-div 53.7%

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

      5. rem-cbrt-cube 71.4%

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

   7. Applied egg-rr 71.4%

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

      1. cbrt-div 74.3%

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

      2. unpow2 74.3%

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

      3. cbrt-prod 80.0%

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

      4. pow2 80.0%

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

      5. unpow2 80.0%

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

      6. cbrt-prod 94.2%

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

      7. pow2 94.2%

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

   9. Applied egg-rr 94.2%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.6 \cdot 10^{-106}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{k}\right)}^{2}}}{t}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4.5 \cdot 10^{-106}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t_m \cdot {\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}^{2}\right)}\\ \mathbf{elif}\;t_m \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t_m}^{1.5}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.5e-106)
    (*
     2.0
     (/
      (pow l 2.0)
      (*
       (pow k 2.0)
       (* t_m (pow (+ k (* 0.08333333333333333 (pow k 3.0))) 2.0)))))
    (if (<= t_m 5.5e+102)
      (/
       2.0
       (*
        (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
        (/ (* (sin k) (/ (pow t_m 3.0) l)) l)))
      (pow (/ l (* k (pow t_m 1.5))) 2.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.5e-106) {
		tmp = 2.0 * (pow(l, 2.0) / (pow(k, 2.0) * (t_m * pow((k + (0.08333333333333333 * pow(k, 3.0))), 2.0))));
	} else if (t_m <= 5.5e+102) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * ((sin(k) * (pow(t_m, 3.0) / l)) / l));
	} else {
		tmp = pow((l / (k * pow(t_m, 1.5))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 4.5d-106) then
        tmp = 2.0d0 * ((l ** 2.0d0) / ((k ** 2.0d0) * (t_m * ((k + (0.08333333333333333d0 * (k ** 3.0d0))) ** 2.0d0))))
    else if (t_m <= 5.5d+102) then
        tmp = 2.0d0 / ((tan(k) * (2.0d0 + ((k / t_m) ** 2.0d0))) * ((sin(k) * ((t_m ** 3.0d0) / l)) / l))
    else
        tmp = (l / (k * (t_m ** 1.5d0))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.5e-106) {
		tmp = 2.0 * (Math.pow(l, 2.0) / (Math.pow(k, 2.0) * (t_m * Math.pow((k + (0.08333333333333333 * Math.pow(k, 3.0))), 2.0))));
	} else if (t_m <= 5.5e+102) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * ((Math.sin(k) * (Math.pow(t_m, 3.0) / l)) / l));
	} else {
		tmp = Math.pow((l / (k * Math.pow(t_m, 1.5))), 2.0);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 4.5e-106:
		tmp = 2.0 * (math.pow(l, 2.0) / (math.pow(k, 2.0) * (t_m * math.pow((k + (0.08333333333333333 * math.pow(k, 3.0))), 2.0))))
	elif t_m <= 5.5e+102:
		tmp = 2.0 / ((math.tan(k) * (2.0 + math.pow((k / t_m), 2.0))) * ((math.sin(k) * (math.pow(t_m, 3.0) / l)) / l))
	else:
		tmp = math.pow((l / (k * math.pow(t_m, 1.5))), 2.0)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.5e-106)
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64((k ^ 2.0) * Float64(t_m * (Float64(k + Float64(0.08333333333333333 * (k ^ 3.0))) ^ 2.0)))));
	elseif (t_m <= 5.5e+102)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l)) / l)));
	else
		tmp = Float64(l / Float64(k * (t_m ^ 1.5))) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 4.5e-106)
		tmp = 2.0 * ((l ^ 2.0) / ((k ^ 2.0) * (t_m * ((k + (0.08333333333333333 * (k ^ 3.0))) ^ 2.0))));
	elseif (t_m <= 5.5e+102)
		tmp = 2.0 / ((tan(k) * (2.0 + ((k / t_m) ^ 2.0))) * ((sin(k) * ((t_m ^ 3.0) / l)) / l));
	else
		tmp = (l / (k * (t_m ^ 1.5))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.5e-106], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[(k + N[(0.08333333333333333 * N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.5e+102], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(l / N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.49999999999999955e-106

   1. Initial program 49.3%

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

      1. associate-/r* 49.7%

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

      2. sqr-neg 49.7%

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

      3. associate-*l* 46.9%

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

      4. sqr-neg 46.9%

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

      5. associate-/r* 51.9%

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

      6. associate-+l+ 51.9%

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

      7. unpow2 51.9%

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

      8. times-frac 39.2%

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

      9. sqr-neg 39.2%

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

      10. times-frac 51.9%

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

      11. unpow2 51.9%

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

   3. Simplified 51.9%

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

      1. add-sqr-sqrt 13.5%

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

      2. pow2 13.5%

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

      3. sqrt-prod 4.3%

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

      4. associate-/l/ 3.8%

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

      5. sqrt-div 3.8%

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

      6. sqrt-pow1 5.3%

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

      7. metadata-eval 5.3%

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

      8. sqrt-unprod 2.1%

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

      9. add-sqr-sqrt 5.8%

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

   6. Applied egg-rr 5.8%

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

   7. Taylor expanded in k around 0 6.4%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-commutative 6.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k + \color{blue}{{k}^{3} \cdot 0.08333333333333333}\right)\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Simplified 6.4%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + {k}^{3} \cdot 0.08333333333333333\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in t around 0 54.9%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}^{2}\right)}} \]

   if 4.49999999999999955e-106 < t < 5.49999999999999981e102

   1. Initial program 71.2%

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

      1. associate-*l* 71.3%

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

      2. sqr-neg 71.3%

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

      3. sqr-neg 71.3%

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

      4. associate-/r* 82.7%

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

      5. distribute-rgt-in 82.7%

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

      6. unpow2 82.7%

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

      7. times-frac 82.7%

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

      8. sqr-neg 82.7%

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

      9. times-frac 82.7%

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

      10. unpow2 82.7%

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

      11. distribute-rgt-in 82.7%

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

   3. Simplified 82.7%

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

      1. associate-*l/ 82.6%

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

   6. Applied egg-rr 82.6%

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

   if 5.49999999999999981e102 < t

   1. Initial program 68.5%

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

      1. associate-/r* 68.5%

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

      2. sqr-neg 68.5%

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

      3. associate-*l* 56.6%

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

      4. sqr-neg 56.6%

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

      5. associate-/r* 59.7%

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

      6. associate-+l+ 59.7%

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

      7. unpow2 59.7%

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

      8. times-frac 39.1%

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

      9. sqr-neg 39.1%

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

      10. times-frac 59.7%

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

      11. unpow2 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in k around 0 56.6%

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

      1. unpow2 56.6%

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

   7. Applied egg-rr 56.6%

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

      1. expm1-log1p-u 56.6%

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

      2. expm1-udef 56.6%

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

      3. add-sqr-sqrt 56.6%

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

      4. pow2 56.6%

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

      5. sqrt-div 56.6%

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

      6. unpow2 56.6%

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

      7. sqrt-prod 41.9%

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

      8. add-sqr-sqrt 59.7%

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

      9. sqrt-prod 59.7%

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

      10. sqrt-prod 41.8%

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

      11. add-sqr-sqrt 71.9%

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

      12. sqrt-pow1 80.6%

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

      13. metadata-eval 80.6%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2}\right)} - 1 \]

   9. Applied egg-rr 80.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 86.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

       2. expm1-log1p 86.0%

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

   11. Simplified 86.0%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-106}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3.8 \cdot 10^{-106}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t_m \cdot {\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}^{2}\right)}\\ \mathbf{elif}\;t_m \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \frac{{t_m}^{3}}{\ell}}{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t_m}^{1.5}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.8e-106)
    (*
     2.0
     (/
      (pow l 2.0)
      (*
       (pow k 2.0)
       (* t_m (pow (+ k (* 0.08333333333333333 (pow k 3.0))) 2.0)))))
    (if (<= t_m 5.5e+102)
      (/
       2.0
       (/
        (* (sin k) (/ (pow t_m 3.0) l))
        (/ l (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))))
      (pow (/ l (* k (pow t_m 1.5))) 2.0)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.8e-106) {
		tmp = 2.0 * (pow(l, 2.0) / (pow(k, 2.0) * (t_m * pow((k + (0.08333333333333333 * pow(k, 3.0))), 2.0))));
	} else if (t_m <= 5.5e+102) {
		tmp = 2.0 / ((sin(k) * (pow(t_m, 3.0) / l)) / (l / (tan(k) * (2.0 + pow((k / t_m), 2.0)))));
	} else {
		tmp = pow((l / (k * pow(t_m, 1.5))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.8d-106) then
        tmp = 2.0d0 * ((l ** 2.0d0) / ((k ** 2.0d0) * (t_m * ((k + (0.08333333333333333d0 * (k ** 3.0d0))) ** 2.0d0))))
    else if (t_m <= 5.5d+102) then
        tmp = 2.0d0 / ((sin(k) * ((t_m ** 3.0d0) / l)) / (l / (tan(k) * (2.0d0 + ((k / t_m) ** 2.0d0)))))
    else
        tmp = (l / (k * (t_m ** 1.5d0))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.8e-106) {
		tmp = 2.0 * (Math.pow(l, 2.0) / (Math.pow(k, 2.0) * (t_m * Math.pow((k + (0.08333333333333333 * Math.pow(k, 3.0))), 2.0))));
	} else if (t_m <= 5.5e+102) {
		tmp = 2.0 / ((Math.sin(k) * (Math.pow(t_m, 3.0) / l)) / (l / (Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))));
	} else {
		tmp = Math.pow((l / (k * Math.pow(t_m, 1.5))), 2.0);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3.8e-106:
		tmp = 2.0 * (math.pow(l, 2.0) / (math.pow(k, 2.0) * (t_m * math.pow((k + (0.08333333333333333 * math.pow(k, 3.0))), 2.0))))
	elif t_m <= 5.5e+102:
		tmp = 2.0 / ((math.sin(k) * (math.pow(t_m, 3.0) / l)) / (l / (math.tan(k) * (2.0 + math.pow((k / t_m), 2.0)))))
	else:
		tmp = math.pow((l / (k * math.pow(t_m, 1.5))), 2.0)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.8e-106)
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64((k ^ 2.0) * Float64(t_m * (Float64(k + Float64(0.08333333333333333 * (k ^ 3.0))) ^ 2.0)))));
	elseif (t_m <= 5.5e+102)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l)) / Float64(l / Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))))));
	else
		tmp = Float64(l / Float64(k * (t_m ^ 1.5))) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3.8e-106)
		tmp = 2.0 * ((l ^ 2.0) / ((k ^ 2.0) * (t_m * ((k + (0.08333333333333333 * (k ^ 3.0))) ^ 2.0))));
	elseif (t_m <= 5.5e+102)
		tmp = 2.0 / ((sin(k) * ((t_m ^ 3.0) / l)) / (l / (tan(k) * (2.0 + ((k / t_m) ^ 2.0)))));
	else
		tmp = (l / (k * (t_m ^ 1.5))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.8e-106], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[(k + N[(0.08333333333333333 * N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.5e+102], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(l / N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(l / N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.7999999999999999e-106

   1. Initial program 49.3%

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

      1. associate-/r* 49.7%

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

      2. sqr-neg 49.7%

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

      3. associate-*l* 46.9%

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

      4. sqr-neg 46.9%

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

      5. associate-/r* 51.9%

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

      6. associate-+l+ 51.9%

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

      7. unpow2 51.9%

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

      8. times-frac 39.2%

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

      9. sqr-neg 39.2%

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

      10. times-frac 51.9%

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

      11. unpow2 51.9%

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

   3. Simplified 51.9%

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

      1. add-sqr-sqrt 13.5%

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

      2. pow2 13.5%

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

      3. sqrt-prod 4.3%

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

      4. associate-/l/ 3.8%

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

      5. sqrt-div 3.8%

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

      6. sqrt-pow1 5.3%

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

      7. metadata-eval 5.3%

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

      8. sqrt-unprod 2.1%

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

      9. add-sqr-sqrt 5.8%

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

   6. Applied egg-rr 5.8%

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

   7. Taylor expanded in k around 0 6.4%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-commutative 6.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k + \color{blue}{{k}^{3} \cdot 0.08333333333333333}\right)\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Simplified 6.4%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + {k}^{3} \cdot 0.08333333333333333\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in t around 0 54.9%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}^{2}\right)}} \]

   if 3.7999999999999999e-106 < t < 5.49999999999999981e102

   1. Initial program 71.2%

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

      1. associate-*l* 71.3%

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

      2. sqr-neg 71.3%

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

      3. sqr-neg 71.3%

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

      4. associate-/r* 82.7%

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

      5. distribute-rgt-in 82.7%

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

      6. unpow2 82.7%

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

      7. times-frac 82.7%

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

      8. sqr-neg 82.7%

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

      9. times-frac 82.7%

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

      10. unpow2 82.7%

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

      11. distribute-rgt-in 82.7%

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

   3. Simplified 82.7%

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

      1. associate-*l/ 82.6%

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

   6. Applied egg-rr 82.6%

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

      1. associate-*l/ 88.1%

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

      2. *-commutative 88.1%

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

   8. Applied egg-rr 88.1%

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

      1. associate-/l* 88.1%

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

   10. Simplified 88.1%

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

   if 5.49999999999999981e102 < t

   1. Initial program 68.5%

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

      1. associate-/r* 68.5%

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

      2. sqr-neg 68.5%

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

      3. associate-*l* 56.6%

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

      4. sqr-neg 56.6%

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

      5. associate-/r* 59.7%

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

      6. associate-+l+ 59.7%

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

      7. unpow2 59.7%

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

      8. times-frac 39.1%

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

      9. sqr-neg 39.1%

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

      10. times-frac 59.7%

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

      11. unpow2 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in k around 0 56.6%

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

      1. unpow2 56.6%

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

   7. Applied egg-rr 56.6%

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

      1. expm1-log1p-u 56.6%

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

      2. expm1-udef 56.6%

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

      3. add-sqr-sqrt 56.6%

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

      4. pow2 56.6%

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

      5. sqrt-div 56.6%

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

      6. unpow2 56.6%

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

      7. sqrt-prod 41.9%

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

      8. add-sqr-sqrt 59.7%

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

      9. sqrt-prod 59.7%

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

      10. sqrt-prod 41.8%

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

      11. add-sqr-sqrt 71.9%

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

      12. sqrt-pow1 80.6%

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

      13. metadata-eval 80.6%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2}\right)} - 1 \]

   9. Applied egg-rr 80.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 86.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

       2. expm1-log1p 86.0%

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

   11. Simplified 86.0%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-106}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\frac{\ell}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{\ell}{k \cdot {t_m}^{1.5}}\right)}^{2}}{2 + {\left(\frac{k}{t_m}\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \frac{{t_m}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \frac{\sin k}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t_m \cdot {\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 5.1e-5)
    (/ (* 2.0 (pow (/ l (* k (pow t_m 1.5))) 2.0)) (+ 2.0 (pow (/ k t_m) 2.0)))
    (if (<= k 2.2e+48)
      (/
       2.0
       (* (/ (* (sin k) (/ (pow t_m 3.0) l)) l) (* 2.0 (/ (sin k) (cos k)))))
      (*
       2.0
       (/
        (pow l 2.0)
        (*
         (pow k 2.0)
         (* t_m (pow (+ k (* 0.08333333333333333 (pow k 3.0))) 2.0)))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5.1e-5) {
		tmp = (2.0 * pow((l / (k * pow(t_m, 1.5))), 2.0)) / (2.0 + pow((k / t_m), 2.0));
	} else if (k <= 2.2e+48) {
		tmp = 2.0 / (((sin(k) * (pow(t_m, 3.0) / l)) / l) * (2.0 * (sin(k) / cos(k))));
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (pow(k, 2.0) * (t_m * pow((k + (0.08333333333333333 * pow(k, 3.0))), 2.0))));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.1d-5) then
        tmp = (2.0d0 * ((l / (k * (t_m ** 1.5d0))) ** 2.0d0)) / (2.0d0 + ((k / t_m) ** 2.0d0))
    else if (k <= 2.2d+48) then
        tmp = 2.0d0 / (((sin(k) * ((t_m ** 3.0d0) / l)) / l) * (2.0d0 * (sin(k) / cos(k))))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) / ((k ** 2.0d0) * (t_m * ((k + (0.08333333333333333d0 * (k ** 3.0d0))) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5.1e-5) {
		tmp = (2.0 * Math.pow((l / (k * Math.pow(t_m, 1.5))), 2.0)) / (2.0 + Math.pow((k / t_m), 2.0));
	} else if (k <= 2.2e+48) {
		tmp = 2.0 / (((Math.sin(k) * (Math.pow(t_m, 3.0) / l)) / l) * (2.0 * (Math.sin(k) / Math.cos(k))));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (Math.pow(k, 2.0) * (t_m * Math.pow((k + (0.08333333333333333 * Math.pow(k, 3.0))), 2.0))));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 5.1e-5:
		tmp = (2.0 * math.pow((l / (k * math.pow(t_m, 1.5))), 2.0)) / (2.0 + math.pow((k / t_m), 2.0))
	elif k <= 2.2e+48:
		tmp = 2.0 / (((math.sin(k) * (math.pow(t_m, 3.0) / l)) / l) * (2.0 * (math.sin(k) / math.cos(k))))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) / (math.pow(k, 2.0) * (t_m * math.pow((k + (0.08333333333333333 * math.pow(k, 3.0))), 2.0))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 5.1e-5)
		tmp = Float64(Float64(2.0 * (Float64(l / Float64(k * (t_m ^ 1.5))) ^ 2.0)) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	elseif (k <= 2.2e+48)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / l)) / l) * Float64(2.0 * Float64(sin(k) / cos(k)))));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64((k ^ 2.0) * Float64(t_m * (Float64(k + Float64(0.08333333333333333 * (k ^ 3.0))) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 5.1e-5)
		tmp = (2.0 * ((l / (k * (t_m ^ 1.5))) ^ 2.0)) / (2.0 + ((k / t_m) ^ 2.0));
	elseif (k <= 2.2e+48)
		tmp = 2.0 / (((sin(k) * ((t_m ^ 3.0) / l)) / l) * (2.0 * (sin(k) / cos(k))));
	else
		tmp = 2.0 * ((l ^ 2.0) / ((k ^ 2.0) * (t_m * ((k + (0.08333333333333333 * (k ^ 3.0))) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 5.1e-5], N[(N[(2.0 * N[Power[N[(l / N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.2e+48], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(2.0 * N[(N[Sin[k], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[(k + N[(0.08333333333333333 * N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 5.09999999999999996e-5

   1. Initial program 58.8%

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

      1. associate-/r* 59.2%

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

      2. sqr-neg 59.2%

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

      3. associate-*l* 53.9%

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

      4. sqr-neg 53.9%

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

      5. associate-/r* 59.1%

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

      6. associate-+l+ 59.1%

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

      7. unpow2 59.1%

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

      8. times-frac 49.2%

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

      9. sqr-neg 49.2%

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

      10. times-frac 59.1%

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

      11. unpow2 59.1%

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

   3. Simplified 59.1%

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

      1. add-sqr-sqrt 24.5%

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

      2. pow2 24.5%

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

      3. sqrt-prod 19.8%

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

      4. associate-/l/ 17.8%

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

      5. sqrt-div 17.8%

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

      6. sqrt-pow1 21.4%

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

      7. metadata-eval 21.4%

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

      8. sqrt-unprod 12.5%

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

      9. add-sqr-sqrt 23.4%

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

   6. Applied egg-rr 23.4%

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

   7. Taylor expanded in k around 0 28.2%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-commutative 28.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k + \color{blue}{{k}^{3} \cdot 0.08333333333333333}\right)\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Simplified 28.2%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + {k}^{3} \cdot 0.08333333333333333\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in k around 0 52.3%

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

       1. associate-/r* 50.9%

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

       2. metadata-eval 50.9%

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

       3. pow-sqr 19.5%

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

       4. associate-/r* 23.1%

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

       5. unpow2 23.1%

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

       6. unpow2 23.1%

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

       7. times-frac 26.1%

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

       8. associate-*l/ 28.3%

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

       9. associate-*r/ 28.7%

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

       10. unpow2 28.7%

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

       11. associate-/r* 28.8%

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

   12. Simplified 28.8%

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

   if 5.09999999999999996e-5 < k < 2.1999999999999999e48

   1. Initial program 39.7%

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

      1. associate-*l* 39.8%

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

      2. sqr-neg 39.8%

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

      3. sqr-neg 39.8%

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

      4. associate-/r* 39.8%

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

      5. distribute-rgt-in 39.8%

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

      6. unpow2 39.8%

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

      7. times-frac 39.8%

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

      8. sqr-neg 39.8%

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

      9. times-frac 39.8%

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

      10. unpow2 39.8%

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

      11. distribute-rgt-in 39.8%

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

   3. Simplified 39.8%

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

      1. associate-*l/ 39.8%

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

   6. Applied egg-rr 39.8%

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

   7. Taylor expanded in t around inf 52.0%

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

   if 2.1999999999999999e48 < k

   1. Initial program 43.2%

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

      1. associate-/r* 43.2%

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

      2. sqr-neg 43.2%

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

      3. associate-*l* 43.2%

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

      4. sqr-neg 43.2%

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

      5. associate-/r* 50.7%

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

      6. associate-+l+ 50.7%

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

      7. unpow2 50.7%

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

      8. times-frac 28.7%

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

      9. sqr-neg 28.7%

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

      10. times-frac 50.7%

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

      11. unpow2 50.7%

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

   3. Simplified 50.7%

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

      1. add-sqr-sqrt 21.0%

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

      2. pow2 21.0%

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

      3. sqrt-prod 5.8%

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

      4. associate-/l/ 4.0%

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

      5. sqrt-div 4.0%

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

      6. sqrt-pow1 5.6%

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

      7. metadata-eval 5.6%

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

      8. sqrt-unprod 3.7%

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

      9. add-sqr-sqrt 7.5%

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

   6. Applied egg-rr 7.5%

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

   7. Taylor expanded in k around 0 17.0%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-commutative 17.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k + \color{blue}{{k}^{3} \cdot 0.08333333333333333}\right)\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Simplified 17.0%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + {k}^{3} \cdot 0.08333333333333333\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in t around 0 58.0%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}^{2}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell} \cdot \left(2 \cdot \frac{\sin k}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.00037:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{\ell}{k \cdot {t_m}^{1.5}}\right)}^{2}}{2 + {\left(\frac{k}{t_m}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{+60}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot {t_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;288 \cdot \frac{{\ell}^{2}}{t_m \cdot {k}^{8}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.00037)
    (/ (* 2.0 (pow (/ l (* k (pow t_m 1.5))) 2.0)) (+ 2.0 (pow (/ k t_m) 2.0)))
    (if (<= k 1.3e+60)
      (/ (* (pow l 2.0) (cos k)) (* (pow k 2.0) (pow t_m 3.0)))
      (* 288.0 (/ (pow l 2.0) (* t_m (pow k 8.0))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.00037) {
		tmp = (2.0 * pow((l / (k * pow(t_m, 1.5))), 2.0)) / (2.0 + pow((k / t_m), 2.0));
	} else if (k <= 1.3e+60) {
		tmp = (pow(l, 2.0) * cos(k)) / (pow(k, 2.0) * pow(t_m, 3.0));
	} else {
		tmp = 288.0 * (pow(l, 2.0) / (t_m * pow(k, 8.0)));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 0.00037d0) then
        tmp = (2.0d0 * ((l / (k * (t_m ** 1.5d0))) ** 2.0d0)) / (2.0d0 + ((k / t_m) ** 2.0d0))
    else if (k <= 1.3d+60) then
        tmp = ((l ** 2.0d0) * cos(k)) / ((k ** 2.0d0) * (t_m ** 3.0d0))
    else
        tmp = 288.0d0 * ((l ** 2.0d0) / (t_m * (k ** 8.0d0)))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.00037) {
		tmp = (2.0 * Math.pow((l / (k * Math.pow(t_m, 1.5))), 2.0)) / (2.0 + Math.pow((k / t_m), 2.0));
	} else if (k <= 1.3e+60) {
		tmp = (Math.pow(l, 2.0) * Math.cos(k)) / (Math.pow(k, 2.0) * Math.pow(t_m, 3.0));
	} else {
		tmp = 288.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 8.0)));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 0.00037:
		tmp = (2.0 * math.pow((l / (k * math.pow(t_m, 1.5))), 2.0)) / (2.0 + math.pow((k / t_m), 2.0))
	elif k <= 1.3e+60:
		tmp = (math.pow(l, 2.0) * math.cos(k)) / (math.pow(k, 2.0) * math.pow(t_m, 3.0))
	else:
		tmp = 288.0 * (math.pow(l, 2.0) / (t_m * math.pow(k, 8.0)))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.00037)
		tmp = Float64(Float64(2.0 * (Float64(l / Float64(k * (t_m ^ 1.5))) ^ 2.0)) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	elseif (k <= 1.3e+60)
		tmp = Float64(Float64((l ^ 2.0) * cos(k)) / Float64((k ^ 2.0) * (t_m ^ 3.0)));
	else
		tmp = Float64(288.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 8.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 0.00037)
		tmp = (2.0 * ((l / (k * (t_m ^ 1.5))) ^ 2.0)) / (2.0 + ((k / t_m) ^ 2.0));
	elseif (k <= 1.3e+60)
		tmp = ((l ^ 2.0) * cos(k)) / ((k ^ 2.0) * (t_m ^ 3.0));
	else
		tmp = 288.0 * ((l ^ 2.0) / (t_m * (k ^ 8.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 0.00037], N[(N[(2.0 * N[Power[N[(l / N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.3e+60], N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(288.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.6999999999999999e-4

   1. Initial program 58.8%

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

      1. associate-/r* 59.2%

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

      2. sqr-neg 59.2%

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

      3. associate-*l* 53.9%

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

      4. sqr-neg 53.9%

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

      5. associate-/r* 59.1%

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

      6. associate-+l+ 59.1%

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

      7. unpow2 59.1%

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

      8. times-frac 49.2%

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

      9. sqr-neg 49.2%

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

      10. times-frac 59.1%

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

      11. unpow2 59.1%

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

   3. Simplified 59.1%

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

      1. add-sqr-sqrt 24.5%

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

      2. pow2 24.5%

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

      3. sqrt-prod 19.8%

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

      4. associate-/l/ 17.8%

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

      5. sqrt-div 17.8%

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

      6. sqrt-pow1 21.4%

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

      7. metadata-eval 21.4%

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

      8. sqrt-unprod 12.5%

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

      9. add-sqr-sqrt 23.4%

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

   6. Applied egg-rr 23.4%

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

   7. Taylor expanded in k around 0 28.2%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-commutative 28.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k + \color{blue}{{k}^{3} \cdot 0.08333333333333333}\right)\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Simplified 28.2%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + {k}^{3} \cdot 0.08333333333333333\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in k around 0 52.3%

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

       1. associate-/r* 50.9%

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

       2. metadata-eval 50.9%

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

       3. pow-sqr 19.5%

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

       4. associate-/r* 23.1%

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

       5. unpow2 23.1%

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

       6. unpow2 23.1%

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

       7. times-frac 26.1%

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

       8. associate-*l/ 28.3%

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

       9. associate-*r/ 28.7%

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

       10. unpow2 28.7%

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

       11. associate-/r* 28.8%

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

   12. Simplified 28.8%

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

   if 3.6999999999999999e-4 < k < 1.30000000000000004e60

   1. Initial program 36.7%

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

      1. associate-/r* 36.7%

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

      2. sqr-neg 36.7%

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

      3. associate-*l* 36.6%

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

      4. sqr-neg 36.6%

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

      5. associate-/r* 36.6%

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

      6. associate-+l+ 36.6%

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

      7. unpow2 36.6%

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

      8. times-frac 36.7%

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

      9. sqr-neg 36.7%

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

      10. times-frac 36.6%

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

      11. unpow2 36.6%

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

   3. Simplified 36.6%

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

   5. Taylor expanded in t around inf 47.7%

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

   6. Taylor expanded in k around 0 47.8%

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

   if 1.30000000000000004e60 < k

   1. Initial program 43.9%

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

      1. associate-/r* 43.9%

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

      2. sqr-neg 43.9%

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

      3. associate-*l* 43.9%

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

      4. sqr-neg 43.9%

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

      5. associate-/r* 51.5%

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

      6. associate-+l+ 51.5%

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

      7. unpow2 51.5%

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

      8. times-frac 29.1%

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

      9. sqr-neg 29.1%

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

      10. times-frac 51.5%

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

      11. unpow2 51.5%

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

   3. Simplified 51.5%

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

      1. add-sqr-sqrt 21.3%

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

      2. pow2 21.3%

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

      3. sqrt-prod 5.8%

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

      4. associate-/l/ 3.9%

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

      5. sqrt-div 3.9%

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

      6. sqrt-pow1 5.6%

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

      7. metadata-eval 5.6%

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

      8. sqrt-unprod 3.8%

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

      9. add-sqr-sqrt 7.5%

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

   6. Applied egg-rr 7.5%

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

   7. Taylor expanded in k around 0 17.3%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-commutative 17.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k + \color{blue}{{k}^{3} \cdot 0.08333333333333333}\right)\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Simplified 17.3%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + {k}^{3} \cdot 0.08333333333333333\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in k around inf 59.1%

       \[\leadsto \color{blue}{288 \cdot \frac{{\ell}^{2}}{{k}^{8} \cdot t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00037:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{+60}:\\ \;\;\;\;\frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;288 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{8}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 8 \cdot 10^{-154}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{{\ell}^{2}}{{k}^{2}}}}{t_m}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{\ell}{k \cdot {t_m}^{1.5}}\right)}^{2}}{2 + {\left(\frac{k}{t_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e-154)
    (pow (/ (cbrt (/ (pow l 2.0) (pow k 2.0))) t_m) 3.0)
    (/
     (* 2.0 (pow (/ l (* k (pow t_m 1.5))) 2.0))
     (+ 2.0 (pow (/ k t_m) 2.0))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e-154) {
		tmp = pow((cbrt((pow(l, 2.0) / pow(k, 2.0))) / t_m), 3.0);
	} else {
		tmp = (2.0 * pow((l / (k * pow(t_m, 1.5))), 2.0)) / (2.0 + pow((k / t_m), 2.0));
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e-154) {
		tmp = Math.pow((Math.cbrt((Math.pow(l, 2.0) / Math.pow(k, 2.0))) / t_m), 3.0);
	} else {
		tmp = (2.0 * Math.pow((l / (k * Math.pow(t_m, 1.5))), 2.0)) / (2.0 + Math.pow((k / t_m), 2.0));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8e-154)
		tmp = Float64(cbrt(Float64((l ^ 2.0) / (k ^ 2.0))) / t_m) ^ 3.0;
	else
		tmp = Float64(Float64(2.0 * (Float64(l / Float64(k * (t_m ^ 1.5))) ^ 2.0)) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8e-154], N[Power[N[(N[Power[N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision], N[(N[(2.0 * N[Power[N[(l / N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.9999999999999998e-154

   1. Initial program 50.1%

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

      1. associate-/r* 50.4%

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

      2. sqr-neg 50.4%

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

      3. associate-*l* 47.6%

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

      4. sqr-neg 47.6%

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

      5. associate-/r* 52.7%

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

      6. associate-+l+ 52.7%

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

      7. unpow2 52.7%

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

      8. times-frac 39.6%

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

      9. sqr-neg 39.6%

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

      10. times-frac 52.7%

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

      11. unpow2 52.7%

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

   3. Simplified 52.7%

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

   5. Taylor expanded in k around 0 48.2%

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

      1. add-cube-cbrt 48.1%

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

      2. pow3 48.1%

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

      3. associate-/r* 46.6%

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

      4. cbrt-div 46.6%

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

      5. rem-cbrt-cube 57.9%

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

   7. Applied egg-rr 57.9%

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

   if 7.9999999999999998e-154 < t

   1. Initial program 66.6%

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

      1. associate-/r* 66.6%

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

      2. sqr-neg 66.6%

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

      3. associate-*l* 59.8%

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

      4. sqr-neg 59.8%

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

      5. associate-/r* 66.2%

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

      6. associate-+l+ 66.2%

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

      7. unpow2 66.2%

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

      8. times-frac 56.6%

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

      9. sqr-neg 56.6%

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

      10. times-frac 66.2%

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

      11. unpow2 66.2%

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

   3. Simplified 66.2%

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

      1. add-sqr-sqrt 50.9%

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

      2. pow2 50.9%

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

      3. sqrt-prod 50.8%

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

      4. associate-/l/ 45.7%

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

      5. sqrt-div 45.7%

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

      6. sqrt-pow1 54.9%

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

      7. metadata-eval 54.9%

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

      8. sqrt-unprod 30.1%

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

      9. add-sqr-sqrt 60.1%

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

   6. Applied egg-rr 60.1%

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

   7. Taylor expanded in k around 0 77.3%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-commutative 77.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k + \color{blue}{{k}^{3} \cdot 0.08333333333333333}\right)\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Simplified 77.3%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + {k}^{3} \cdot 0.08333333333333333\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in k around 0 55.6%

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

       1. associate-/r* 55.6%

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

       2. metadata-eval 55.6%

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

       3. pow-sqr 55.6%

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

       4. associate-/r* 63.6%

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

       5. unpow2 63.6%

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

       6. unpow2 63.6%

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

       7. times-frac 71.6%

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

       8. associate-*l/ 77.4%

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

       9. associate-*r/ 78.6%

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

       10. unpow2 78.6%

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

       11. associate-/r* 78.7%

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

   12. Simplified 78.7%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-154}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{{\ell}^{2}}{{k}^{2}}}}{t}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2 \cdot 10^{-153}:\\ \;\;\;\;288 \cdot \frac{{\ell}^{2}}{t_m \cdot {k}^{8}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{\ell}{k \cdot {t_m}^{1.5}}\right)}^{2}}{2 + {\left(\frac{k}{t_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2e-153)
    (* 288.0 (/ (pow l 2.0) (* t_m (pow k 8.0))))
    (/
     (* 2.0 (pow (/ l (* k (pow t_m 1.5))) 2.0))
     (+ 2.0 (pow (/ k t_m) 2.0))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2e-153) {
		tmp = 288.0 * (pow(l, 2.0) / (t_m * pow(k, 8.0)));
	} else {
		tmp = (2.0 * pow((l / (k * pow(t_m, 1.5))), 2.0)) / (2.0 + pow((k / t_m), 2.0));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2d-153) then
        tmp = 288.0d0 * ((l ** 2.0d0) / (t_m * (k ** 8.0d0)))
    else
        tmp = (2.0d0 * ((l / (k * (t_m ** 1.5d0))) ** 2.0d0)) / (2.0d0 + ((k / t_m) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2e-153) {
		tmp = 288.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 8.0)));
	} else {
		tmp = (2.0 * Math.pow((l / (k * Math.pow(t_m, 1.5))), 2.0)) / (2.0 + Math.pow((k / t_m), 2.0));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2e-153:
		tmp = 288.0 * (math.pow(l, 2.0) / (t_m * math.pow(k, 8.0)))
	else:
		tmp = (2.0 * math.pow((l / (k * math.pow(t_m, 1.5))), 2.0)) / (2.0 + math.pow((k / t_m), 2.0))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2e-153)
		tmp = Float64(288.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 8.0))));
	else
		tmp = Float64(Float64(2.0 * (Float64(l / Float64(k * (t_m ^ 1.5))) ^ 2.0)) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2e-153)
		tmp = 288.0 * ((l ^ 2.0) / (t_m * (k ^ 8.0)));
	else
		tmp = (2.0 * ((l / (k * (t_m ^ 1.5))) ^ 2.0)) / (2.0 + ((k / t_m) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2e-153], N[(288.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Power[N[(l / N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.00000000000000008e-153

   1. Initial program 50.1%

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

      1. associate-/r* 50.4%

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

      2. sqr-neg 50.4%

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

      3. associate-*l* 47.6%

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

      4. sqr-neg 47.6%

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

      5. associate-/r* 52.7%

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

      6. associate-+l+ 52.7%

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

      7. unpow2 52.7%

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

      8. times-frac 39.6%

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

      9. sqr-neg 39.6%

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

      10. times-frac 52.7%

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

      11. unpow2 52.7%

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

   3. Simplified 52.7%

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

      1. add-sqr-sqrt 13.3%

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

      2. pow2 13.3%

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

      3. sqrt-prod 3.8%

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

      4. associate-/l/ 3.3%

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

      5. sqrt-div 3.3%

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

      6. sqrt-pow1 3.8%

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

      7. metadata-eval 3.8%

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

      8. sqrt-unprod 2.2%

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

      9. add-sqr-sqrt 4.4%

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

   6. Applied egg-rr 4.4%

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

   7. Taylor expanded in k around 0 4.9%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-commutative 4.9%

         \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k + \color{blue}{{k}^{3} \cdot 0.08333333333333333}\right)\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Simplified 4.9%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + {k}^{3} \cdot 0.08333333333333333\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in k around inf 51.5%

       \[\leadsto \color{blue}{288 \cdot \frac{{\ell}^{2}}{{k}^{8} \cdot t}} \]

   if 2.00000000000000008e-153 < t

   1. Initial program 66.6%

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

      1. associate-/r* 66.6%

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

      2. sqr-neg 66.6%

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

      3. associate-*l* 59.8%

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

      4. sqr-neg 59.8%

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

      5. associate-/r* 66.2%

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

      6. associate-+l+ 66.2%

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

      7. unpow2 66.2%

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

      8. times-frac 56.6%

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

      9. sqr-neg 56.6%

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

      10. times-frac 66.2%

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

      11. unpow2 66.2%

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

   3. Simplified 66.2%

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

      1. add-sqr-sqrt 50.9%

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

      2. pow2 50.9%

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

      3. sqrt-prod 50.8%

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

      4. associate-/l/ 45.7%

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

      5. sqrt-div 45.7%

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

      6. sqrt-pow1 54.9%

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

      7. metadata-eval 54.9%

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

      8. sqrt-unprod 30.1%

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

      9. add-sqr-sqrt 60.1%

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

   6. Applied egg-rr 60.1%

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

   7. Taylor expanded in k around 0 77.3%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-commutative 77.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k + \color{blue}{{k}^{3} \cdot 0.08333333333333333}\right)\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Simplified 77.3%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + {k}^{3} \cdot 0.08333333333333333\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in k around 0 55.6%

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

       1. associate-/r* 55.6%

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

       2. metadata-eval 55.6%

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

       3. pow-sqr 55.6%

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

       4. associate-/r* 63.6%

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

       5. unpow2 63.6%

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

       6. unpow2 63.6%

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

       7. times-frac 71.6%

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

       8. associate-*l/ 77.4%

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

       9. associate-*r/ 78.6%

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

       10. unpow2 78.6%

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

       11. associate-/r* 78.7%

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

   12. Simplified 78.7%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-153}:\\ \;\;\;\;288 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{8}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{+25}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t_m}^{1.5}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;288 \cdot \frac{{\ell}^{2}}{t_m \cdot {k}^{8}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9.5e+25)
    (pow (/ l (* k (pow t_m 1.5))) 2.0)
    (* 288.0 (/ (pow l 2.0) (* t_m (pow k 8.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9.5e+25) {
		tmp = pow((l / (k * pow(t_m, 1.5))), 2.0);
	} else {
		tmp = 288.0 * (pow(l, 2.0) / (t_m * pow(k, 8.0)));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9.5d+25) then
        tmp = (l / (k * (t_m ** 1.5d0))) ** 2.0d0
    else
        tmp = 288.0d0 * ((l ** 2.0d0) / (t_m * (k ** 8.0d0)))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9.5e+25) {
		tmp = Math.pow((l / (k * Math.pow(t_m, 1.5))), 2.0);
	} else {
		tmp = 288.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 8.0)));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 9.5e+25:
		tmp = math.pow((l / (k * math.pow(t_m, 1.5))), 2.0)
	else:
		tmp = 288.0 * (math.pow(l, 2.0) / (t_m * math.pow(k, 8.0)))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 9.5e+25)
		tmp = Float64(l / Float64(k * (t_m ^ 1.5))) ^ 2.0;
	else
		tmp = Float64(288.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 8.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 9.5e+25)
		tmp = (l / (k * (t_m ^ 1.5))) ^ 2.0;
	else
		tmp = 288.0 * ((l ^ 2.0) / (t_m * (k ^ 8.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 9.5e+25], N[Power[N[(l / N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(288.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.5000000000000005e25

   1. Initial program 58.3%

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

      1. associate-/r* 58.6%

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

      2. sqr-neg 58.6%

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

      3. associate-*l* 53.5%

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

      4. sqr-neg 53.5%

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

      5. associate-/r* 58.5%

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

      6. associate-+l+ 58.5%

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

      7. unpow2 58.5%

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

      8. times-frac 49.0%

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

      9. sqr-neg 49.0%

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

      10. times-frac 58.5%

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

      11. unpow2 58.5%

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

   3. Simplified 58.5%

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

   5. Taylor expanded in k around 0 53.5%

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

      1. unpow2 53.5%

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

   7. Applied egg-rr 53.5%

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

      1. expm1-log1p-u 35.1%

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

      2. expm1-udef 34.4%

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

      3. add-sqr-sqrt 34.0%

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

      4. pow2 34.0%

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

      5. sqrt-div 21.3%

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

      6. unpow2 21.3%

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

      7. sqrt-prod 12.2%

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

      8. add-sqr-sqrt 22.5%

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

      9. sqrt-prod 22.9%

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

      10. sqrt-prod 10.8%

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

      11. add-sqr-sqrt 26.0%

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

      12. sqrt-pow1 26.9%

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

      13. metadata-eval 26.9%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2}\right)} - 1 \]

   9. Applied egg-rr 26.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 28.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

       2. expm1-log1p 28.4%

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

   11. Simplified 28.4%

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

   if 9.5000000000000005e25 < k

   1. Initial program 42.6%

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

      1. associate-/r* 42.7%

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

      2. sqr-neg 42.7%

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

      3. associate-*l* 42.7%

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

      4. sqr-neg 42.7%

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

      5. associate-/r* 49.8%

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

      6. associate-+l+ 49.8%

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

      7. unpow2 49.8%

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

      8. times-frac 28.9%

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

      9. sqr-neg 28.9%

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

      10. times-frac 49.8%

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

      11. unpow2 49.8%

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

   3. Simplified 49.8%

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

      1. add-sqr-sqrt 21.7%

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

      2. pow2 21.7%

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

      3. sqrt-prod 7.2%

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

      4. associate-/l/ 5.5%

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

      5. sqrt-div 5.4%

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

      6. sqrt-pow1 7.0%

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

      7. metadata-eval 7.0%

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

      8. sqrt-unprod 3.5%

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

      9. add-sqr-sqrt 8.8%

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

   6. Applied egg-rr 8.8%

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

   7. Taylor expanded in k around 0 16.2%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-commutative 16.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k + \color{blue}{{k}^{3} \cdot 0.08333333333333333}\right)\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Simplified 16.2%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + {k}^{3} \cdot 0.08333333333333333\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in k around inf 56.8%

       \[\leadsto \color{blue}{288 \cdot \frac{{\ell}^{2}}{{k}^{8} \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{+25}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;288 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{8}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.48 \cdot 10^{+26}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{k}}{{t_m}^{1.5}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;288 \cdot \frac{{\ell}^{2}}{t_m \cdot {k}^{8}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.48e+26)
    (pow (/ (/ l k) (pow t_m 1.5)) 2.0)
    (* 288.0 (/ (pow l 2.0) (* t_m (pow k 8.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.48e+26) {
		tmp = pow(((l / k) / pow(t_m, 1.5)), 2.0);
	} else {
		tmp = 288.0 * (pow(l, 2.0) / (t_m * pow(k, 8.0)));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.48d+26) then
        tmp = ((l / k) / (t_m ** 1.5d0)) ** 2.0d0
    else
        tmp = 288.0d0 * ((l ** 2.0d0) / (t_m * (k ** 8.0d0)))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.48e+26) {
		tmp = Math.pow(((l / k) / Math.pow(t_m, 1.5)), 2.0);
	} else {
		tmp = 288.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 8.0)));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.48e+26:
		tmp = math.pow(((l / k) / math.pow(t_m, 1.5)), 2.0)
	else:
		tmp = 288.0 * (math.pow(l, 2.0) / (t_m * math.pow(k, 8.0)))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.48e+26)
		tmp = Float64(Float64(l / k) / (t_m ^ 1.5)) ^ 2.0;
	else
		tmp = Float64(288.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 8.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.48e+26)
		tmp = ((l / k) / (t_m ^ 1.5)) ^ 2.0;
	else
		tmp = 288.0 * ((l ^ 2.0) / (t_m * (k ^ 8.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.48e+26], N[Power[N[(N[(l / k), $MachinePrecision] / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(288.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.48e26

   1. Initial program 58.3%

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

      1. associate-/r* 58.6%

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

      2. sqr-neg 58.6%

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

      3. associate-*l* 53.5%

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

      4. sqr-neg 53.5%

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

      5. associate-/r* 58.5%

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

      6. associate-+l+ 58.5%

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

      7. unpow2 58.5%

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

      8. times-frac 49.0%

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

      9. sqr-neg 49.0%

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

      10. times-frac 58.5%

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

      11. unpow2 58.5%

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

   3. Simplified 58.5%

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

   5. Taylor expanded in k around 0 53.5%

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

      1. unpow2 53.5%

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

   7. Applied egg-rr 53.5%

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

      1. expm1-log1p-u 35.1%

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

      2. expm1-udef 34.4%

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

      3. add-sqr-sqrt 34.0%

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

      4. pow2 34.0%

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

      5. sqrt-div 21.3%

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

      6. unpow2 21.3%

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

      7. sqrt-prod 12.2%

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

      8. add-sqr-sqrt 22.5%

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

      9. sqrt-prod 22.9%

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

      10. sqrt-prod 10.8%

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

      11. add-sqr-sqrt 26.0%

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

      12. sqrt-pow1 26.9%

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

      13. metadata-eval 26.9%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2}\right)} - 1 \]

   9. Applied egg-rr 26.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 28.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

       2. expm1-log1p 28.4%

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

   11. Simplified 28.4%

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

       1. *-un-lft-identity 28.4%

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

       2. associate-/r* 28.4%

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

   13. Applied egg-rr 28.4%

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

   if 1.48e26 < k

   1. Initial program 42.6%

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

      1. associate-/r* 42.7%

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

      2. sqr-neg 42.7%

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

      3. associate-*l* 42.7%

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

      4. sqr-neg 42.7%

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

      5. associate-/r* 49.8%

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

      6. associate-+l+ 49.8%

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

      7. unpow2 49.8%

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

      8. times-frac 28.9%

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

      9. sqr-neg 28.9%

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

      10. times-frac 49.8%

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

      11. unpow2 49.8%

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

   3. Simplified 49.8%

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

      1. add-sqr-sqrt 21.7%

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

      2. pow2 21.7%

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

      3. sqrt-prod 7.2%

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

      4. associate-/l/ 5.5%

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

      5. sqrt-div 5.4%

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

      6. sqrt-pow1 7.0%

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

      7. metadata-eval 7.0%

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

      8. sqrt-unprod 3.5%

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

      9. add-sqr-sqrt 8.8%

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

   6. Applied egg-rr 8.8%

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

   7. Taylor expanded in k around 0 16.2%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + 0.08333333333333333 \cdot {k}^{3}\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-commutative 16.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k + \color{blue}{{k}^{3} \cdot 0.08333333333333333}\right)\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   9. Simplified 16.2%

      \[\leadsto \frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k + {k}^{3} \cdot 0.08333333333333333\right)}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   10. Taylor expanded in k around inf 56.8%

       \[\leadsto \color{blue}{288 \cdot \frac{{\ell}^{2}}{{k}^{8} \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.48 \cdot 10^{+26}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{k}}{{t}^{1.5}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;288 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{8}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot {\left(\frac{\ell}{k \cdot {t_m}^{1.5}}\right)}^{2} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (pow (/ l (* k (pow t_m 1.5))) 2.0)))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * pow((l / (k * pow(t_m, 1.5))), 2.0);
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((l / (k * (t_m ** 1.5d0))) ** 2.0d0)
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * Math.pow((l / (k * Math.pow(t_m, 1.5))), 2.0);
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * math.pow((l / (k * math.pow(t_m, 1.5))), 2.0)

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * (Float64(l / Float64(k * (t_m ^ 1.5))) ^ 2.0))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((l / (k * (t_m ^ 1.5))) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. associate-/r* 55.0%

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

   2. sqr-neg 55.0%

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

   3. associate-*l* 51.1%

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

   4. sqr-neg 51.1%

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

   5. associate-/r* 56.6%

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

   6. associate-+l+ 56.6%

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

   7. unpow2 56.6%

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

   8. times-frac 44.5%

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

   9. sqr-neg 44.5%

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

   10. times-frac 56.6%

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

   11. unpow2 56.6%

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

3. Simplified 56.6%

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

5. Taylor expanded in k around 0 50.3%

   \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
6. Step-by-step derivation

   1. unpow2 50.3%

      \[\leadsto \frac{{\ell}^{2}}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

7. Applied egg-rr 50.3%

   \[\leadsto \frac{{\ell}^{2}}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]
8. Step-by-step derivation

   1. expm1-log1p-u 36.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot {t}^{3}}\right)\right)} \]

   2. expm1-udef 35.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot {t}^{3}}\right)} - 1} \]

   3. add-sqr-sqrt 35.2%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot {t}^{3}}} \cdot \sqrt{\frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot {t}^{3}}}}\right)} - 1 \]

   4. pow2 35.2%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{\frac{{\ell}^{2}}{\left(k \cdot k\right) \cdot {t}^{3}}}\right)}^{2}}\right)} - 1 \]

   5. sqrt-div 19.4%

      \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(\frac{\sqrt{{\ell}^{2}}}{\sqrt{\left(k \cdot k\right) \cdot {t}^{3}}}\right)}}^{2}\right)} - 1 \]

   6. unpow2 19.4%

      \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\sqrt{\color{blue}{\ell \cdot \ell}}}{\sqrt{\left(k \cdot k\right) \cdot {t}^{3}}}\right)}^{2}\right)} - 1 \]

   7. sqrt-prod 12.3%

      \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}{\sqrt{\left(k \cdot k\right) \cdot {t}^{3}}}\right)}^{2}\right)} - 1 \]

   8. add-sqr-sqrt 20.8%

      \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\color{blue}{\ell}}{\sqrt{\left(k \cdot k\right) \cdot {t}^{3}}}\right)}^{2}\right)} - 1 \]

   9. sqrt-prod 21.1%

      \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\ell}{\color{blue}{\sqrt{k \cdot k} \cdot \sqrt{{t}^{3}}}}\right)}^{2}\right)} - 1 \]

   10. sqrt-prod 11.8%

       \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\ell}{\color{blue}{\left(\sqrt{k} \cdot \sqrt{k}\right)} \cdot \sqrt{{t}^{3}}}\right)}^{2}\right)} - 1 \]

   11. add-sqr-sqrt 23.6%

       \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\ell}{\color{blue}{k} \cdot \sqrt{{t}^{3}}}\right)}^{2}\right)} - 1 \]

   12. sqrt-pow1 24.3%

       \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot \color{blue}{{t}^{\left(\frac{3}{2}\right)}}}\right)}^{2}\right)} - 1 \]

   13. metadata-eval 24.3%

       \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{\color{blue}{1.5}}}\right)}^{2}\right)} - 1 \]

9. Applied egg-rr 24.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 25.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\right)\right)} \]

    2. expm1-log1p 25.4%

       \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]

11. Simplified 25.4%

    \[\leadsto \color{blue}{{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}} \]

12. Final simplification 25.4%

    \[\leadsto {\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024017 
(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))))