Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.3% → 90.0%

Time: 24.5s

Alternatives: 17

Speedup: 3.8×

• 27.2% 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: 54.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.4 \cdot 10^{-81}:\\ \;\;\;\;\frac{2}{\frac{\left(t_m \cdot \frac{{k}^{2}}{\cos k}\right) \cdot \frac{{\sin k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)}\right)\right)}^{3}}\\ \end{array} \end{array} \]

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 1.4e-81)
    (/ 2.0 (/ (* (* t_m (/ (pow k 2.0) (cos k))) (/ (pow (sin k) 2.0) l)) l))
    (/
     2.0
     (pow
      (*
       (/ t_m (pow (cbrt l) 2.0))
       (* (cbrt (sin k)) (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 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 tmp;
	if (t_m <= 1.4e-81) {
		tmp = 2.0 / (((t_m * (pow(k, 2.0) / cos(k))) * (pow(sin(k), 2.0) / l)) / l);
	} else {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * (cbrt(sin(k)) * cbrt((tan(k) * (2.0 + pow((k / t_m), 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 tmp;
	if (t_m <= 1.4e-81) {
		tmp = 2.0 / (((t_m * (Math.pow(k, 2.0) / Math.cos(k))) * (Math.pow(Math.sin(k), 2.0) / l)) / l);
	} else {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * (Math.cbrt(Math.sin(k)) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 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)
	tmp = 0.0
	if (t_m <= 1.4e-81)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64((k ^ 2.0) / cos(k))) * Float64((sin(k) ^ 2.0) / l)) / l));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * Float64(cbrt(sin(k)) * cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.4e-81], N[(2.0 / N[(N[(N[(t$95$m * N[(N[Power[k, 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.3999999999999999e-81

   1. Initial program 56.4%

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

      1. associate-*l* 56.4%

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

      2. sqr-neg 56.4%

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

      3. sqr-neg 56.4%

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

      4. associate-/r* 61.2%

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

      5. distribute-rgt-in 61.2%

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

      6. unpow2 61.2%

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

      7. times-frac 38.4%

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

      8. sqr-neg 38.4%

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

      9. times-frac 61.2%

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

      10. unpow2 61.2%

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

      11. distribute-rgt-in 61.2%

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

   3. Simplified 61.2%

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

      1. associate-*r* 61.2%

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

      2. associate-*r* 60.0%

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

      3. *-commutative 60.0%

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

      4. associate-*l/ 60.6%

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

      5. associate-*r/ 60.6%

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

      6. *-commutative 60.6%

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

      7. associate-*r* 61.8%

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

   6. Applied egg-rr 61.8%

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

   7. Taylor expanded in k around inf 73.4%

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

      1. associate-*r* 73.4%

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

      2. *-commutative 73.4%

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

      3. times-frac 73.6%

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

   9. Simplified 73.6%

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

       1. expm1-log1p-u 57.5%

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

       2. expm1-udef 36.1%

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

       3. associate-/l* 36.1%

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

   11. Applied egg-rr 36.1%

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

       1. expm1-def 57.5%

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

       2. expm1-log1p 73.6%

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

       3. associate-/r/ 73.6%

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

   13. Simplified 73.6%

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

   if 1.3999999999999999e-81 < t

   1. Initial program 69.5%

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

      1. associate-*l* 69.5%

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

      2. sqr-neg 69.5%

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

      3. sqr-neg 69.5%

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

      4. associate-/r* 73.7%

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

      5. distribute-rgt-in 73.7%

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

      6. unpow2 73.7%

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

      7. times-frac 62.5%

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

      8. sqr-neg 62.5%

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

      9. times-frac 73.7%

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

      10. unpow2 73.7%

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

      11. distribute-rgt-in 73.7%

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

   3. Simplified 73.7%

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

      1. add-cube-cbrt 73.6%

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

      2. pow3 73.6%

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

   6. Applied egg-rr 83.7%

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

      1. associate-*l* 83.7%

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

      2. cbrt-prod 96.6%

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

   8. Applied egg-rr 96.6%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-81}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{{k}^{2}}{\cos k}\right) \cdot \frac{{\sin k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{\frac{\left(t_m \cdot \frac{{k}^{2}}{\cos k}\right) \cdot \frac{{\sin k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right) \cdot {\left(\frac{t_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \end{array} \end{array} \]

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 4.5e-70)
    (/ 2.0 (/ (* (* t_m (/ (pow k 2.0) (cos k))) (/ (pow (sin k) 2.0) l)) l))
    (/
     2.0
     (*
      (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
      (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (sin k))) 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 <= 4.5e-70) {
		tmp = 2.0 / (((t_m * (pow(k, 2.0) / cos(k))) * (pow(sin(k), 2.0) / l)) / l);
	} else {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(sin(k))), 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 <= 4.5e-70) {
		tmp = 2.0 / (((t_m * (Math.pow(k, 2.0) / Math.cos(k))) * (Math.pow(Math.sin(k), 2.0) / l)) / l);
	} else {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k))), 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 <= 4.5e-70)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64((k ^ 2.0) / cos(k))) * Float64((sin(k) ^ 2.0) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(sin(k))) ^ 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, 4.5e-70], N[(2.0 / N[(N[(N[(t$95$m * N[(N[Power[k, 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 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[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.50000000000000022e-70

   1. Initial program 56.6%

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

      1. associate-*l* 56.6%

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

      2. sqr-neg 56.6%

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

      3. sqr-neg 56.6%

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

      4. associate-/r* 61.4%

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

      5. distribute-rgt-in 61.4%

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

      6. unpow2 61.4%

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

      7. times-frac 38.7%

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

      8. sqr-neg 38.7%

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

      9. times-frac 61.4%

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

      10. unpow2 61.4%

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

      11. distribute-rgt-in 61.4%

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

   3. Simplified 61.4%

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

      1. associate-*r* 61.4%

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

      2. associate-*r* 60.3%

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

      3. *-commutative 60.3%

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

      4. associate-*l/ 60.8%

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

      5. associate-*r/ 60.9%

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

      6. *-commutative 60.9%

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

      7. associate-*r* 62.0%

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

   6. Applied egg-rr 62.0%

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

   7. Taylor expanded in k around inf 73.6%

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

      1. associate-*r* 73.6%

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

      2. *-commutative 73.6%

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

      3. times-frac 73.7%

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

   9. Simplified 73.7%

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

       1. expm1-log1p-u 57.8%

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

       2. expm1-udef 36.5%

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

       3. associate-/l* 36.5%

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

   11. Applied egg-rr 36.5%

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

       1. expm1-def 57.8%

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

       2. expm1-log1p 73.7%

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

       3. associate-/r/ 73.8%

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

   13. Simplified 73.8%

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

   if 4.50000000000000022e-70 < t

   1. Initial program 69.2%

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

      1. associate-*l* 69.2%

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

      2. sqr-neg 69.2%

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

      3. sqr-neg 69.2%

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

      4. associate-/r* 73.4%

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

      5. distribute-rgt-in 73.4%

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

      6. unpow2 73.4%

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

      7. times-frac 62.1%

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

      8. sqr-neg 62.1%

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

      9. times-frac 73.4%

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

      10. unpow2 73.4%

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

      11. distribute-rgt-in 73.4%

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

   3. Simplified 73.4%

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

      1. associate-/r* 69.2%

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

      2. add-cube-cbrt 69.0%

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

      3. pow3 69.1%

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

      4. associate-/r* 73.3%

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

      5. *-commutative 73.3%

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

      6. cbrt-prod 73.2%

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

      7. associate-/r* 69.0%

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

      8. cbrt-div 69.0%

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

      9. rem-cbrt-cube 74.6%

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

      10. cbrt-prod 92.1%

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

      11. pow2 92.1%

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

   6. Applied egg-rr 92.1%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{{k}^{2}}{\cos k}\right) \cdot \frac{{\sin k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 6.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{\left(t_m \cdot \frac{{k}^{2}}{\cos k}\right) \cdot \frac{{\sin k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot {\left(\frac{t_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.8e-18)
    (/ 2.0 (/ (* (* t_m (/ (pow k 2.0) (cos k))) (/ (pow (sin k) 2.0) l)) l))
    (/
     2.0
     (*
      (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
      (* (sin k) (pow (/ 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 tmp;
	if (t_m <= 6.8e-18) {
		tmp = 2.0 / (((t_m * (pow(k, 2.0) / cos(k))) * (pow(sin(k), 2.0) / l)) / l);
	} else {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * (sin(k) * pow((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 tmp;
	if (t_m <= 6.8e-18) {
		tmp = 2.0 / (((t_m * (Math.pow(k, 2.0) / Math.cos(k))) * (Math.pow(Math.sin(k), 2.0) / l)) / l);
	} else {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * (Math.sin(k) * Math.pow((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)
	tmp = 0.0
	if (t_m <= 6.8e-18)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64((k ^ 2.0) / cos(k))) * Float64((sin(k) ^ 2.0) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(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_] := N[(t$95$s * If[LessEqual[t$95$m, 6.8e-18], N[(2.0 / N[(N[(N[(t$95$m * N[(N[Power[k, 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 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[Sin[k], $MachinePrecision] * N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6.80000000000000002e-18

   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-*l* 58.3%

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

      2. sqr-neg 58.3%

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

      3. sqr-neg 58.3%

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

      4. associate-/r* 62.8%

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

      5. distribute-rgt-in 62.8%

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

      6. unpow2 62.8%

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

      7. times-frac 41.6%

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

      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 62.8%

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

      10. unpow2 62.8%

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

      11. distribute-rgt-in 62.8%

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

   3. Simplified 62.8%

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

      1. associate-*r* 62.8%

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

      2. associate-*r* 61.8%

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

      3. *-commutative 61.8%

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

      4. associate-*l/ 62.3%

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

      5. associate-*r/ 62.3%

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

      6. *-commutative 62.3%

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

      7. associate-*r* 63.4%

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

   6. Applied egg-rr 63.4%

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

   7. Taylor expanded in k around inf 73.8%

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

      1. associate-*r* 73.8%

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

      2. *-commutative 73.8%

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

      3. times-frac 73.9%

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

   9. Simplified 73.9%

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

       1. expm1-log1p-u 58.3%

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

       2. expm1-udef 37.9%

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

       3. associate-/l* 37.9%

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

   11. Applied egg-rr 37.9%

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

       1. expm1-def 58.4%

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

       2. expm1-log1p 73.9%

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

       3. associate-/r/ 74.0%

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

   13. Simplified 74.0%

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

   if 6.80000000000000002e-18 < 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-*l* 66.6%

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

      2. sqr-neg 66.6%

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

      3. sqr-neg 66.6%

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

      4. associate-/r* 71.6%

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

      5. distribute-rgt-in 71.6%

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

      6. unpow2 71.6%

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

      7. times-frac 58.3%

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

      8. sqr-neg 58.3%

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

      9. times-frac 71.6%

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

      10. unpow2 71.6%

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

      11. distribute-rgt-in 71.6%

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

   3. Simplified 71.6%

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

      1. add-cube-cbrt 71.5%

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

      2. pow3 71.5%

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

      3. associate-/r* 66.6%

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

      4. cbrt-div 66.4%

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

      5. rem-cbrt-cube 70.2%

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

      6. cbrt-prod 89.5%

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

      7. pow2 89.5%

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

   6. Applied egg-rr 89.5%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{{k}^{2}}{\cos k}\right) \cdot \frac{{\sin k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{\frac{\left(t_m \cdot \frac{{k}^{2}}{\cos k}\right) \cdot \frac{{\sin k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t_m \leq 1.22 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t_m}^{2}}{\ell} \cdot \frac{t_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \left(t_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2e-17)
    (/ 2.0 (/ (* (* t_m (/ (pow k 2.0) (cos k))) (/ (pow (sin k) 2.0) l)) l))
    (if (<= t_m 1.22e+154)
      (/
       2.0
       (*
        (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
        (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))))
      (/
       2.0
       (*
        (pow (* (cbrt (sin k)) (* t_m (pow (cbrt l) -2.0))) 3.0)
        (* 2.0 k)))))))

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-17) {
		tmp = 2.0 / (((t_m * (pow(k, 2.0) / cos(k))) * (pow(sin(k), 2.0) / l)) / l);
	} else if (t_m <= 1.22e+154) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t_m * pow(cbrt(l), -2.0))), 3.0) * (2.0 * k));
	}
	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 <= 2e-17) {
		tmp = 2.0 / (((t_m * (Math.pow(k, 2.0) / Math.cos(k))) * (Math.pow(Math.sin(k), 2.0) / l)) / l);
	} else if (t_m <= 1.22e+154) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t_m * Math.pow(Math.cbrt(l), -2.0))), 3.0) * (2.0 * k));
	}
	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-17)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64((k ^ 2.0) / cos(k))) * Float64((sin(k) ^ 2.0) / l)) / l));
	elseif (t_m <= 1.22e+154)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t_m * (cbrt(l) ^ -2.0))) ^ 3.0) * Float64(2.0 * k)));
	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, 2e-17], N[(2.0 / N[(N[(N[(t$95$m * N[(N[Power[k, 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.22e+154], 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[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(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] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.00000000000000014e-17

   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-*l* 58.3%

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

      2. sqr-neg 58.3%

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

      3. sqr-neg 58.3%

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

      4. associate-/r* 62.8%

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

      5. distribute-rgt-in 62.8%

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

      6. unpow2 62.8%

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

      7. times-frac 41.6%

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

      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 62.8%

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

      10. unpow2 62.8%

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

      11. distribute-rgt-in 62.8%

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

   3. Simplified 62.8%

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

      1. associate-*r* 62.8%

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

      2. associate-*r* 61.8%

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

      3. *-commutative 61.8%

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

      4. associate-*l/ 62.3%

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

      5. associate-*r/ 62.3%

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

      6. *-commutative 62.3%

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

      7. associate-*r* 63.4%

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

   6. Applied egg-rr 63.4%

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

   7. Taylor expanded in k around inf 73.8%

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

      1. associate-*r* 73.8%

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

      2. *-commutative 73.8%

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

      3. times-frac 73.9%

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

   9. Simplified 73.9%

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

       1. expm1-log1p-u 58.3%

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

       2. expm1-udef 37.9%

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

       3. associate-/l* 37.9%

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

   11. Applied egg-rr 37.9%

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

       1. expm1-def 58.4%

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

       2. expm1-log1p 73.9%

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

       3. associate-/r/ 74.0%

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

   13. Simplified 74.0%

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

   if 2.00000000000000014e-17 < t < 1.22e154

   1. Initial program 69.4%

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

      1. associate-*l* 69.4%

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

      2. sqr-neg 69.4%

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

      3. sqr-neg 69.4%

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

      4. associate-/r* 75.8%

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

      5. distribute-rgt-in 75.8%

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

      6. unpow2 75.8%

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

      7. times-frac 72.8%

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

      8. sqr-neg 72.8%

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

      9. times-frac 75.8%

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

      10. unpow2 75.8%

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

      11. distribute-rgt-in 75.8%

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

   3. Simplified 75.8%

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

      1. associate-/r* 69.4%

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

      2. unpow3 69.3%

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

      3. times-frac 90.8%

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

      4. pow2 90.8%

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

   6. Applied egg-rr 90.8%

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

   if 1.22e154 < t

   1. Initial program 64.1%

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

      1. *-commutative 64.1%

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

      2. sqr-neg 64.1%

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

      3. *-commutative 64.1%

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

      4. associate-*l* 64.1%

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

      5. *-commutative 64.1%

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

      6. sqr-neg 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in k around 0 64.1%

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

      1. *-commutative 64.1%

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

   7. Simplified 64.1%

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

      1. *-commutative 64.1%

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

      2. rem-cube-cbrt 64.1%

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

      3. rem-cube-cbrt 64.1%

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

      4. cbrt-div 64.1%

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

      5. rem-cbrt-cube 67.9%

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

      6. cbrt-prod 83.9%

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

      7. unpow2 83.9%

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

      8. unpow-prod-down 89.0%

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

      9. rem-cube-cbrt 88.8%

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

      10. rem-cbrt-cube 89.0%

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

      11. div-inv 89.1%

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

      12. pow-flip 89.1%

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

      13. metadata-eval 89.1%

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

   9. Applied egg-rr 89.1%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{{k}^{2}}{\cos k}\right) \cdot \frac{{\sin k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.9 \cdot 10^{-21}:\\ \;\;\;\;\frac{2}{\frac{\left(t_m \cdot \frac{{k}^{2}}{\cos k}\right) \cdot \frac{{\sin k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t_m}^{2}}{\ell} \cdot \frac{t_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(t_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.9e-21)
    (/ 2.0 (/ (* (* t_m (/ (pow k 2.0) (cos k))) (/ (pow (sin k) 2.0) l)) l))
    (if (<= t_m 1.35e+154)
      (/
       2.0
       (*
        (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
        (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))))
      (/
       2.0
       (* (* 2.0 k) (* (sin k) (pow (* 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 tmp;
	if (t_m <= 1.9e-21) {
		tmp = 2.0 / (((t_m * (pow(k, 2.0) / cos(k))) * (pow(sin(k), 2.0) / l)) / l);
	} else if (t_m <= 1.35e+154) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * pow((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 tmp;
	if (t_m <= 1.9e-21) {
		tmp = 2.0 / (((t_m * (Math.pow(k, 2.0) / Math.cos(k))) * (Math.pow(Math.sin(k), 2.0) / l)) / l);
	} else if (t_m <= 1.35e+154) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * Math.pow((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)
	tmp = 0.0
	if (t_m <= 1.9e-21)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64((k ^ 2.0) / cos(k))) * Float64((sin(k) ^ 2.0) / l)) / l));
	elseif (t_m <= 1.35e+154)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.9e-21], N[(2.0 / N[(N[(N[(t$95$m * N[(N[Power[k, 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.35e+154], 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[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.8999999999999999e-21

   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-*l* 58.3%

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

      2. sqr-neg 58.3%

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

      3. sqr-neg 58.3%

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

      4. associate-/r* 62.8%

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

      5. distribute-rgt-in 62.8%

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

      6. unpow2 62.8%

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

      7. times-frac 41.6%

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

      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 62.8%

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

      10. unpow2 62.8%

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

      11. distribute-rgt-in 62.8%

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

   3. Simplified 62.8%

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

      1. associate-*r* 62.8%

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

      2. associate-*r* 61.8%

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

      3. *-commutative 61.8%

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

      4. associate-*l/ 62.3%

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

      5. associate-*r/ 62.3%

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

      6. *-commutative 62.3%

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

      7. associate-*r* 63.4%

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

   6. Applied egg-rr 63.4%

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

   7. Taylor expanded in k around inf 73.8%

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

      1. associate-*r* 73.8%

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

      2. *-commutative 73.8%

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

      3. times-frac 73.9%

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

   9. Simplified 73.9%

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

       1. expm1-log1p-u 58.3%

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

       2. expm1-udef 37.9%

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

       3. associate-/l* 37.9%

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

   11. Applied egg-rr 37.9%

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

       1. expm1-def 58.4%

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

       2. expm1-log1p 73.9%

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

       3. associate-/r/ 74.0%

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

   13. Simplified 74.0%

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

   if 1.8999999999999999e-21 < t < 1.35000000000000003e154

   1. Initial program 69.4%

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

      1. associate-*l* 69.4%

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

      2. sqr-neg 69.4%

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

      3. sqr-neg 69.4%

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

      4. associate-/r* 75.8%

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

      5. distribute-rgt-in 75.8%

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

      6. unpow2 75.8%

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

      7. times-frac 72.8%

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

      8. sqr-neg 72.8%

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

      9. times-frac 75.8%

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

      10. unpow2 75.8%

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

      11. distribute-rgt-in 75.8%

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

   3. Simplified 75.8%

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

      1. associate-/r* 69.4%

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

      2. unpow3 69.3%

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

      3. times-frac 90.8%

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

      4. pow2 90.8%

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

   6. Applied egg-rr 90.8%

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

   if 1.35000000000000003e154 < t

   1. Initial program 64.1%

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

      1. *-commutative 64.1%

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

      2. sqr-neg 64.1%

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

      3. *-commutative 64.1%

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

      4. associate-*l* 64.1%

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

      5. *-commutative 64.1%

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

      6. sqr-neg 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in k around 0 64.1%

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

      1. *-commutative 64.1%

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

   7. Simplified 64.1%

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

      1. rem-cube-cbrt 64.1%

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

      2. cbrt-div 64.1%

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

      3. rem-cbrt-cube 67.9%

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

      4. cbrt-prod 83.8%

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

      5. unpow2 83.8%

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

      6. cube-mult 83.7%

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

      7. div-inv 83.8%

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

      8. pow-flip 83.8%

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

      9. metadata-eval 83.8%

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

      10. pow2 83.8%

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

      11. div-inv 83.8%

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

      12. pow-flip 83.8%

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

      13. metadata-eval 83.8%

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

   9. Applied egg-rr 83.8%

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

       1. unpow2 83.8%

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

       2. cube-mult 83.8%

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

   11. Simplified 83.8%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{-21}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{{k}^{2}}{\cos k}\right) \cdot \frac{{\sin k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.35 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\frac{\left(t_m \cdot \frac{{k}^{2}}{\cos k}\right) \cdot \frac{{\sin k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t_m \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \frac{\frac{{t_m}^{3}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(t_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.35e-16)
    (/ 2.0 (/ (* (* t_m (/ (pow k 2.0) (cos k))) (/ (pow (sin k) 2.0) l)) l))
    (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))))
      (/
       2.0
       (* (* 2.0 k) (* (sin k) (pow (* 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 tmp;
	if (t_m <= 1.35e-16) {
		tmp = 2.0 / (((t_m * (pow(k, 2.0) / cos(k))) * (pow(sin(k), 2.0) / l)) / l);
	} 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 = 2.0 / ((2.0 * k) * (sin(k) * pow((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 tmp;
	if (t_m <= 1.35e-16) {
		tmp = 2.0 / (((t_m * (Math.pow(k, 2.0) / Math.cos(k))) * (Math.pow(Math.sin(k), 2.0) / l)) / l);
	} 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 = 2.0 / ((2.0 * k) * (Math.sin(k) * Math.pow((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)
	tmp = 0.0
	if (t_m <= 1.35e-16)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64((k ^ 2.0) / cos(k))) * Float64((sin(k) ^ 2.0) / l)) / l));
	elseif (t_m <= 5.5e+102)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(sin(k) * Float64(Float64((t_m ^ 3.0) / l) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(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_] := N[(t$95$s * If[LessEqual[t$95$m, 1.35e-16], N[(2.0 / N[(N[(N[(t$95$m * N[(N[Power[k, 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $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[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.35e-16

   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-*l* 58.3%

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

      2. sqr-neg 58.3%

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

      3. sqr-neg 58.3%

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

      4. associate-/r* 62.8%

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

      5. distribute-rgt-in 62.8%

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

      6. unpow2 62.8%

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

      7. times-frac 41.6%

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

      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 62.8%

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

      10. unpow2 62.8%

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

      11. distribute-rgt-in 62.8%

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

   3. Simplified 62.8%

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

      1. associate-*r* 62.8%

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

      2. associate-*r* 61.8%

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

      3. *-commutative 61.8%

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

      4. associate-*l/ 62.3%

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

      5. associate-*r/ 62.3%

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

      6. *-commutative 62.3%

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

      7. associate-*r* 63.4%

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

   6. Applied egg-rr 63.4%

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

   7. Taylor expanded in k around inf 73.8%

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

      1. associate-*r* 73.8%

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

      2. *-commutative 73.8%

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

      3. times-frac 73.9%

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

   9. Simplified 73.9%

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

       1. expm1-log1p-u 58.3%

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

       2. expm1-udef 37.9%

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

       3. associate-/l* 37.9%

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

   11. Applied egg-rr 37.9%

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

       1. expm1-def 58.4%

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

       2. expm1-log1p 73.9%

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

       3. associate-/r/ 74.0%

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

   13. Simplified 74.0%

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

   if 1.35e-16 < t < 5.49999999999999981e102

   1. Initial program 89.7%

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

      1. associate-*l* 89.7%

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

      2. sqr-neg 89.7%

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

      3. sqr-neg 89.7%

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

      4. associate-/r* 99.8%

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

      5. distribute-rgt-in 99.8%

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

      6. unpow2 99.8%

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

      7. times-frac 94.7%

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

      8. sqr-neg 94.7%

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

      9. times-frac 99.8%

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

      10. unpow2 99.8%

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

      11. distribute-rgt-in 99.8%

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

   3. Simplified 99.8%

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

   if 5.49999999999999981e102 < t

   1. Initial program 57.6%

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

      1. *-commutative 57.6%

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

      2. sqr-neg 57.6%

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

      3. *-commutative 57.6%

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

      4. associate-*l* 57.6%

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

      5. *-commutative 57.6%

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

      6. sqr-neg 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in k around 0 57.6%

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

      1. *-commutative 57.6%

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

   7. Simplified 57.6%

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

      1. rem-cube-cbrt 57.6%

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

      2. cbrt-div 57.6%

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

      3. rem-cbrt-cube 62.6%

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

      4. cbrt-prod 76.2%

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

      5. unpow2 76.2%

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

      6. cube-mult 76.2%

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

      7. div-inv 76.2%

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

      8. pow-flip 76.2%

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

      9. metadata-eval 76.2%

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

      10. pow2 76.2%

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

      11. div-inv 76.3%

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

      12. pow-flip 76.2%

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

      13. metadata-eval 76.2%

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

   9. Applied egg-rr 76.2%

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

       1. unpow2 76.2%

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

       2. cube-mult 76.2%

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

   11. Simplified 76.2%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{{k}^{2}}{\cos k}\right) \cdot \frac{{\sin k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 3.8 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{\frac{\left(t_m \cdot \frac{{k}^{2}}{\cos k}\right) \cdot \frac{{\sin k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t_m \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t_m}\right)}^{2}\right)\right) \cdot \frac{\frac{{t_m}^{3}}{\ell}}{\frac{\ell}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(t_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.8e-70)
    (/ 2.0 (/ (* (* t_m (/ (pow k 2.0) (cos k))) (/ (pow (sin k) 2.0) l)) l))
    (if (<= t_m 5.5e+102)
      (/
       2.0
       (*
        (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
        (/ (/ (pow t_m 3.0) l) (/ l (sin k)))))
      (/
       2.0
       (* (* 2.0 k) (* (sin k) (pow (* 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 tmp;
	if (t_m <= 3.8e-70) {
		tmp = 2.0 / (((t_m * (pow(k, 2.0) / cos(k))) * (pow(sin(k), 2.0) / l)) / l);
	} else if (t_m <= 5.5e+102) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * ((pow(t_m, 3.0) / l) / (l / sin(k))));
	} else {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * pow((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 tmp;
	if (t_m <= 3.8e-70) {
		tmp = 2.0 / (((t_m * (Math.pow(k, 2.0) / Math.cos(k))) * (Math.pow(Math.sin(k), 2.0) / l)) / l);
	} else if (t_m <= 5.5e+102) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * ((Math.pow(t_m, 3.0) / l) / (l / Math.sin(k))));
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * Math.pow((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)
	tmp = 0.0
	if (t_m <= 3.8e-70)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64((k ^ 2.0) / cos(k))) * Float64((sin(k) ^ 2.0) / l)) / l));
	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((t_m ^ 3.0) / l) / Float64(l / sin(k)))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(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_] := N[(t$95$s * If[LessEqual[t$95$m, 3.8e-70], N[(2.0 / N[(N[(N[(t$95$m * N[(N[Power[k, 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $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[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.7999999999999998e-70

   1. Initial program 56.6%

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

      1. associate-*l* 56.6%

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

      2. sqr-neg 56.6%

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

      3. sqr-neg 56.6%

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

      4. associate-/r* 61.4%

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

      5. distribute-rgt-in 61.4%

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

      6. unpow2 61.4%

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

      7. times-frac 38.7%

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

      8. sqr-neg 38.7%

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

      9. times-frac 61.4%

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

      10. unpow2 61.4%

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

      11. distribute-rgt-in 61.4%

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

   3. Simplified 61.4%

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

      1. associate-*r* 61.4%

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

      2. associate-*r* 60.3%

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

      3. *-commutative 60.3%

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

      4. associate-*l/ 60.8%

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

      5. associate-*r/ 60.9%

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

      6. *-commutative 60.9%

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

      7. associate-*r* 62.0%

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

   6. Applied egg-rr 62.0%

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

   7. Taylor expanded in k around inf 73.6%

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

      1. associate-*r* 73.6%

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

      2. *-commutative 73.6%

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

      3. times-frac 73.7%

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

   9. Simplified 73.7%

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

       1. expm1-log1p-u 57.8%

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

       2. expm1-udef 36.5%

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

       3. associate-/l* 36.5%

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

   11. Applied egg-rr 36.5%

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

       1. expm1-def 57.8%

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

       2. expm1-log1p 73.7%

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

       3. associate-/r/ 73.8%

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

   13. Simplified 73.8%

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

   if 3.7999999999999998e-70 < t < 5.49999999999999981e102

   1. Initial program 87.4%

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

      1. associate-*l* 87.4%

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

      2. sqr-neg 87.4%

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

      3. sqr-neg 87.4%

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

      4. associate-/r* 93.6%

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

      5. distribute-rgt-in 93.6%

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

      6. unpow2 93.6%

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

      7. times-frac 90.5%

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

      8. sqr-neg 90.5%

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

      9. times-frac 93.6%

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

      10. unpow2 93.6%

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

      11. distribute-rgt-in 93.6%

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

   3. Simplified 93.6%

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

      1. associate-*l/ 93.5%

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

      2. associate-/l* 93.5%

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

   6. Applied egg-rr 93.5%

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

   if 5.49999999999999981e102 < t

   1. Initial program 57.6%

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

      1. *-commutative 57.6%

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

      2. sqr-neg 57.6%

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

      3. *-commutative 57.6%

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

      4. associate-*l* 57.6%

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

      5. *-commutative 57.6%

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

      6. sqr-neg 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in k around 0 57.6%

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

      1. *-commutative 57.6%

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

   7. Simplified 57.6%

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

      1. rem-cube-cbrt 57.6%

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

      2. cbrt-div 57.6%

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

      3. rem-cbrt-cube 62.6%

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

      4. cbrt-prod 76.2%

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

      5. unpow2 76.2%

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

      6. cube-mult 76.2%

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

      7. div-inv 76.2%

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

      8. pow-flip 76.2%

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

      9. metadata-eval 76.2%

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

      10. pow2 76.2%

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

      11. div-inv 76.3%

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

      12. pow-flip 76.2%

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

      13. metadata-eval 76.2%

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

   9. Applied egg-rr 76.2%

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

       1. unpow2 76.2%

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

       2. cube-mult 76.2%

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

   11. Simplified 76.2%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-70}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{{k}^{2}}{\cos k}\right) \cdot \frac{{\sin k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\frac{\ell}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{2}{\frac{\left(t_m \cdot \frac{{k}^{2}}{\cos k}\right) \cdot \frac{{\sin k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left({t_m}^{1.5} \cdot \frac{\sqrt{k}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

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 2.5e+35)
    (/ 2.0 (/ (* (* t_m (/ (pow k 2.0) (cos k))) (/ (pow (sin k) 2.0) l)) l))
    (/ 2.0 (* (* 2.0 k) (pow (* (pow t_m 1.5) (/ (sqrt k) l)) 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 <= 2.5e+35) {
		tmp = 2.0 / (((t_m * (pow(k, 2.0) / cos(k))) * (pow(sin(k), 2.0) / l)) / l);
	} else {
		tmp = 2.0 / ((2.0 * k) * pow((pow(t_m, 1.5) * (sqrt(k) / l)), 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 <= 2.5d+35) then
        tmp = 2.0d0 / (((t_m * ((k ** 2.0d0) / cos(k))) * ((sin(k) ** 2.0d0) / l)) / l)
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (((t_m ** 1.5d0) * (sqrt(k) / l)) ** 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 <= 2.5e+35) {
		tmp = 2.0 / (((t_m * (Math.pow(k, 2.0) / Math.cos(k))) * (Math.pow(Math.sin(k), 2.0) / l)) / l);
	} else {
		tmp = 2.0 / ((2.0 * k) * Math.pow((Math.pow(t_m, 1.5) * (Math.sqrt(k) / l)), 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 <= 2.5e+35:
		tmp = 2.0 / (((t_m * (math.pow(k, 2.0) / math.cos(k))) * (math.pow(math.sin(k), 2.0) / l)) / l)
	else:
		tmp = 2.0 / ((2.0 * k) * math.pow((math.pow(t_m, 1.5) * (math.sqrt(k) / l)), 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 <= 2.5e+35)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64((k ^ 2.0) / cos(k))) * Float64((sin(k) ^ 2.0) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64((t_m ^ 1.5) * Float64(sqrt(k) / l)) ^ 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 <= 2.5e+35)
		tmp = 2.0 / (((t_m * ((k ^ 2.0) / cos(k))) * ((sin(k) ^ 2.0) / l)) / l);
	else
		tmp = 2.0 / ((2.0 * k) * (((t_m ^ 1.5) * (sqrt(k) / l)) ^ 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, 2.5e+35], N[(2.0 / N[(N[(N[(t$95$m * N[(N[Power[k, 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(N[Sqrt[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.50000000000000011e35

   1. Initial program 60.2%

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

      1. associate-*l* 60.2%

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

      2. sqr-neg 60.2%

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

      3. sqr-neg 60.2%

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

      4. associate-/r* 64.5%

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

      5. distribute-rgt-in 64.5%

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

      6. unpow2 64.5%

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

      7. times-frac 44.3%

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

      8. sqr-neg 44.3%

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

      9. times-frac 64.5%

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

      10. unpow2 64.5%

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

      11. distribute-rgt-in 64.5%

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

   3. Simplified 64.5%

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

      1. associate-*r* 64.5%

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

      2. associate-*r* 63.5%

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

      3. *-commutative 63.5%

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

      4. associate-*l/ 64.0%

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

      5. associate-*r/ 64.0%

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

      6. *-commutative 64.0%

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

      7. associate-*r* 65.1%

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

   6. Applied egg-rr 65.1%

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

   7. Taylor expanded in k around inf 75.0%

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

      1. associate-*r* 75.0%

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

      2. *-commutative 75.0%

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

      3. times-frac 75.1%

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

   9. Simplified 75.1%

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

       1. expm1-log1p-u 58.7%

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

       2. expm1-udef 38.6%

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

       3. associate-/l* 38.6%

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

   11. Applied egg-rr 38.6%

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

       1. expm1-def 58.7%

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

       2. expm1-log1p 75.1%

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

       3. associate-/r/ 75.1%

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

   13. Simplified 75.1%

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

   if 2.50000000000000011e35 < t

   1. Initial program 61.5%

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

      1. *-commutative 61.5%

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

      2. sqr-neg 61.5%

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

      3. *-commutative 61.5%

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

      4. associate-*l* 61.5%

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

      5. *-commutative 61.5%

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

      6. sqr-neg 61.5%

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

   3. Simplified 61.5%

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

   5. Taylor expanded in k around 0 57.3%

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

      1. *-commutative 57.3%

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

   7. Simplified 57.3%

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

   8. Taylor expanded in k around 0 57.1%

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

      1. associate-/l* 57.2%

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

   10. Simplified 57.2%

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

       1. add-sqr-sqrt 24.6%

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

       2. sqrt-div 24.6%

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

       3. sqrt-div 24.6%

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

       4. unpow2 24.6%

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

       5. sqrt-prod 10.8%

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

       6. add-sqr-sqrt 24.5%

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

       7. sqrt-pow1 24.5%

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

       8. metadata-eval 24.5%

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

       9. sqrt-div 24.5%

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

       10. sqrt-div 24.5%

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

       11. unpow2 24.5%

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

       12. sqrt-prod 11.0%

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

       13. add-sqr-sqrt 24.9%

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

       14. sqrt-pow1 31.5%

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

       15. metadata-eval 31.5%

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

   12. Applied egg-rr 31.5%

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

       1. unpow2 31.5%

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

       2. associate-/r/ 30.9%

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

       3. *-commutative 30.9%

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

   14. Simplified 30.9%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \frac{{k}^{2}}{\cos k}\right) \cdot \frac{{\sin k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left({t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.3 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2}}{\ell} \cdot \frac{t_m \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(t_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.3e+80)
    (/ 2.0 (/ (* (/ (pow k 2.0) l) (/ (* t_m (pow (sin k) 2.0)) (cos k))) l))
    (/ 2.0 (* (* 2.0 k) (* (sin k) (pow (* 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 tmp;
	if (t_m <= 2.3e+80) {
		tmp = 2.0 / (((pow(k, 2.0) / l) * ((t_m * pow(sin(k), 2.0)) / cos(k))) / l);
	} else {
		tmp = 2.0 / ((2.0 * k) * (sin(k) * pow((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 tmp;
	if (t_m <= 2.3e+80) {
		tmp = 2.0 / (((Math.pow(k, 2.0) / l) * ((t_m * Math.pow(Math.sin(k), 2.0)) / Math.cos(k))) / l);
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(k) * Math.pow((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)
	tmp = 0.0
	if (t_m <= 2.3e+80)
		tmp = Float64(2.0 / Float64(Float64(Float64((k ^ 2.0) / l) * Float64(Float64(t_m * (sin(k) ^ 2.0)) / cos(k))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(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_] := N[(t$95$s * If[LessEqual[t$95$m, 2.3e+80], N[(2.0 / N[(N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.30000000000000004e80

   1. Initial program 61.1%

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

      1. associate-*l* 61.1%

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

      2. sqr-neg 61.1%

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

      3. sqr-neg 61.1%

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

      4. associate-/r* 65.7%

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

      5. distribute-rgt-in 65.7%

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

      6. unpow2 65.7%

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

      7. times-frac 46.2%

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

      8. sqr-neg 46.2%

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

      9. times-frac 65.7%

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

      10. unpow2 65.7%

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

      11. distribute-rgt-in 65.7%

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

   3. Simplified 65.7%

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

      1. associate-*r* 65.7%

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

      2. associate-*r* 64.7%

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

      3. *-commutative 64.7%

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

      4. associate-*l/ 65.2%

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

      5. associate-*r/ 65.2%

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

      6. *-commutative 65.2%

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

      7. associate-*r* 66.2%

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

   6. Applied egg-rr 66.2%

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

   7. Taylor expanded in k around inf 73.9%

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

      1. times-frac 74.6%

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

   9. Simplified 74.6%

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

   if 2.30000000000000004e80 < t

   1. Initial program 58.2%

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

      1. *-commutative 58.2%

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

      2. sqr-neg 58.2%

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

      3. *-commutative 58.2%

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

      4. associate-*l* 58.2%

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

      5. *-commutative 58.2%

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

      6. sqr-neg 58.2%

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

   3. Simplified 58.2%

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

   5. Taylor expanded in k around 0 58.2%

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

      1. *-commutative 58.2%

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

   7. Simplified 58.2%

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

      1. rem-cube-cbrt 58.2%

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

      2. cbrt-div 58.2%

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

      3. rem-cbrt-cube 62.9%

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

      4. cbrt-prod 75.8%

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

      5. unpow2 75.8%

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

      6. cube-mult 75.8%

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

      7. div-inv 75.8%

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

      8. pow-flip 75.8%

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

      9. metadata-eval 75.8%

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

      10. pow2 75.8%

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

      11. div-inv 75.9%

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

      12. pow-flip 75.8%

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

      13. metadata-eval 75.8%

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

   9. Applied egg-rr 75.8%

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

       1. unpow2 75.8%

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

       2. cube-mult 75.8%

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

   11. Simplified 75.8%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2}}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 8.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{3}}{\frac{\ell}{t_m \cdot \sin k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left({t_m}^{1.5} \cdot \frac{\sqrt{k}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

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 8.8e-28)
    (/ 2.0 (/ (/ (pow k 3.0) (/ l (* t_m (sin k)))) l))
    (/ 2.0 (* (* 2.0 k) (pow (* (pow t_m 1.5) (/ (sqrt k) l)) 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 <= 8.8e-28) {
		tmp = 2.0 / ((pow(k, 3.0) / (l / (t_m * sin(k)))) / l);
	} else {
		tmp = 2.0 / ((2.0 * k) * pow((pow(t_m, 1.5) * (sqrt(k) / l)), 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 <= 8.8d-28) then
        tmp = 2.0d0 / (((k ** 3.0d0) / (l / (t_m * sin(k)))) / l)
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (((t_m ** 1.5d0) * (sqrt(k) / l)) ** 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 <= 8.8e-28) {
		tmp = 2.0 / ((Math.pow(k, 3.0) / (l / (t_m * Math.sin(k)))) / l);
	} else {
		tmp = 2.0 / ((2.0 * k) * Math.pow((Math.pow(t_m, 1.5) * (Math.sqrt(k) / l)), 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 <= 8.8e-28:
		tmp = 2.0 / ((math.pow(k, 3.0) / (l / (t_m * math.sin(k)))) / l)
	else:
		tmp = 2.0 / ((2.0 * k) * math.pow((math.pow(t_m, 1.5) * (math.sqrt(k) / l)), 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 <= 8.8e-28)
		tmp = Float64(2.0 / Float64(Float64((k ^ 3.0) / Float64(l / Float64(t_m * sin(k)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64((t_m ^ 1.5) * Float64(sqrt(k) / l)) ^ 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 <= 8.8e-28)
		tmp = 2.0 / (((k ^ 3.0) / (l / (t_m * sin(k)))) / l);
	else
		tmp = 2.0 / ((2.0 * k) * (((t_m ^ 1.5) * (sqrt(k) / l)) ^ 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, 8.8e-28], N[(2.0 / N[(N[(N[Power[k, 3.0], $MachinePrecision] / N[(l / N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[(N[Sqrt[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.79999999999999984e-28

   1. Initial program 58.4%

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

      1. associate-*l* 58.4%

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

      2. sqr-neg 58.4%

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

      3. sqr-neg 58.4%

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

      4. associate-/r* 62.9%

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

      5. distribute-rgt-in 62.9%

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

      6. unpow2 62.9%

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

      7. times-frac 41.5%

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

      8. sqr-neg 41.5%

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

      9. times-frac 62.9%

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

      10. unpow2 62.9%

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

      11. distribute-rgt-in 62.9%

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

   3. Simplified 62.9%

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

      1. associate-*r* 62.9%

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

      2. associate-*r* 61.9%

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

      3. *-commutative 61.9%

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

      4. associate-*l/ 62.4%

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

      5. associate-*r/ 62.4%

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

      6. *-commutative 62.4%

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

      7. associate-*r* 63.5%

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

   6. Applied egg-rr 63.5%

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

   7. Taylor expanded in k around 0 59.0%

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

      1. associate-/l* 58.9%

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

   9. Simplified 58.9%

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

   10. Taylor expanded in k around inf 66.7%

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

       1. associate-/l* 67.8%

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

       2. *-commutative 67.8%

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

   12. Simplified 67.8%

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

   if 8.79999999999999984e-28 < t

   1. Initial program 66.2%

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

      1. *-commutative 66.2%

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

      2. sqr-neg 66.2%

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

      3. *-commutative 66.2%

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

      4. associate-*l* 66.2%

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

      5. *-commutative 66.2%

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

      6. sqr-neg 66.2%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in k around 0 61.3%

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

      1. *-commutative 61.3%

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

   7. Simplified 61.3%

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

   8. Taylor expanded in k around 0 61.1%

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

      1. associate-/l* 61.2%

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

   10. Simplified 61.2%

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

       1. add-sqr-sqrt 25.1%

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

       2. sqrt-div 25.1%

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

       3. sqrt-div 25.1%

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

       4. unpow2 25.1%

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

       5. sqrt-prod 12.0%

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

       6. add-sqr-sqrt 23.6%

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

       7. sqrt-pow1 23.6%

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

       8. metadata-eval 23.6%

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

       9. sqrt-div 23.6%

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

       10. sqrt-div 23.6%

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

       11. unpow2 23.6%

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

       12. sqrt-prod 12.2%

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

       13. add-sqr-sqrt 25.4%

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

       14. sqrt-pow1 31.0%

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

       15. metadata-eval 31.0%

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

   12. Applied egg-rr 31.0%

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

       1. unpow2 31.0%

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

       2. associate-/r/ 30.4%

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

       3. *-commutative 30.4%

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

   14. Simplified 30.4%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{3}}{\frac{\ell}{t \cdot \sin k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left({t}^{1.5} \cdot \frac{\sqrt{k}}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 4.9 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{3}}{\frac{\ell}{t_m \cdot \sin k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k}{\frac{{\ell}^{2}}{{t_m}^{3}}}}\\ \end{array} \end{array} \]

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 4.9e-27)
    (/ 2.0 (/ (/ (pow k 3.0) (/ l (* t_m (sin k)))) l))
    (/ 2.0 (* (* 2.0 k) (/ k (/ (pow l 2.0) (pow 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 <= 4.9e-27) {
		tmp = 2.0 / ((pow(k, 3.0) / (l / (t_m * sin(k)))) / l);
	} else {
		tmp = 2.0 / ((2.0 * k) * (k / (pow(l, 2.0) / pow(t_m, 3.0))));
	}
	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 <= 4.9d-27) then
        tmp = 2.0d0 / (((k ** 3.0d0) / (l / (t_m * sin(k)))) / l)
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (k / ((l ** 2.0d0) / (t_m ** 3.0d0))))
    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 <= 4.9e-27) {
		tmp = 2.0 / ((Math.pow(k, 3.0) / (l / (t_m * Math.sin(k)))) / l);
	} else {
		tmp = 2.0 / ((2.0 * k) * (k / (Math.pow(l, 2.0) / Math.pow(t_m, 3.0))));
	}
	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 <= 4.9e-27:
		tmp = 2.0 / ((math.pow(k, 3.0) / (l / (t_m * math.sin(k)))) / l)
	else:
		tmp = 2.0 / ((2.0 * k) * (k / (math.pow(l, 2.0) / math.pow(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 <= 4.9e-27)
		tmp = Float64(2.0 / Float64(Float64((k ^ 3.0) / Float64(l / Float64(t_m * sin(k)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(k / Float64((l ^ 2.0) / (t_m ^ 3.0)))));
	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 <= 4.9e-27)
		tmp = 2.0 / (((k ^ 3.0) / (l / (t_m * sin(k)))) / l);
	else
		tmp = 2.0 / ((2.0 * k) * (k / ((l ^ 2.0) / (t_m ^ 3.0))));
	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, 4.9e-27], N[(2.0 / N[(N[(N[Power[k, 3.0], $MachinePrecision] / N[(l / N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(k / N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.89999999999999976e-27

   1. Initial program 58.4%

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

      1. associate-*l* 58.4%

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

      2. sqr-neg 58.4%

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

      3. sqr-neg 58.4%

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

      4. associate-/r* 62.9%

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

      5. distribute-rgt-in 62.9%

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

      6. unpow2 62.9%

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

      7. times-frac 41.5%

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

      8. sqr-neg 41.5%

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

      9. times-frac 62.9%

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

      10. unpow2 62.9%

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

      11. distribute-rgt-in 62.9%

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

   3. Simplified 62.9%

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

      1. associate-*r* 62.9%

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

      2. associate-*r* 61.9%

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

      3. *-commutative 61.9%

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

      4. associate-*l/ 62.4%

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

      5. associate-*r/ 62.4%

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

      6. *-commutative 62.4%

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

      7. associate-*r* 63.5%

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

   6. Applied egg-rr 63.5%

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

   7. Taylor expanded in k around 0 59.0%

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

      1. associate-/l* 58.9%

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

   9. Simplified 58.9%

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

   10. Taylor expanded in k around inf 66.7%

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

       1. associate-/l* 67.8%

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

       2. *-commutative 67.8%

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

   12. Simplified 67.8%

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

   if 4.89999999999999976e-27 < t

   1. Initial program 66.2%

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

      1. *-commutative 66.2%

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

      2. sqr-neg 66.2%

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

      3. *-commutative 66.2%

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

      4. associate-*l* 66.2%

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

      5. *-commutative 66.2%

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

      6. sqr-neg 66.2%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in k around 0 61.3%

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

      1. *-commutative 61.3%

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

   7. Simplified 61.3%

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

   8. Taylor expanded in k around 0 61.1%

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

      1. associate-/l* 61.2%

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

   10. Simplified 61.2%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.9 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{3}}{\frac{\ell}{t \cdot \sin k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 5.6 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{3}}{\frac{\ell}{t_m \cdot \sin k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k}{{\left(\frac{\ell}{{t_m}^{1.5}}\right)}^{2}}}\\ \end{array} \end{array} \]

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 5.6e-27)
    (/ 2.0 (/ (/ (pow k 3.0) (/ l (* t_m (sin k)))) l))
    (/ 2.0 (* (* 2.0 k) (/ k (pow (/ l (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 <= 5.6e-27) {
		tmp = 2.0 / ((pow(k, 3.0) / (l / (t_m * sin(k)))) / l);
	} else {
		tmp = 2.0 / ((2.0 * k) * (k / pow((l / 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 <= 5.6d-27) then
        tmp = 2.0d0 / (((k ** 3.0d0) / (l / (t_m * sin(k)))) / l)
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (k / ((l / (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 <= 5.6e-27) {
		tmp = 2.0 / ((Math.pow(k, 3.0) / (l / (t_m * Math.sin(k)))) / l);
	} else {
		tmp = 2.0 / ((2.0 * k) * (k / Math.pow((l / 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 <= 5.6e-27:
		tmp = 2.0 / ((math.pow(k, 3.0) / (l / (t_m * math.sin(k)))) / l)
	else:
		tmp = 2.0 / ((2.0 * k) * (k / math.pow((l / 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 <= 5.6e-27)
		tmp = Float64(2.0 / Float64(Float64((k ^ 3.0) / Float64(l / Float64(t_m * sin(k)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(k / (Float64(l / (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 <= 5.6e-27)
		tmp = 2.0 / (((k ^ 3.0) / (l / (t_m * sin(k)))) / l);
	else
		tmp = 2.0 / ((2.0 * k) * (k / ((l / (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, 5.6e-27], N[(2.0 / N[(N[(N[Power[k, 3.0], $MachinePrecision] / N[(l / N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[(k / N[Power[N[(l / N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.5999999999999999e-27

   1. Initial program 58.4%

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

      1. associate-*l* 58.4%

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

      2. sqr-neg 58.4%

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

      3. sqr-neg 58.4%

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

      4. associate-/r* 62.9%

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

      5. distribute-rgt-in 62.9%

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

      6. unpow2 62.9%

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

      7. times-frac 41.5%

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

      8. sqr-neg 41.5%

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

      9. times-frac 62.9%

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

      10. unpow2 62.9%

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

      11. distribute-rgt-in 62.9%

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

   3. Simplified 62.9%

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

      1. associate-*r* 62.9%

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

      2. associate-*r* 61.9%

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

      3. *-commutative 61.9%

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

      4. associate-*l/ 62.4%

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

      5. associate-*r/ 62.4%

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

      6. *-commutative 62.4%

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

      7. associate-*r* 63.5%

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

   6. Applied egg-rr 63.5%

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

   7. Taylor expanded in k around 0 59.0%

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

      1. associate-/l* 58.9%

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

   9. Simplified 58.9%

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

   10. Taylor expanded in k around inf 66.7%

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

       1. associate-/l* 67.8%

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

       2. *-commutative 67.8%

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

   12. Simplified 67.8%

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

   if 5.5999999999999999e-27 < t

   1. Initial program 66.2%

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

      1. *-commutative 66.2%

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

      2. sqr-neg 66.2%

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

      3. *-commutative 66.2%

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

      4. associate-*l* 66.2%

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

      5. *-commutative 66.2%

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

      6. sqr-neg 66.2%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in k around 0 61.3%

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

      1. *-commutative 61.3%

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

   7. Simplified 61.3%

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

   8. Taylor expanded in k around 0 61.1%

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

      1. associate-/l* 61.2%

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

   10. Simplified 61.2%

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

       1. add-sqr-sqrt 61.2%

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

       2. pow2 61.2%

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

       3. sqrt-div 61.2%

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

       4. unpow2 61.2%

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

       5. sqrt-prod 38.2%

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

       6. add-sqr-sqrt 63.3%

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

       7. sqrt-pow1 71.7%

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

       8. metadata-eval 71.7%

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

   12. Applied egg-rr 71.7%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.6 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{3}}{\frac{\ell}{t \cdot \sin k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \frac{k}{{\left(\frac{\ell}{{t}^{1.5}}\right)}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.3 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{3}}{\frac{\ell}{t_m \cdot \sin k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{k}}{{t_m}^{3} \cdot \frac{k}{{\ell}^{2}}}\\ \end{array} \end{array} \]

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 2.3e-30)
    (/ 2.0 (/ (/ (pow k 3.0) (/ l (* t_m (sin k)))) l))
    (/ (/ 1.0 k) (* (pow t_m 3.0) (/ k (pow l 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 <= 2.3e-30) {
		tmp = 2.0 / ((pow(k, 3.0) / (l / (t_m * sin(k)))) / l);
	} else {
		tmp = (1.0 / k) / (pow(t_m, 3.0) * (k / pow(l, 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 <= 2.3d-30) then
        tmp = 2.0d0 / (((k ** 3.0d0) / (l / (t_m * sin(k)))) / l)
    else
        tmp = (1.0d0 / k) / ((t_m ** 3.0d0) * (k / (l ** 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 <= 2.3e-30) {
		tmp = 2.0 / ((Math.pow(k, 3.0) / (l / (t_m * Math.sin(k)))) / l);
	} else {
		tmp = (1.0 / k) / (Math.pow(t_m, 3.0) * (k / Math.pow(l, 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 <= 2.3e-30:
		tmp = 2.0 / ((math.pow(k, 3.0) / (l / (t_m * math.sin(k)))) / l)
	else:
		tmp = (1.0 / k) / (math.pow(t_m, 3.0) * (k / math.pow(l, 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 <= 2.3e-30)
		tmp = Float64(2.0 / Float64(Float64((k ^ 3.0) / Float64(l / Float64(t_m * sin(k)))) / l));
	else
		tmp = Float64(Float64(1.0 / k) / Float64((t_m ^ 3.0) * Float64(k / (l ^ 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 <= 2.3e-30)
		tmp = 2.0 / (((k ^ 3.0) / (l / (t_m * sin(k)))) / l);
	else
		tmp = (1.0 / k) / ((t_m ^ 3.0) * (k / (l ^ 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, 2.3e-30], N[(2.0 / N[(N[(N[Power[k, 3.0], $MachinePrecision] / N[(l / N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(k / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.29999999999999984e-30

   1. Initial program 58.2%

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

      1. associate-*l* 58.2%

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

      2. sqr-neg 58.2%

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

      3. sqr-neg 58.2%

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

      4. associate-/r* 62.7%

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

      5. distribute-rgt-in 62.7%

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

      6. unpow2 62.7%

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

      7. times-frac 41.2%

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

      8. sqr-neg 41.2%

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

      9. times-frac 62.7%

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

      10. unpow2 62.7%

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

      11. distribute-rgt-in 62.7%

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

   3. Simplified 62.7%

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

      1. associate-*r* 62.7%

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

      2. associate-*r* 61.6%

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

      3. *-commutative 61.6%

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

      4. associate-*l/ 62.2%

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

      5. associate-*r/ 62.2%

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

      6. *-commutative 62.2%

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

      7. associate-*r* 63.3%

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

   6. Applied egg-rr 63.3%

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

   7. Taylor expanded in k around 0 58.8%

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

      1. associate-/l* 58.7%

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

   9. Simplified 58.7%

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

   10. Taylor expanded in k around inf 66.6%

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

       1. associate-/l* 67.6%

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

       2. *-commutative 67.6%

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

   12. Simplified 67.6%

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

   if 2.29999999999999984e-30 < t

   1. Initial program 66.7%

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

      1. *-commutative 66.7%

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

      2. sqr-neg 66.7%

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

      3. *-commutative 66.7%

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

      4. associate-*l* 66.7%

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

      5. *-commutative 66.7%

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

      6. sqr-neg 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in k around 0 60.6%

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

      1. *-commutative 60.6%

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

   7. Simplified 60.6%

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

   8. Taylor expanded in k around 0 60.3%

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

      1. associate-/l* 60.4%

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

   10. Simplified 60.4%

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

       1. expm1-log1p-u 60.3%

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

       2. expm1-udef 61.5%

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

       3. associate-/r/ 61.4%

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

   12. Applied egg-rr 61.4%

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

       1. expm1-def 61.5%

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

       2. expm1-log1p 61.7%

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

       3. *-commutative 61.7%

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot 2\right) \cdot \left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)}} \]

       4. associate-/r* 61.6%

          \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot 2}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}}} \]

       5. *-commutative 61.6%

          \[\leadsto \frac{\frac{2}{\color{blue}{2 \cdot k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       6. associate-/r* 61.6%

          \[\leadsto \frac{\color{blue}{\frac{\frac{2}{2}}{k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       7. metadata-eval 61.6%

          \[\leadsto \frac{\frac{\color{blue}{1}}{k}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       8. *-commutative 61.6%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

   14. Simplified 61.6%

       \[\leadsto \color{blue}{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{3}}{\frac{\ell}{t \cdot \sin k}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 2.75 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{4}}{\frac{\ell}{t_m}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{k}}{{t_m}^{3} \cdot \left(k \cdot {\ell}^{-2}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.75e-95)
    (/ 2.0 (/ (/ (pow k 4.0) (/ l t_m)) l))
    (/ (/ 1.0 k) (* (pow t_m 3.0) (* k (pow l -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 <= 2.75e-95) {
		tmp = 2.0 / ((pow(k, 4.0) / (l / t_m)) / l);
	} else {
		tmp = (1.0 / k) / (pow(t_m, 3.0) * (k * pow(l, -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 <= 2.75d-95) then
        tmp = 2.0d0 / (((k ** 4.0d0) / (l / t_m)) / l)
    else
        tmp = (1.0d0 / k) / ((t_m ** 3.0d0) * (k * (l ** (-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 <= 2.75e-95) {
		tmp = 2.0 / ((Math.pow(k, 4.0) / (l / t_m)) / l);
	} else {
		tmp = (1.0 / k) / (Math.pow(t_m, 3.0) * (k * Math.pow(l, -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 <= 2.75e-95:
		tmp = 2.0 / ((math.pow(k, 4.0) / (l / t_m)) / l)
	else:
		tmp = (1.0 / k) / (math.pow(t_m, 3.0) * (k * math.pow(l, -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 <= 2.75e-95)
		tmp = Float64(2.0 / Float64(Float64((k ^ 4.0) / Float64(l / t_m)) / l));
	else
		tmp = Float64(Float64(1.0 / k) / Float64((t_m ^ 3.0) * Float64(k * (l ^ -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 <= 2.75e-95)
		tmp = 2.0 / (((k ^ 4.0) / (l / t_m)) / l);
	else
		tmp = (1.0 / k) / ((t_m ^ 3.0) * (k * (l ^ -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, 2.75e-95], N[(2.0 / N[(N[(N[Power[k, 4.0], $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(k * N[Power[l, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.75000000000000001e-95

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 55.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 55.9%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 55.9%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 60.9%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 60.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 60.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 37.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 37.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 60.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 60.9%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 60.9%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 60.9%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 60.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      2. associate-*r* 59.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      3. *-commutative 59.7%

         \[\leadsto \frac{2}{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      4. associate-*l/ 60.3%

         \[\leadsto \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}}} \]

      5. associate-*r/ 60.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}} \]

      6. *-commutative 60.3%

         \[\leadsto \frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\tan k \cdot \sin k\right)}\right)}{\ell}} \]

      7. associate-*r* 61.5%

         \[\leadsto \frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\frac{{t}^{3}}{\ell} \cdot \tan k\right) \cdot \sin k\right)}}{\ell}} \]

   6. Applied egg-rr 61.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell} \cdot \tan k\right) \cdot \sin k\right)}{\ell}}} \]

   7. Taylor expanded in k around inf 72.8%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
   8. Step-by-step derivation

      1. associate-*r* 72.8%

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{\ell \cdot \cos k}}{\ell}} \]

      2. *-commutative 72.8%

         \[\leadsto \frac{2}{\frac{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\cos k \cdot \ell}}}{\ell}} \]

      3. times-frac 73.0%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell}}}{\ell}} \]

   9. Simplified 73.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell}}}{\ell}} \]

   10. Taylor expanded in k around 0 64.4%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4} \cdot t}{\ell}}}{\ell}} \]
   11. Step-by-step derivation

       1. associate-/l* 65.9%

          \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4}}{\frac{\ell}{t}}}}{\ell}} \]

   12. Simplified 65.9%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4}}{\frac{\ell}{t}}}}{\ell}} \]

   if 2.75000000000000001e-95 < t

   1. Initial program 69.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 69.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. sqr-neg 69.8%

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative 69.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* 69.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      5. *-commutative 69.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. sqr-neg 69.8%

         \[\leadsto \frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 60.2%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 60.2%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   7. Simplified 60.2%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Taylor expanded in k around 0 61.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(k \cdot 2\right)} \]
   9. Step-by-step derivation

      1. associate-/l* 61.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]

   10. Simplified 61.3%

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 61.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(k \cdot 2\right)}\right)\right)} \]

       2. expm1-udef 64.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(k \cdot 2\right)}\right)} - 1} \]

       3. associate-/r/ 64.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)} \cdot \left(k \cdot 2\right)}\right)} - 1 \]

   12. Applied egg-rr 64.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 63.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}\right)\right)} \]

       2. expm1-log1p 63.5%

          \[\leadsto \color{blue}{\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}} \]

       3. *-commutative 63.5%

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot 2\right) \cdot \left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)}} \]

       4. associate-/r* 63.4%

          \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot 2}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}}} \]

       5. *-commutative 63.4%

          \[\leadsto \frac{\frac{2}{\color{blue}{2 \cdot k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       6. associate-/r* 63.4%

          \[\leadsto \frac{\color{blue}{\frac{\frac{2}{2}}{k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       7. metadata-eval 63.4%

          \[\leadsto \frac{\frac{\color{blue}{1}}{k}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       8. *-commutative 63.4%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

   14. Simplified 63.4%

       \[\leadsto \color{blue}{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]
   15. Step-by-step derivation

       1. expm1-log1p-u 39.6%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({t}^{3} \cdot \frac{k}{{\ell}^{2}}\right)\right)}} \]

       2. expm1-udef 27.4%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{e^{\mathsf{log1p}\left({t}^{3} \cdot \frac{k}{{\ell}^{2}}\right)} - 1}} \]

       3. div-inv 27.4%

          \[\leadsto \frac{\frac{1}{k}}{e^{\mathsf{log1p}\left({t}^{3} \cdot \color{blue}{\left(k \cdot \frac{1}{{\ell}^{2}}\right)}\right)} - 1} \]

       4. pow-flip 27.4%

          \[\leadsto \frac{\frac{1}{k}}{e^{\mathsf{log1p}\left({t}^{3} \cdot \left(k \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)\right)} - 1} \]

       5. metadata-eval 27.4%

          \[\leadsto \frac{\frac{1}{k}}{e^{\mathsf{log1p}\left({t}^{3} \cdot \left(k \cdot {\ell}^{\color{blue}{-2}}\right)\right)} - 1} \]

   16. Applied egg-rr 27.4%

       \[\leadsto \frac{\frac{1}{k}}{\color{blue}{e^{\mathsf{log1p}\left({t}^{3} \cdot \left(k \cdot {\ell}^{-2}\right)\right)} - 1}} \]
   17. Step-by-step derivation

       1. expm1-def 39.6%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({t}^{3} \cdot \left(k \cdot {\ell}^{-2}\right)\right)\right)}} \]

       2. expm1-log1p 63.4%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{t}^{3} \cdot \left(k \cdot {\ell}^{-2}\right)}} \]

   18. Simplified 63.4%

       \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{t}^{3} \cdot \left(k \cdot {\ell}^{-2}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.75 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{4}}{\frac{\ell}{t}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{k}}{{t}^{3} \cdot \left(k \cdot {\ell}^{-2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;t_m \leq 1.5 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{4}}{\frac{\ell}{t_m}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{k}}{{t_m}^{3} \cdot \frac{k}{{\ell}^{2}}}\\ \end{array} \end{array} \]

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 1.5e-95)
    (/ 2.0 (/ (/ (pow k 4.0) (/ l t_m)) l))
    (/ (/ 1.0 k) (* (pow t_m 3.0) (/ k (pow l 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 <= 1.5e-95) {
		tmp = 2.0 / ((pow(k, 4.0) / (l / t_m)) / l);
	} else {
		tmp = (1.0 / k) / (pow(t_m, 3.0) * (k / pow(l, 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 <= 1.5d-95) then
        tmp = 2.0d0 / (((k ** 4.0d0) / (l / t_m)) / l)
    else
        tmp = (1.0d0 / k) / ((t_m ** 3.0d0) * (k / (l ** 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 <= 1.5e-95) {
		tmp = 2.0 / ((Math.pow(k, 4.0) / (l / t_m)) / l);
	} else {
		tmp = (1.0 / k) / (Math.pow(t_m, 3.0) * (k / Math.pow(l, 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 <= 1.5e-95:
		tmp = 2.0 / ((math.pow(k, 4.0) / (l / t_m)) / l)
	else:
		tmp = (1.0 / k) / (math.pow(t_m, 3.0) * (k / math.pow(l, 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 <= 1.5e-95)
		tmp = Float64(2.0 / Float64(Float64((k ^ 4.0) / Float64(l / t_m)) / l));
	else
		tmp = Float64(Float64(1.0 / k) / Float64((t_m ^ 3.0) * Float64(k / (l ^ 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 <= 1.5e-95)
		tmp = 2.0 / (((k ^ 4.0) / (l / t_m)) / l);
	else
		tmp = (1.0 / k) / ((t_m ^ 3.0) * (k / (l ^ 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, 1.5e-95], N[(2.0 / N[(N[(N[Power[k, 4.0], $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] / N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(k / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.5e-95

   1. Initial program 55.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 55.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. sqr-neg 55.9%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      3. sqr-neg 55.9%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      4. associate-/r* 60.9%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      5. distribute-rgt-in 60.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

      6. unpow2 60.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      7. times-frac 37.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      8. sqr-neg 37.5%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      9. times-frac 60.9%

         \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      10. unpow2 60.9%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

      11. distribute-rgt-in 60.9%

          \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 60.9%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 60.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      2. associate-*r* 59.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

      3. *-commutative 59.7%

         \[\leadsto \frac{2}{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      4. associate-*l/ 60.3%

         \[\leadsto \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}}} \]

      5. associate-*r/ 60.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}} \]

      6. *-commutative 60.3%

         \[\leadsto \frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\tan k \cdot \sin k\right)}\right)}{\ell}} \]

      7. associate-*r* 61.5%

         \[\leadsto \frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\frac{{t}^{3}}{\ell} \cdot \tan k\right) \cdot \sin k\right)}}{\ell}} \]

   6. Applied egg-rr 61.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell} \cdot \tan k\right) \cdot \sin k\right)}{\ell}}} \]

   7. Taylor expanded in k around inf 72.8%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
   8. Step-by-step derivation

      1. associate-*r* 72.8%

         \[\leadsto \frac{2}{\frac{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{\ell \cdot \cos k}}{\ell}} \]

      2. *-commutative 72.8%

         \[\leadsto \frac{2}{\frac{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\cos k \cdot \ell}}}{\ell}} \]

      3. times-frac 73.0%

         \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell}}}{\ell}} \]

   9. Simplified 73.0%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell}}}{\ell}} \]

   10. Taylor expanded in k around 0 64.4%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4} \cdot t}{\ell}}}{\ell}} \]
   11. Step-by-step derivation

       1. associate-/l* 65.9%

          \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4}}{\frac{\ell}{t}}}}{\ell}} \]

   12. Simplified 65.9%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4}}{\frac{\ell}{t}}}}{\ell}} \]

   if 1.5e-95 < t

   1. Initial program 69.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 69.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. sqr-neg 69.8%

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative 69.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      4. associate-*l* 69.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      5. *-commutative 69.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)}\right)} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

      6. sqr-neg 69.8%

         \[\leadsto \frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 60.2%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 60.2%

         \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   7. Simplified 60.2%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   8. Taylor expanded in k around 0 61.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(k \cdot 2\right)} \]
   9. Step-by-step derivation

      1. associate-/l* 61.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]

   10. Simplified 61.3%

       \[\leadsto \frac{2}{\color{blue}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}}} \cdot \left(k \cdot 2\right)} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 61.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(k \cdot 2\right)}\right)\right)} \]

       2. expm1-udef 64.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\frac{k}{\frac{{\ell}^{2}}{{t}^{3}}} \cdot \left(k \cdot 2\right)}\right)} - 1} \]

       3. associate-/r/ 64.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)} \cdot \left(k \cdot 2\right)}\right)} - 1 \]

   12. Applied egg-rr 64.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 63.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}\right)\right)} \]

       2. expm1-log1p 63.5%

          \[\leadsto \color{blue}{\frac{2}{\left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right) \cdot \left(k \cdot 2\right)}} \]

       3. *-commutative 63.5%

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot 2\right) \cdot \left(\frac{k}{{\ell}^{2}} \cdot {t}^{3}\right)}} \]

       4. associate-/r* 63.4%

          \[\leadsto \color{blue}{\frac{\frac{2}{k \cdot 2}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}}} \]

       5. *-commutative 63.4%

          \[\leadsto \frac{\frac{2}{\color{blue}{2 \cdot k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       6. associate-/r* 63.4%

          \[\leadsto \frac{\color{blue}{\frac{\frac{2}{2}}{k}}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       7. metadata-eval 63.4%

          \[\leadsto \frac{\frac{\color{blue}{1}}{k}}{\frac{k}{{\ell}^{2}} \cdot {t}^{3}} \]

       8. *-commutative 63.4%

          \[\leadsto \frac{\frac{1}{k}}{\color{blue}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]

   14. Simplified 63.4%

       \[\leadsto \color{blue}{\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{4}}{\frac{\ell}{t}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{k}}{{t}^{3} \cdot \frac{k}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \frac{2}{\frac{\frac{t_m \cdot {k}^{4}}{\ell}}{\ell}} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (/ (/ (* t_m (pow k 4.0)) l) l))))

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 * (2.0 / (((t_m * pow(k, 4.0)) / l) / l));
}

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 * (2.0d0 / (((t_m * (k ** 4.0d0)) / l) / l))
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 * (2.0 / (((t_m * Math.pow(k, 4.0)) / l) / l));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / (((t_m * math.pow(k, 4.0)) / l) / l))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 4.0)) / l) / l)))
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 * (2.0 / (((t_m * (k ^ 4.0)) / l) / l));
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[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 60.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   2. sqr-neg 60.5%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. sqr-neg 60.5%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   4. associate-/r* 65.2%

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   5. distribute-rgt-in 65.2%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

   6. unpow2 65.2%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   7. times-frac 46.0%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   8. sqr-neg 46.0%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   9. times-frac 65.2%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   10. unpow2 65.2%

       \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   11. distribute-rgt-in 65.2%

       \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 65.2%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r* 65.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

   2. associate-*r* 62.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. *-commutative 62.3%

      \[\leadsto \frac{2}{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   4. associate-*l/ 62.7%

      \[\leadsto \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}}} \]

   5. associate-*r/ 62.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}} \]

   6. *-commutative 62.7%

      \[\leadsto \frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\tan k \cdot \sin k\right)}\right)}{\ell}} \]

   7. associate-*r* 65.6%

      \[\leadsto \frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\frac{{t}^{3}}{\ell} \cdot \tan k\right) \cdot \sin k\right)}}{\ell}} \]

6. Applied egg-rr 65.6%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell} \cdot \tan k\right) \cdot \sin k\right)}{\ell}}} \]

7. Taylor expanded in k around inf 67.6%

   \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
8. Step-by-step derivation

   1. associate-*r* 67.6%

      \[\leadsto \frac{2}{\frac{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{\ell \cdot \cos k}}{\ell}} \]

   2. *-commutative 67.6%

      \[\leadsto \frac{2}{\frac{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\cos k \cdot \ell}}}{\ell}} \]

   3. times-frac 67.7%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell}}}{\ell}} \]

9. Simplified 67.7%

   \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell}}}{\ell}} \]

10. Taylor expanded in k around 0 60.8%

    \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4} \cdot t}{\ell}}}{\ell}} \]

11. Final simplification 60.8%

    \[\leadsto \frac{2}{\frac{\frac{t \cdot {k}^{4}}{\ell}}{\ell}} \]
12. Add Preprocessing

Alternative 17: 56.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \frac{2}{\frac{\frac{{k}^{4}}{\frac{\ell}{t_m}}}{\ell}} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (/ (/ (pow k 4.0) (/ l t_m)) l))))

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 * (2.0 / ((pow(k, 4.0) / (l / t_m)) / l));
}

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 * (2.0d0 / (((k ** 4.0d0) / (l / t_m)) / l))
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 * (2.0 / ((Math.pow(k, 4.0) / (l / t_m)) / l));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / ((math.pow(k, 4.0) / (l / t_m)) / l))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64(Float64((k ^ 4.0) / Float64(l / t_m)) / l)))
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 * (2.0 / (((k ^ 4.0) / (l / t_m)) / l));
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[(2.0 / N[(N[(N[Power[k, 4.0], $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.5%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 60.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   2. sqr-neg 60.5%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   3. sqr-neg 60.5%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   4. associate-/r* 65.2%

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]

   5. distribute-rgt-in 65.2%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k + 1 \cdot \tan k\right)}} \]

   6. unpow2 65.2%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   7. times-frac 46.0%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   8. sqr-neg 46.0%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \frac{k \cdot k}{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   9. times-frac 65.2%

      \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{\frac{k}{-t} \cdot \frac{k}{-t}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   10. unpow2 65.2%

       \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\left(1 + \color{blue}{{\left(\frac{k}{-t}\right)}^{2}}\right) \cdot \tan k + 1 \cdot \tan k\right)} \]

   11. distribute-rgt-in 65.2%

       \[\leadsto \frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1\right)\right)}} \]

3. Simplified 65.2%

   \[\leadsto \color{blue}{\frac{2}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r* 65.1%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

   2. associate-*r* 62.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. *-commutative 62.3%

      \[\leadsto \frac{2}{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   4. associate-*l/ 62.7%

      \[\leadsto \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}{\ell}}} \]

   5. associate-*r/ 62.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right)}{\ell}}} \]

   6. *-commutative 62.7%

      \[\leadsto \frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \color{blue}{\left(\tan k \cdot \sin k\right)}\right)}{\ell}} \]

   7. associate-*r* 65.6%

      \[\leadsto \frac{2}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\frac{{t}^{3}}{\ell} \cdot \tan k\right) \cdot \sin k\right)}}{\ell}} \]

6. Applied egg-rr 65.6%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell} \cdot \tan k\right) \cdot \sin k\right)}{\ell}}} \]

7. Taylor expanded in k around inf 67.6%

   \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}}{\ell}} \]
8. Step-by-step derivation

   1. associate-*r* 67.6%

      \[\leadsto \frac{2}{\frac{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{\ell \cdot \cos k}}{\ell}} \]

   2. *-commutative 67.6%

      \[\leadsto \frac{2}{\frac{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\cos k \cdot \ell}}}{\ell}} \]

   3. times-frac 67.7%

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell}}}{\ell}} \]

9. Simplified 67.7%

   \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{2} \cdot t}{\cos k} \cdot \frac{{\sin k}^{2}}{\ell}}}{\ell}} \]

10. Taylor expanded in k around 0 60.8%

    \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4} \cdot t}{\ell}}}{\ell}} \]
11. Step-by-step derivation

    1. associate-/l* 61.7%

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4}}{\frac{\ell}{t}}}}{\ell}} \]

12. Simplified 61.7%

    \[\leadsto \frac{2}{\frac{\color{blue}{\frac{{k}^{4}}{\frac{\ell}{t}}}}{\ell}} \]

13. Final simplification 61.7%

    \[\leadsto \frac{2}{\frac{\frac{{k}^{4}}{\frac{\ell}{t}}}{\ell}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(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))))