Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.6% → 89.3%

Time: 25.1s

Alternatives: 20

Speedup: 2.0×

• 27.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.15 \cdot 10^{-145}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t\_m \cdot {k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\right)}^{3}}\\ \end{array} \end{array} \]

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 1.15e-145)
    (/ 2.0 (/ (* (/ (pow (sin k) 2.0) (cos k)) (/ (* t_m (pow k 2.0)) l)) l))
    (/
     2.0
     (pow
      (*
       (* (* t_m (pow (cbrt l) -2.0)) (cbrt (sin k)))
       (cbrt (* (tan k) (+ 1.0 (+ 1.0 (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 <= 1.15e-145) {
		tmp = 2.0 / (((pow(sin(k), 2.0) / cos(k)) * ((t_m * pow(k, 2.0)) / l)) / l);
	} else {
		tmp = 2.0 / pow((((t_m * pow(cbrt(l), -2.0)) * cbrt(sin(k))) * cbrt((tan(k) * (1.0 + (1.0 + 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 <= 1.15e-145) {
		tmp = 2.0 / (((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((t_m * Math.pow(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) * (1.0 + (1.0 + 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 <= 1.15e-145)
		tmp = Float64(2.0 / Float64(Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(t_m * (k ^ 2.0)) / l)) / l));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * cbrt(sin(k))) * cbrt(Float64(tan(k) * Float64(1.0 + Float64(1.0 + (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, 1.15e-145], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(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] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.15000000000000004e-145

   1. Initial program 50.3%

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

      1. associate-/r* 57.8%

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

      2. +-commutative 57.8%

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

      3. associate-+r+ 57.8%

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

      4. metadata-eval 57.8%

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

      5. associate-*r* 57.8%

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

      6. *-commutative 57.8%

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

      7. associate-*l/ 58.3%

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

      8. associate-*r/ 58.4%

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

   4. Applied egg-rr 58.4%

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

   5. Taylor expanded in k around inf 72.3%

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

      1. associate-*r* 72.3%

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

      2. *-commutative 72.3%

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

      3. *-commutative 72.3%

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

      4. times-frac 72.6%

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

   7. Simplified 72.6%

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

   if 1.15000000000000004e-145 < t

   1. Initial program 67.9%

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

      1. associate-/r* 73.1%

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

      2. add-cube-cbrt 72.9%

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

      3. pow3 72.9%

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

      4. *-commutative 72.9%

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

      5. cbrt-prod 72.9%

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

      6. associate-/r* 67.9%

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

      7. cbrt-div 67.8%

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

      8. rem-cbrt-cube 72.5%

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

      9. cbrt-prod 84.7%

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

      10. pow2 84.7%

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

   4. Applied egg-rr 84.7%

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

      1. expm1-log1p-u 65.1%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)\right)\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. expm1-udef 40.4%

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

   6. Applied egg-rr 40.4%

      \[\leadsto \frac{2}{\left({\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)} - 1\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 65.1%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)\right)\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. expm1-log1p 84.7%

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

      3. *-commutative 84.7%

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

   8. Simplified 84.7%

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

      1. add-cube-cbrt 84.6%

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

      2. pow3 84.7%

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

   10. Applied egg-rr 88.9%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{-145}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t \cdot {k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left(1 + t\_2\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)\right)} \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(2 + t\_2\right)}{\frac{\ell}{\sin k \cdot \frac{{t\_m}^{3}}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t\_m \cdot {k}^{2}}{\ell}}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<=
         (/
          2.0
          (*
           (+ 1.0 (+ 1.0 t_2))
           (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l))))))
         2e+256)
      (/ 2.0 (/ (* (tan k) (+ 2.0 t_2)) (/ l (* (sin k) (/ (pow t_m 3.0) l)))))
      (/
       2.0
       (/ (* (/ (pow (sin k) 2.0) (cos k)) (/ (* t_m (pow k 2.0)) 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) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (1.0 + t_2)) * (tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l)))))) <= 2e+256) {
		tmp = 2.0 / ((tan(k) * (2.0 + t_2)) / (l / (sin(k) * (pow(t_m, 3.0) / l))));
	} else {
		tmp = 2.0 / (((pow(sin(k), 2.0) / cos(k)) * ((t_m * pow(k, 2.0)) / l)) / l);
	}
	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) :: t_2
    real(8) :: tmp
    t_2 = (k / t_m) ** 2.0d0
    if ((2.0d0 / ((1.0d0 + (1.0d0 + t_2)) * (tan(k) * (sin(k) * ((t_m ** 3.0d0) / (l * l)))))) <= 2d+256) then
        tmp = 2.0d0 / ((tan(k) * (2.0d0 + t_2)) / (l / (sin(k) * ((t_m ** 3.0d0) / l))))
    else
        tmp = 2.0d0 / ((((sin(k) ** 2.0d0) / cos(k)) * ((t_m * (k ** 2.0d0)) / l)) / l)
    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 t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (1.0 + t_2)) * (Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l)))))) <= 2e+256) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + t_2)) / (l / (Math.sin(k) * (Math.pow(t_m, 3.0) / l))));
	} else {
		tmp = 2.0 / (((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((t_m * Math.pow(k, 2.0)) / l)) / l);
	}
	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):
	t_2 = math.pow((k / t_m), 2.0)
	tmp = 0
	if (2.0 / ((1.0 + (1.0 + t_2)) * (math.tan(k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l)))))) <= 2e+256:
		tmp = 2.0 / ((math.tan(k) * (2.0 + t_2)) / (l / (math.sin(k) * (math.pow(t_m, 3.0) / l))))
	else:
		tmp = 2.0 / (((math.pow(math.sin(k), 2.0) / math.cos(k)) * ((t_m * math.pow(k, 2.0)) / l)) / l)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + t_2)) * Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))))) <= 2e+256)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + t_2)) / Float64(l / Float64(sin(k) * Float64((t_m ^ 3.0) / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(t_m * (k ^ 2.0)) / l)) / l));
	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)
	t_2 = (k / t_m) ^ 2.0;
	tmp = 0.0;
	if ((2.0 / ((1.0 + (1.0 + t_2)) * (tan(k) * (sin(k) * ((t_m ^ 3.0) / (l * l)))))) <= 2e+256)
		tmp = 2.0 / ((tan(k) * (2.0 + t_2)) / (l / (sin(k) * ((t_m ^ 3.0) / l))));
	else
		tmp = 2.0 / ((((sin(k) ^ 2.0) / cos(k)) * ((t_m * (k ^ 2.0)) / l)) / l);
	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_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(1.0 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+256], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision] / N[(l / N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 2.0000000000000001e256

   1. Initial program 85.1%

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

      1. associate-/r* 89.2%

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

      2. +-commutative 89.2%

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

      3. associate-+r+ 89.2%

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

      4. metadata-eval 89.2%

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

      5. associate-*r* 89.2%

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

      6. *-commutative 89.2%

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

      7. associate-*l/ 91.3%

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

      8. associate-*r/ 89.9%

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

   4. Applied egg-rr 89.9%

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

      1. expm1-log1p-u 44.6%

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

      2. expm1-udef 22.8%

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

      3. associate-/l* 23.4%

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

      4. *-commutative 23.4%

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

   6. Applied egg-rr 23.4%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{\sin k \cdot \frac{{t}^{3}}{\ell}}}\right)} - 1}} \]
   7. Step-by-step derivation

      1. expm1-def 47.4%

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

      2. expm1-log1p 92.0%

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

   8. Simplified 92.0%

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

   if 2.0000000000000001e256 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 25.9%

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

      1. associate-/r* 35.3%

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

      2. +-commutative 35.3%

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

      3. associate-+r+ 35.3%

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

      4. metadata-eval 35.3%

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

      5. associate-*r* 35.3%

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

      6. *-commutative 35.3%

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

      7. associate-*l/ 35.3%

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

      8. associate-*r/ 36.2%

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

   4. Applied egg-rr 36.2%

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

   5. Taylor expanded in k around inf 67.4%

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

      1. associate-*r* 67.4%

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

      2. *-commutative 67.4%

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

      3. *-commutative 67.4%

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

      4. times-frac 67.8%

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

   7. Simplified 67.8%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{\sin k \cdot \frac{{t}^{3}}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t \cdot {k}^{2}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.15 \cdot 10^{-131}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t\_m \cdot {k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \end{array} \]

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 2.15e-131)
    (/ 2.0 (/ (* (/ (pow (sin k) 2.0) (cos k)) (/ (* t_m (pow k 2.0)) l)) l))
    (/
     2.0
     (*
      (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0)))
      (* (tan k) (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.15e-131) {
		tmp = 2.0 / (((pow(sin(k), 2.0) / cos(k)) * ((t_m * pow(k, 2.0)) / l)) / l);
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + pow((k / t_m), 2.0))) * (tan(k) * pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0)));
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.15e-131) {
		tmp = 2.0 / (((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((t_m * Math.pow(k, 2.0)) / l)) / l);
	} else {
		tmp = 2.0 / ((1.0 + (1.0 + Math.pow((k / t_m), 2.0))) * (Math.tan(k) * Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0)));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.15e-131)
		tmp = Float64(2.0 / Float64(Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(t_m * (k ^ 2.0)) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0))) * Float64(tan(k) * (Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.15e-131], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.15000000000000009e-131

   1. Initial program 49.7%

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

      1. associate-/r* 57.1%

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

      2. +-commutative 57.1%

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

      3. associate-+r+ 57.1%

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

      4. metadata-eval 57.1%

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

      5. associate-*r* 57.1%

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

      6. *-commutative 57.1%

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

      7. associate-*l/ 57.7%

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

      8. associate-*r/ 57.7%

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

   4. Applied egg-rr 57.7%

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

   5. Taylor expanded in k around inf 71.6%

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

      1. associate-*r* 71.6%

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

      2. *-commutative 71.6%

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

      3. *-commutative 71.6%

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

      4. times-frac 71.8%

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

   7. Simplified 71.8%

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

   if 2.15000000000000009e-131 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/r* 74.8%

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

      2. add-cube-cbrt 74.6%

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

      3. pow3 74.6%

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

      4. *-commutative 74.6%

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

      5. cbrt-prod 74.6%

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

      6. associate-/r* 69.4%

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

      7. cbrt-div 69.4%

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

      8. rem-cbrt-cube 74.2%

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

      9. cbrt-prod 86.6%

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

      10. pow2 86.6%

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

   4. Applied egg-rr 86.6%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{-131}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t \cdot {k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.4 \cdot 10^{-130}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t\_m \cdot {k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot {\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right)}^{3}\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

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 1.4e-130)
    (/ 2.0 (/ (* (/ (pow (sin k) 2.0) (cos k)) (/ (* t_m (pow k 2.0)) l)) l))
    (/
     2.0
     (*
      (* (tan k) (pow (* (* t_m (pow (cbrt l) -2.0)) (cbrt (sin k))) 3.0))
      (+ 1.0 (+ 1.0 (pow (/ k t_m) 2.0))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.4e-130) {
		tmp = 2.0 / (((pow(sin(k), 2.0) / cos(k)) * ((t_m * pow(k, 2.0)) / l)) / l);
	} else {
		tmp = 2.0 / ((tan(k) * pow(((t_m * pow(cbrt(l), -2.0)) * cbrt(sin(k))), 3.0)) * (1.0 + (1.0 + pow((k / t_m), 2.0))));
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.4e-130) {
		tmp = 2.0 / (((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((t_m * Math.pow(k, 2.0)) / l)) / l);
	} else {
		tmp = 2.0 / ((Math.tan(k) * Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt(Math.sin(k))), 3.0)) * (1.0 + (1.0 + Math.pow((k / t_m), 2.0))));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.4e-130)
		tmp = Float64(2.0 / Float64(Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(t_m * (k ^ 2.0)) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * cbrt(sin(k))) ^ 3.0)) * Float64(1.0 + Float64(1.0 + (Float64(k / t_m) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.4e-130], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $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] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.40000000000000008e-130

   1. Initial program 49.7%

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

      1. associate-/r* 57.1%

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

      2. +-commutative 57.1%

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

      3. associate-+r+ 57.1%

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

      4. metadata-eval 57.1%

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

      5. associate-*r* 57.1%

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

      6. *-commutative 57.1%

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

      7. associate-*l/ 57.7%

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

      8. associate-*r/ 57.7%

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

   4. Applied egg-rr 57.7%

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

   5. Taylor expanded in k around inf 71.6%

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

      1. associate-*r* 71.6%

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

      2. *-commutative 71.6%

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

      3. *-commutative 71.6%

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

      4. times-frac 71.8%

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

   7. Simplified 71.8%

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

   if 1.40000000000000008e-130 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/r* 74.8%

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

      2. add-cube-cbrt 74.6%

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

      3. pow3 74.6%

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

      4. *-commutative 74.6%

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

      5. cbrt-prod 74.6%

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

      6. associate-/r* 69.4%

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

      7. cbrt-div 69.4%

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

      8. rem-cbrt-cube 74.2%

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

      9. cbrt-prod 86.6%

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

      10. pow2 86.6%

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

   4. Applied egg-rr 86.6%

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

      1. expm1-log1p-u 66.6%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)\right)\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. expm1-udef 41.4%

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

   6. Applied egg-rr 41.3%

      \[\leadsto \frac{2}{\left({\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)} - 1\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 66.6%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)\right)\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. expm1-log1p 86.6%

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

      3. *-commutative 86.6%

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

   8. Simplified 86.6%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-130}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t \cdot {k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot {\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right)}^{3}\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t\_m \cdot {k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 1.48 \cdot 10^{+134}:\\ \;\;\;\;\frac{2}{\frac{t\_2 \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(t\_2 \cdot {\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))
   (*
    t_s
    (if (<= t_m 5.6e-48)
      (/ 2.0 (/ (* (/ (pow (sin k) 2.0) (cos k)) (/ (* t_m (pow k 2.0)) l)) l))
      (if (<= t_m 1.48e+134)
        (/ 2.0 (/ (* t_2 (pow (* (cbrt (sin k)) (/ t_m (cbrt l))) 3.0)) l))
        (/ 2.0 (* (sin k) (* t_2 (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 t_2 = tan(k) * (2.0 + pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 5.6e-48) {
		tmp = 2.0 / (((pow(sin(k), 2.0) / cos(k)) * ((t_m * pow(k, 2.0)) / l)) / l);
	} else if (t_m <= 1.48e+134) {
		tmp = 2.0 / ((t_2 * pow((cbrt(sin(k)) * (t_m / cbrt(l))), 3.0)) / l);
	} else {
		tmp = 2.0 / (sin(k) * (t_2 * 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 t_2 = Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0));
	double tmp;
	if (t_m <= 5.6e-48) {
		tmp = 2.0 / (((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((t_m * Math.pow(k, 2.0)) / l)) / l);
	} else if (t_m <= 1.48e+134) {
		tmp = 2.0 / ((t_2 * Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.cbrt(l))), 3.0)) / l);
	} else {
		tmp = 2.0 / (Math.sin(k) * (t_2 * 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)
	t_2 = Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))
	tmp = 0.0
	if (t_m <= 5.6e-48)
		tmp = Float64(2.0 / Float64(Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(t_m * (k ^ 2.0)) / l)) / l));
	elseif (t_m <= 1.48e+134)
		tmp = Float64(2.0 / Float64(Float64(t_2 * (Float64(cbrt(sin(k)) * Float64(t_m / cbrt(l))) ^ 3.0)) / l));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(t_2 * (Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 5.6e-48], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.48e+134], N[(2.0 / N[(N[(t$95$2 * N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(t$95$2 * 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 < 5.6000000000000001e-48

   1. Initial program 51.3%

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

      1. associate-/r* 58.9%

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

      2. +-commutative 58.9%

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

      3. associate-+r+ 58.9%

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

      4. metadata-eval 58.9%

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

      5. associate-*r* 58.9%

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

      6. *-commutative 58.9%

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

      7. associate-*l/ 59.9%

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

      8. associate-*r/ 58.9%

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

   4. Applied egg-rr 58.9%

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

   5. Taylor expanded in k around inf 71.5%

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

      1. associate-*r* 71.5%

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

      2. *-commutative 71.5%

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

      3. *-commutative 71.5%

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

      4. times-frac 71.8%

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

   7. Simplified 71.8%

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

   if 5.6000000000000001e-48 < t < 1.48000000000000009e134

   1. Initial program 72.7%

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

      1. associate-/r* 76.9%

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

      2. +-commutative 76.9%

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

      3. associate-+r+ 76.9%

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

      4. metadata-eval 76.9%

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

      5. associate-*r* 76.9%

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

      6. *-commutative 76.9%

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

      7. associate-*l/ 79.0%

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

      8. associate-*r/ 81.2%

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

   4. Applied egg-rr 81.2%

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

      1. add-cube-cbrt 81.2%

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

      2. pow3 81.2%

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

      3. *-commutative 81.2%

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

      4. cbrt-prod 81.0%

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

      5. cbrt-div 80.8%

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

      6. unpow3 80.9%

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

      7. add-cbrt-cube 87.2%

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

   6. Applied egg-rr 87.2%

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

   if 1.48000000000000009e134 < t

   1. Initial program 65.3%

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

      1. associate-/r* 69.3%

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

      2. add-cube-cbrt 69.3%

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

      3. pow3 69.3%

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

      4. *-commutative 69.3%

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

      5. cbrt-prod 69.3%

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

      6. associate-/r* 65.3%

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

      7. cbrt-div 65.3%

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

      8. rem-cbrt-cube 74.8%

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

      9. cbrt-prod 98.0%

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

      10. pow2 98.0%

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

   4. Applied egg-rr 98.0%

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

      1. expm1-log1p-u 55.1%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)\right)\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. expm1-udef 50.7%

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

   6. Applied egg-rr 50.6%

      \[\leadsto \frac{2}{\left({\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)} - 1\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 54.9%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)\right)\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. expm1-log1p 98.0%

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

      3. *-commutative 98.0%

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

   8. Simplified 98.0%

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

      1. distribute-lft-in 98.0%

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

      2. *-commutative 98.0%

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

      3. unpow-prod-down 86.0%

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

      4. pow3 85.9%

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

      5. add-cube-cbrt 86.1%

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

   10. Applied egg-rr 85.9%

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

       1. distribute-lft-out 85.9%

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

       2. +-commutative 85.9%

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

       3. associate-*r* 85.9%

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

       4. associate-*l* 85.9%

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

       5. associate-+r+ 85.9%

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

       6. metadata-eval 85.9%

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

   12. Simplified 85.9%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t \cdot {k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 1.48 \cdot 10^{+134}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 57000000:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t\_m \cdot {k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 1.15 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}{\frac{\ell}{\sin k \cdot \frac{{t\_m}^{3}}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{2 \cdot k}\right)}^{3}}\\ \end{array} \end{array} \]

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 57000000.0)
    (/ 2.0 (/ (* (/ (pow (sin k) 2.0) (cos k)) (/ (* t_m (pow k 2.0)) l)) l))
    (if (<= t_m 1.15e+102)
      (/
       2.0
       (/
        (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
        (/ l (* (sin k) (/ (pow t_m 3.0) l)))))
      (/
       2.0
       (pow
        (* (* (* t_m (pow (cbrt l) -2.0)) (cbrt (sin k))) (cbrt (* 2.0 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 <= 57000000.0) {
		tmp = 2.0 / (((pow(sin(k), 2.0) / cos(k)) * ((t_m * pow(k, 2.0)) / l)) / l);
	} else if (t_m <= 1.15e+102) {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) / (l / (sin(k) * (pow(t_m, 3.0) / l))));
	} else {
		tmp = 2.0 / pow((((t_m * pow(cbrt(l), -2.0)) * cbrt(sin(k))) * cbrt((2.0 * 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 <= 57000000.0) {
		tmp = 2.0 / (((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((t_m * Math.pow(k, 2.0)) / l)) / l);
	} else if (t_m <= 1.15e+102) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) / (l / (Math.sin(k) * (Math.pow(t_m, 3.0) / l))));
	} else {
		tmp = 2.0 / Math.pow((((t_m * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt(Math.sin(k))) * Math.cbrt((2.0 * 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 <= 57000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(t_m * (k ^ 2.0)) / l)) / l));
	elseif (t_m <= 1.15e+102)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) / Float64(l / Float64(sin(k) * Float64((t_m ^ 3.0) / l)))));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * cbrt(sin(k))) * cbrt(Float64(2.0 * 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, 57000000.0], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.15e+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[(l / N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(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] * N[Power[N[(2.0 * k), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.7e7

   1. Initial program 53.4%

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

      1. associate-/r* 60.4%

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

      2. +-commutative 60.4%

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

      3. associate-+r+ 60.4%

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

      4. metadata-eval 60.4%

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

      5. associate-*r* 60.4%

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

      6. *-commutative 60.4%

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

      7. associate-*l/ 61.3%

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

      8. associate-*r/ 60.9%

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

   4. Applied egg-rr 60.9%

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

   5. Taylor expanded in k around inf 72.5%

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

      1. associate-*r* 72.5%

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

      2. *-commutative 72.5%

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

      3. *-commutative 72.5%

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

      4. times-frac 72.7%

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

   7. Simplified 72.7%

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

   if 5.7e7 < t < 1.1499999999999999e102

   1. Initial program 77.1%

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

      1. associate-/r* 86.3%

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

      2. +-commutative 86.3%

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

      3. associate-+r+ 86.3%

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

      4. metadata-eval 86.3%

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

      5. associate-*r* 86.4%

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

      6. *-commutative 86.4%

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

      7. associate-*l/ 91.0%

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

      8. associate-*r/ 90.8%

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

   4. Applied egg-rr 90.8%

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

      1. expm1-log1p-u 56.7%

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

      2. expm1-udef 47.3%

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

      3. associate-/l* 47.3%

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

      4. *-commutative 47.3%

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

   6. Applied egg-rr 47.3%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{\sin k \cdot \frac{{t}^{3}}{\ell}}}\right)} - 1}} \]
   7. Step-by-step derivation

      1. expm1-def 61.3%

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

      2. expm1-log1p 95.0%

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

   8. Simplified 95.0%

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

   if 1.1499999999999999e102 < t

   1. Initial program 62.1%

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

      1. associate-/r* 65.0%

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

      2. add-cube-cbrt 65.0%

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

      3. pow3 65.0%

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

      4. *-commutative 65.0%

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

      5. cbrt-prod 65.0%

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

      6. associate-/r* 62.1%

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

      7. cbrt-div 62.1%

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

      8. rem-cbrt-cube 75.4%

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

      9. cbrt-prod 92.6%

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

      10. pow2 92.6%

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

   4. Applied egg-rr 92.6%

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

      1. expm1-log1p-u 50.6%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)\right)\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. expm1-udef 44.2%

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

   6. Applied egg-rr 44.1%

      \[\leadsto \frac{2}{\left({\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)} - 1\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 50.5%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)\right)\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. expm1-log1p 92.5%

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

      3. *-commutative 92.5%

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

   8. Simplified 92.5%

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

      1. add-cube-cbrt 92.4%

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

      2. pow3 92.4%

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

   10. Applied egg-rr 98.1%

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

   11. Taylor expanded in k around 0 95.0%

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

       1. *-commutative 95.0%

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

   13. Simplified 95.0%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 57000000:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t \cdot {k}^{2}}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{\sin k \cdot \frac{{t}^{3}}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{2 \cdot k}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1950000000000:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t\_m \cdot {k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot {\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \end{array} \]

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 1950000000000.0)
    (/ 2.0 (/ (* (/ (pow (sin k) 2.0) (cos k)) (/ (* t_m (pow k 2.0)) l)) l))
    (/
     2.0
     (*
      (sin k)
      (*
       (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
       (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 <= 1950000000000.0) {
		tmp = 2.0 / (((pow(sin(k), 2.0) / cos(k)) * ((t_m * pow(k, 2.0)) / l)) / l);
	} else {
		tmp = 2.0 / (sin(k) * ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * 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 <= 1950000000000.0) {
		tmp = 2.0 / (((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((t_m * Math.pow(k, 2.0)) / l)) / l);
	} else {
		tmp = 2.0 / (Math.sin(k) * ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * 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 <= 1950000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(t_m * (k ^ 2.0)) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * (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, 1950000000000.0], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * 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[(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 < 1.95e12

   1. Initial program 53.4%

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

      1. associate-/r* 60.3%

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

      2. +-commutative 60.3%

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

      3. associate-+r+ 60.3%

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

      4. metadata-eval 60.3%

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

      5. associate-*r* 60.3%

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

      6. *-commutative 60.3%

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

      7. associate-*l/ 61.6%

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

      8. associate-*r/ 60.8%

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

   4. Applied egg-rr 60.8%

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

   5. Taylor expanded in k around inf 72.3%

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

      1. associate-*r* 72.3%

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

      2. *-commutative 72.3%

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

      3. *-commutative 72.3%

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

      4. times-frac 72.5%

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

   7. Simplified 72.5%

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

   if 1.95e12 < t

   1. Initial program 68.9%

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

      1. associate-/r* 74.5%

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

      2. add-cube-cbrt 74.5%

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

      3. pow3 74.5%

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

      4. *-commutative 74.5%

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

      5. cbrt-prod 74.5%

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

      6. associate-/r* 68.9%

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

      7. cbrt-div 68.8%

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

      8. rem-cbrt-cube 77.0%

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

      9. cbrt-prod 91.3%

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

      10. pow2 91.3%

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

   4. Applied egg-rr 91.3%

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

      1. expm1-log1p-u 59.2%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)\right)\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. expm1-udef 43.3%

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

   6. Applied egg-rr 43.3%

      \[\leadsto \frac{2}{\left({\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)} - 1\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 59.1%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)\right)\right)}}^{3} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      2. expm1-log1p 91.3%

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

      3. *-commutative 91.3%

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

   8. Simplified 91.3%

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

      1. distribute-lft-in 91.3%

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

      2. *-commutative 91.3%

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

      3. unpow-prod-down 83.9%

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

      4. pow3 83.9%

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

      5. add-cube-cbrt 84.0%

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

   10. Applied egg-rr 84.0%

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

       1. distribute-lft-out 84.0%

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

       2. +-commutative 84.0%

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

       3. associate-*r* 84.0%

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

       4. associate-*l* 84.0%

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

       5. associate-+r+ 84.0%

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

       6. metadata-eval 84.0%

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

   12. Simplified 84.0%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1950000000000:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t \cdot {k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1900000000000:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t\_m \cdot {k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

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 1900000000000.0)
    (/ 2.0 (/ (* (/ (pow (sin k) 2.0) (cos k)) (/ (* t_m (pow k 2.0)) l)) l))
    (/
     2.0
     (*
      (+ 2.0 (pow (/ k t_m) 2.0))
      (* (tan k) (* (sin k) (pow (/ (pow t_m 1.5) 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 <= 1900000000000.0) {
		tmp = 2.0 / (((pow(sin(k), 2.0) / cos(k)) * ((t_m * pow(k, 2.0)) / l)) / l);
	} else {
		tmp = 2.0 / ((2.0 + pow((k / t_m), 2.0)) * (tan(k) * (sin(k) * pow((pow(t_m, 1.5) / 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 <= 1900000000000.0d0) then
        tmp = 2.0d0 / ((((sin(k) ** 2.0d0) / cos(k)) * ((t_m * (k ** 2.0d0)) / l)) / l)
    else
        tmp = 2.0d0 / ((2.0d0 + ((k / t_m) ** 2.0d0)) * (tan(k) * (sin(k) * (((t_m ** 1.5d0) / 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 <= 1900000000000.0) {
		tmp = 2.0 / (((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((t_m * Math.pow(k, 2.0)) / l)) / l);
	} else {
		tmp = 2.0 / ((2.0 + Math.pow((k / t_m), 2.0)) * (Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / 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 <= 1900000000000.0:
		tmp = 2.0 / (((math.pow(math.sin(k), 2.0) / math.cos(k)) * ((t_m * math.pow(k, 2.0)) / l)) / l)
	else:
		tmp = 2.0 / ((2.0 + math.pow((k / t_m), 2.0)) * (math.tan(k) * (math.sin(k) * math.pow((math.pow(t_m, 1.5) / 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 <= 1900000000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(t_m * (k ^ 2.0)) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 + (Float64(k / t_m) ^ 2.0)) * Float64(tan(k) * Float64(sin(k) * (Float64((t_m ^ 1.5) / 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 <= 1900000000000.0)
		tmp = 2.0 / ((((sin(k) ^ 2.0) / cos(k)) * ((t_m * (k ^ 2.0)) / l)) / l);
	else
		tmp = 2.0 / ((2.0 + ((k / t_m) ^ 2.0)) * (tan(k) * (sin(k) * (((t_m ^ 1.5) / 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, 1900000000000.0], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.9e12

   1. Initial program 53.4%

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

      1. associate-/r* 60.3%

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

      2. +-commutative 60.3%

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

      3. associate-+r+ 60.3%

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

      4. metadata-eval 60.3%

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

      5. associate-*r* 60.3%

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

      6. *-commutative 60.3%

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

      7. associate-*l/ 61.6%

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

      8. associate-*r/ 60.8%

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

   4. Applied egg-rr 60.8%

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

   5. Taylor expanded in k around inf 72.3%

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

      1. associate-*r* 72.3%

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

      2. *-commutative 72.3%

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

      3. *-commutative 72.3%

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

      4. times-frac 72.5%

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

   7. Simplified 72.5%

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

   if 1.9e12 < t

   1. Initial program 68.9%

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

      1. associate-/r* 74.5%

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

      2. add-sqr-sqrt 36.5%

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

      3. pow2 36.5%

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

      4. *-commutative 36.5%

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

      5. sqrt-prod 36.5%

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

      6. associate-/r* 34.4%

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

      7. sqrt-div 34.4%

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

      8. sqrt-pow1 36.4%

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

      9. metadata-eval 36.4%

         \[\leadsto \frac{2}{\left({\left(\sqrt{\sin k} \cdot \frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      10. sqrt-prod 26.3%

          \[\leadsto \frac{2}{\left({\left(\sqrt{\sin k} \cdot \frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

      11. add-sqr-sqrt 38.4%

          \[\leadsto \frac{2}{\left({\left(\sqrt{\sin k} \cdot \frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   4. Applied egg-rr 38.4%

      \[\leadsto \frac{2}{\left(\color{blue}{{\left(\sqrt{\sin k} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   5. Step-by-step derivation

      1. *-commutative 38.4%

         \[\leadsto \frac{2}{\left({\color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}}^{2} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   6. Simplified 38.4%

      \[\leadsto \frac{2}{\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in 38.4%

         \[\leadsto \frac{2}{\color{blue}{\left({\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2} \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left({\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2} \cdot \tan k\right) \cdot 1}} \]

      2. *-commutative 38.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}\right)} \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left({\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2} \cdot \tan k\right) \cdot 1} \]

      3. *-commutative 38.4%

         \[\leadsto \frac{2}{\left(\tan k \cdot {\color{blue}{\left(\sqrt{\sin k} \cdot \frac{{t}^{1.5}}{\ell}\right)}}^{2}\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left({\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2} \cdot \tan k\right) \cdot 1} \]

      4. unpow-prod-down 36.5%

         \[\leadsto \frac{2}{\left(\tan k \cdot \color{blue}{\left({\left(\sqrt{\sin k}\right)}^{2} \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left({\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2} \cdot \tan k\right) \cdot 1} \]

      5. pow2 36.5%

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\color{blue}{\left(\sqrt{\sin k} \cdot \sqrt{\sin k}\right)} \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left({\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2} \cdot \tan k\right) \cdot 1} \]

      6. add-sqr-sqrt 36.5%

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\color{blue}{\sin k} \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left({\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2} \cdot \tan k\right) \cdot 1} \]

   8. Applied egg-rr 80.4%

      \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot 1}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 80.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      2. +-commutative 80.4%

         \[\leadsto \frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      3. *-commutative 80.4%

         \[\leadsto \frac{2}{\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}} \]

      4. associate-+r+ 80.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)} \]

      5. metadata-eval 80.4%

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

   10. Simplified 80.4%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1900000000000:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t \cdot {k}^{2}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{\ell}{\sqrt{2}}} \cdot \sqrt{{t\_m}^{3}}\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m}\right)}^{3}}{{k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t\_m \cdot {k}^{2}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.1e-146)
    (/ 2.0 (pow (* (/ k (/ l (sqrt 2.0))) (sqrt (pow t_m 3.0))) 2.0))
    (if (<= k 2.8e-25)
      (/ (pow (/ (pow (cbrt l) 2.0) t_m) 3.0) (pow k 2.0))
      (/
       2.0
       (/ (* (/ (pow (sin k) 2.0) (cos k)) (/ (* t_m (pow k 2.0)) 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) {
	double tmp;
	if (k <= 1.1e-146) {
		tmp = 2.0 / pow(((k / (l / sqrt(2.0))) * sqrt(pow(t_m, 3.0))), 2.0);
	} else if (k <= 2.8e-25) {
		tmp = pow((pow(cbrt(l), 2.0) / t_m), 3.0) / pow(k, 2.0);
	} else {
		tmp = 2.0 / (((pow(sin(k), 2.0) / cos(k)) * ((t_m * pow(k, 2.0)) / l)) / l);
	}
	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 (k <= 1.1e-146) {
		tmp = 2.0 / Math.pow(((k / (l / Math.sqrt(2.0))) * Math.sqrt(Math.pow(t_m, 3.0))), 2.0);
	} else if (k <= 2.8e-25) {
		tmp = Math.pow((Math.pow(Math.cbrt(l), 2.0) / t_m), 3.0) / Math.pow(k, 2.0);
	} else {
		tmp = 2.0 / (((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * ((t_m * Math.pow(k, 2.0)) / l)) / l);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.1e-146)
		tmp = Float64(2.0 / (Float64(Float64(k / Float64(l / sqrt(2.0))) * sqrt((t_m ^ 3.0))) ^ 2.0));
	elseif (k <= 2.8e-25)
		tmp = Float64((Float64((cbrt(l) ^ 2.0) / t_m) ^ 3.0) / (k ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(Float64(t_m * (k ^ 2.0)) / l)) / l));
	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[k, 1.1e-146], N[(2.0 / N[Power[N[(N[(k / N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[t$95$m, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.8e-25], N[(N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.1e-146

   1. Initial program 59.6%

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

      1. associate-/r* 68.0%

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

      2. +-commutative 68.0%

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

      3. associate-+r+ 68.0%

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

      4. metadata-eval 68.0%

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

      5. associate-*r* 68.0%

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

      6. *-commutative 68.0%

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

      7. associate-*l/ 69.9%

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

      8. associate-*r/ 69.3%

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

   4. Applied egg-rr 69.3%

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

      1. add-sqr-sqrt 30.7%

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

      2. pow2 30.7%

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

      3. associate-/l* 32.5%

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

      4. *-commutative 32.5%

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

   6. Applied egg-rr 32.5%

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

   7. Taylor expanded in k around 0 33.2%

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

      1. associate-/l* 33.2%

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

   9. Simplified 33.2%

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

   if 1.1e-146 < k < 2.79999999999999988e-25

   1. Initial program 58.8%

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

      1. associate-/r* 58.8%

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

      2. sqr-neg 58.8%

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

      3. associate-*l* 58.8%

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

      4. sqr-neg 58.8%

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

      5. associate-/r* 59.7%

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

      6. associate-+l+ 59.7%

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

      7. unpow2 59.7%

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

      8. times-frac 50.9%

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

      9. sqr-neg 50.9%

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

      10. times-frac 59.7%

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

      11. unpow2 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in k around 0 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-/r* 63.3%

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

   7. Simplified 63.3%

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

      1. add-cube-cbrt 63.4%

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

      2. pow2 63.4%

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

      3. cbrt-div 63.2%

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

      4. unpow2 63.2%

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

      5. cbrt-prod 63.2%

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

      6. unpow2 63.2%

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

      7. unpow3 63.2%

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

      8. add-cbrt-cube 63.2%

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

      9. cbrt-div 63.2%

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

      10. unpow2 63.2%

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

      11. cbrt-prod 64.1%

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

      12. unpow2 64.1%

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

      13. unpow3 64.1%

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

      14. add-cbrt-cube 80.2%

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

   9. Applied egg-rr 80.2%

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

       1. pow-plus 80.3%

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

       2. metadata-eval 80.3%

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

   11. Simplified 80.3%

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

   if 2.79999999999999988e-25 < k

   1. Initial program 49.2%

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

      1. associate-/r* 54.3%

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

      2. +-commutative 54.3%

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

      3. associate-+r+ 54.3%

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

      4. metadata-eval 54.3%

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

      5. associate-*r* 54.3%

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

      6. *-commutative 54.3%

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

      7. associate-*l/ 54.3%

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

      8. associate-*r/ 54.4%

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

   4. Applied egg-rr 54.4%

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

   5. Taylor expanded in k around inf 77.7%

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

      1. associate-*r* 77.7%

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

      2. *-commutative 77.7%

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

      3. *-commutative 77.7%

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

      4. times-frac 77.7%

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

   7. Simplified 77.7%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-146}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{\ell}{\sqrt{2}}} \cdot \sqrt{{t}^{3}}\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}{{k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{\sin k}^{2}}{\cos k} \cdot \frac{t \cdot {k}^{2}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.7 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{\ell}{\sin k}} \cdot \sqrt{\frac{t\_m}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 1.7 \cdot 10^{+107}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{\ell}{\sqrt{2}}} \cdot \sqrt{{t\_m}^{3}}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 1.32 \cdot 10^{+199}:\\ \;\;\;\;\frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m}\right)}^{3}}{{k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \frac{k}{\frac{\ell}{{t\_m}^{3}}}}{\ell}}\\ \end{array} \end{array} \]

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 4.7e-48)
    (/ 2.0 (pow (* (/ k (/ l (sin k))) (sqrt (/ t_m (cos k)))) 2.0))
    (if (<= t_m 1.7e+107)
      (/ 2.0 (pow (* (/ k (/ l (sqrt 2.0))) (sqrt (pow t_m 3.0))) 2.0))
      (if (<= t_m 1.32e+199)
        (/ (pow (/ (pow (cbrt l) 2.0) t_m) 3.0) (pow k 2.0))
        (/
         2.0
         (/
          (* (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))) (/ k (/ l (pow t_m 3.0))))
          l)))))))

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 <= 4.7e-48) {
		tmp = 2.0 / pow(((k / (l / sin(k))) * sqrt((t_m / cos(k)))), 2.0);
	} else if (t_m <= 1.7e+107) {
		tmp = 2.0 / pow(((k / (l / sqrt(2.0))) * sqrt(pow(t_m, 3.0))), 2.0);
	} else if (t_m <= 1.32e+199) {
		tmp = pow((pow(cbrt(l), 2.0) / t_m), 3.0) / pow(k, 2.0);
	} else {
		tmp = 2.0 / (((tan(k) * (2.0 + pow((k / t_m), 2.0))) * (k / (l / pow(t_m, 3.0)))) / l);
	}
	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 <= 4.7e-48) {
		tmp = 2.0 / Math.pow(((k / (l / Math.sin(k))) * Math.sqrt((t_m / Math.cos(k)))), 2.0);
	} else if (t_m <= 1.7e+107) {
		tmp = 2.0 / Math.pow(((k / (l / Math.sqrt(2.0))) * Math.sqrt(Math.pow(t_m, 3.0))), 2.0);
	} else if (t_m <= 1.32e+199) {
		tmp = Math.pow((Math.pow(Math.cbrt(l), 2.0) / t_m), 3.0) / Math.pow(k, 2.0);
	} else {
		tmp = 2.0 / (((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * (k / (l / Math.pow(t_m, 3.0)))) / l);
	}
	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 <= 4.7e-48)
		tmp = Float64(2.0 / (Float64(Float64(k / Float64(l / sin(k))) * sqrt(Float64(t_m / cos(k)))) ^ 2.0));
	elseif (t_m <= 1.7e+107)
		tmp = Float64(2.0 / (Float64(Float64(k / Float64(l / sqrt(2.0))) * sqrt((t_m ^ 3.0))) ^ 2.0));
	elseif (t_m <= 1.32e+199)
		tmp = Float64((Float64((cbrt(l) ^ 2.0) / t_m) ^ 3.0) / (k ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(k / Float64(l / (t_m ^ 3.0)))) / l));
	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, 4.7e-48], N[(2.0 / N[Power[N[(N[(k / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.7e+107], N[(2.0 / N[Power[N[(N[(k / N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[t$95$m, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.32e+199], N[(N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k / N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < 4.6999999999999998e-48

   1. Initial program 51.3%

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

      1. associate-/r* 58.9%

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

      2. +-commutative 58.9%

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

      3. associate-+r+ 58.9%

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

      4. metadata-eval 58.9%

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

      5. associate-*r* 58.9%

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

      6. *-commutative 58.9%

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

      7. associate-*l/ 59.9%

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

      8. associate-*r/ 58.9%

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

   4. Applied egg-rr 58.9%

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

      1. add-sqr-sqrt 22.1%

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

      2. pow2 22.1%

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

      3. associate-/l* 23.6%

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

      4. *-commutative 23.6%

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

   6. Applied egg-rr 23.6%

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

   7. Taylor expanded in k around inf 33.8%

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

      1. associate-/l* 33.8%

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

   9. Simplified 33.8%

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

   if 4.6999999999999998e-48 < t < 1.6999999999999998e107

   1. Initial program 75.6%

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

      1. associate-/r* 80.5%

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

      2. +-commutative 80.5%

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

      3. associate-+r+ 80.5%

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

      4. metadata-eval 80.5%

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

      5. associate-*r* 80.5%

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

      6. *-commutative 80.5%

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

      7. associate-*l/ 82.9%

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

      8. associate-*r/ 85.4%

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

   4. Applied egg-rr 85.4%

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

      1. add-sqr-sqrt 52.9%

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

      2. pow2 52.9%

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

      3. associate-/l* 52.6%

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

      4. *-commutative 52.6%

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

   6. Applied egg-rr 52.6%

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

   7. Taylor expanded in k around 0 78.3%

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

      1. associate-/l* 78.3%

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

   9. Simplified 78.3%

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

   if 1.6999999999999998e107 < t < 1.31999999999999998e199

   1. Initial program 60.0%

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

      1. associate-/r* 60.0%

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

      2. sqr-neg 60.0%

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

      3. associate-*l* 59.8%

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

      4. sqr-neg 59.8%

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

      5. associate-/r* 60.2%

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

      6. associate-+l+ 60.2%

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

      7. unpow2 60.2%

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

      8. times-frac 43.5%

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

      9. sqr-neg 43.5%

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

      10. times-frac 60.2%

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

      11. unpow2 60.2%

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

   3. Simplified 60.2%

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

   5. Taylor expanded in k around 0 59.8%

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

      1. *-commutative 59.8%

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

      2. associate-/r* 59.8%

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

   7. Simplified 59.8%

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

      1. add-cube-cbrt 59.8%

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

      2. pow2 59.8%

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

      3. cbrt-div 59.8%

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

      4. unpow2 59.8%

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

      5. cbrt-prod 59.8%

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

      6. unpow2 59.8%

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

      7. unpow3 59.8%

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

      8. add-cbrt-cube 59.8%

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

      9. cbrt-div 59.8%

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

      10. unpow2 59.8%

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

      11. cbrt-prod 60.2%

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

      12. unpow2 60.2%

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

      13. unpow3 60.2%

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

      14. add-cbrt-cube 84.7%

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

   9. Applied egg-rr 84.7%

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

       1. pow-plus 84.7%

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

       2. metadata-eval 84.7%

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

   11. Simplified 84.7%

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

   if 1.31999999999999998e199 < t

   1. Initial program 64.8%

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

      1. associate-/r* 69.9%

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

      2. +-commutative 69.9%

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

      3. associate-+r+ 69.9%

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

      4. metadata-eval 69.9%

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

      5. associate-*r* 69.9%

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

      6. *-commutative 69.9%

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

      7. associate-*l/ 69.9%

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

      8. associate-*r/ 69.9%

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

   4. Applied egg-rr 69.9%

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

   5. Taylor expanded in k around 0 69.9%

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

      1. associate-/l* 69.9%

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

   7. Simplified 69.9%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.7 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{\ell}{\sin k}} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+107}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{\ell}{\sqrt{2}}} \cdot \sqrt{{t}^{3}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{+199}:\\ \;\;\;\;\frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}{{k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{k}{\frac{\ell}{{t}^{3}}}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{+132}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{\ell}{\sqrt{2}}} \cdot \sqrt{{t\_m}^{3}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m}\right)}^{3}}{{k}^{2}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 4e+132)
    (/ 2.0 (pow (* (/ k (/ l (sqrt 2.0))) (sqrt (pow t_m 3.0))) 2.0))
    (/ (pow (/ (pow (cbrt l) 2.0) t_m) 3.0) (pow k 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 (l <= 4e+132) {
		tmp = 2.0 / pow(((k / (l / sqrt(2.0))) * sqrt(pow(t_m, 3.0))), 2.0);
	} else {
		tmp = pow((pow(cbrt(l), 2.0) / t_m), 3.0) / pow(k, 2.0);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 4e+132) {
		tmp = 2.0 / Math.pow(((k / (l / Math.sqrt(2.0))) * Math.sqrt(Math.pow(t_m, 3.0))), 2.0);
	} else {
		tmp = Math.pow((Math.pow(Math.cbrt(l), 2.0) / t_m), 3.0) / Math.pow(k, 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 (l <= 4e+132)
		tmp = Float64(2.0 / (Float64(Float64(k / Float64(l / sqrt(2.0))) * sqrt((t_m ^ 3.0))) ^ 2.0));
	else
		tmp = Float64((Float64((cbrt(l) ^ 2.0) / t_m) ^ 3.0) / (k ^ 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 4e+132], N[(2.0 / N[Power[N[(N[(k / N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[t$95$m, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.99999999999999996e132

   1. Initial program 59.3%

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

      1. associate-/r* 65.6%

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

      2. +-commutative 65.6%

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

      3. associate-+r+ 65.6%

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

      4. metadata-eval 65.6%

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

      5. associate-*r* 65.6%

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

      6. *-commutative 65.6%

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

      7. associate-*l/ 66.9%

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

      8. associate-*r/ 66.5%

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

   4. Applied egg-rr 66.5%

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

      1. add-sqr-sqrt 26.9%

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

      2. pow2 26.9%

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

      3. associate-/l* 28.2%

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

      4. *-commutative 28.2%

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

   6. Applied egg-rr 28.2%

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

   7. Taylor expanded in k around 0 30.8%

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

      1. associate-/l* 30.8%

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

   9. Simplified 30.8%

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

   if 3.99999999999999996e132 < l

   1. Initial program 42.6%

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

      1. associate-/r* 42.6%

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

      2. sqr-neg 42.6%

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

      3. associate-*l* 42.6%

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

      4. sqr-neg 42.6%

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

      5. associate-/r* 51.1%

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

      6. associate-+l+ 51.1%

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

      7. unpow2 51.1%

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

      8. times-frac 37.8%

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

      9. sqr-neg 37.8%

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

      10. times-frac 51.1%

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

      11. unpow2 51.1%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in k around 0 44.9%

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

      1. *-commutative 44.9%

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

      2. associate-/r* 44.9%

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

   7. Simplified 44.9%

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

      1. add-cube-cbrt 44.9%

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

      2. pow2 44.9%

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

      3. cbrt-div 44.9%

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

      4. unpow2 44.9%

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

      5. cbrt-prod 44.9%

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

      6. unpow2 44.9%

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

      7. unpow3 44.9%

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

      8. add-cbrt-cube 44.9%

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

      9. cbrt-div 44.9%

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

      10. unpow2 44.9%

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

      11. cbrt-prod 49.3%

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

      12. unpow2 49.3%

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

      13. unpow3 49.3%

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

      14. add-cbrt-cube 57.9%

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

   9. Applied egg-rr 57.9%

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

       1. pow-plus 58.0%

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

       2. metadata-eval 58.0%

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

   11. Simplified 58.0%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{+132}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\frac{\ell}{\sqrt{2}}} \cdot \sqrt{{t}^{3}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}{{k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 8.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \frac{k \cdot {t\_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m}\right)}^{3} \cdot {k}^{-2}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 8.2e-8)
    (/
     2.0
     (/
      (* (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))) (/ (* k (pow t_m 3.0)) l))
      l))
    (* (pow (/ (pow (cbrt l) 2.0) t_m) 3.0) (pow k -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 (l <= 8.2e-8) {
		tmp = 2.0 / (((tan(k) * (2.0 + pow((k / t_m), 2.0))) * ((k * pow(t_m, 3.0)) / l)) / l);
	} else {
		tmp = pow((pow(cbrt(l), 2.0) / t_m), 3.0) * pow(k, -2.0);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 8.2e-8) {
		tmp = 2.0 / (((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * ((k * Math.pow(t_m, 3.0)) / l)) / l);
	} else {
		tmp = Math.pow((Math.pow(Math.cbrt(l), 2.0) / t_m), 3.0) * Math.pow(k, -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 (l <= 8.2e-8)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(k * (t_m ^ 3.0)) / l)) / l));
	else
		tmp = Float64((Float64((cbrt(l) ^ 2.0) / t_m) ^ 3.0) * (k ^ -2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 8.2e-8], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision] * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 8.20000000000000063e-8

   1. Initial program 58.8%

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

      1. associate-/r* 65.8%

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

      2. +-commutative 65.8%

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

      3. associate-+r+ 65.8%

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

      4. metadata-eval 65.8%

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

      5. associate-*r* 65.8%

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

      6. *-commutative 65.8%

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

      7. associate-*l/ 67.2%

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

      8. associate-*r/ 66.8%

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

   4. Applied egg-rr 66.8%

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

   5. Taylor expanded in k around 0 63.3%

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

   if 8.20000000000000063e-8 < l

   1. Initial program 49.7%

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

      1. associate-/r* 49.7%

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

      2. sqr-neg 49.7%

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

      3. associate-*l* 49.7%

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

      4. sqr-neg 49.7%

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

      5. associate-/r* 55.4%

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

      6. associate-+l+ 55.4%

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

      7. unpow2 55.4%

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

      8. times-frac 43.5%

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

      9. sqr-neg 43.5%

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

      10. times-frac 55.4%

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

      11. unpow2 55.4%

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

   3. Simplified 55.4%

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

   5. Taylor expanded in k around 0 54.5%

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

      1. *-commutative 54.5%

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

      2. associate-/r* 54.4%

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

   7. Simplified 54.4%

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

      1. add-cube-cbrt 54.4%

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

      2. pow2 54.4%

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

      3. div-inv 54.4%

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

      4. cbrt-prod 54.4%

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

      5. cbrt-div 54.4%

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

      6. unpow2 54.4%

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

      7. cbrt-prod 54.4%

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

      8. unpow2 54.4%

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

      9. unpow3 54.4%

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

      10. add-cbrt-cube 54.4%

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

      11. pow-flip 54.4%

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

      12. metadata-eval 54.4%

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

      13. div-inv 54.4%

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

   9. Applied egg-rr 64.1%

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

       1. pow-plus 64.1%

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

       2. metadata-eval 64.1%

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

       3. cube-prod 63.2%

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

       4. rem-cube-cbrt 63.1%

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

   11. Simplified 63.1%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{k \cdot {t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \cdot {k}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 7.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \frac{k \cdot {t\_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t\_m}\right)}^{3}}{{k}^{2}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= l 7.2e-8)
    (/
     2.0
     (/
      (* (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))) (/ (* k (pow t_m 3.0)) l))
      l))
    (/ (pow (/ (pow (cbrt l) 2.0) t_m) 3.0) (pow k 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 (l <= 7.2e-8) {
		tmp = 2.0 / (((tan(k) * (2.0 + pow((k / t_m), 2.0))) * ((k * pow(t_m, 3.0)) / l)) / l);
	} else {
		tmp = pow((pow(cbrt(l), 2.0) / t_m), 3.0) / pow(k, 2.0);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (l <= 7.2e-8) {
		tmp = 2.0 / (((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * ((k * Math.pow(t_m, 3.0)) / l)) / l);
	} else {
		tmp = Math.pow((Math.pow(Math.cbrt(l), 2.0) / t_m), 3.0) / Math.pow(k, 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 (l <= 7.2e-8)
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(k * (t_m ^ 3.0)) / l)) / l));
	else
		tmp = Float64((Float64((cbrt(l) ^ 2.0) / t_m) ^ 3.0) / (k ^ 2.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[l, 7.2e-8], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 7.19999999999999962e-8

   1. Initial program 58.8%

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

      1. associate-/r* 65.8%

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

      2. +-commutative 65.8%

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

      3. associate-+r+ 65.8%

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

      4. metadata-eval 65.8%

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

      5. associate-*r* 65.8%

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

      6. *-commutative 65.8%

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

      7. associate-*l/ 67.2%

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

      8. associate-*r/ 66.8%

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

   4. Applied egg-rr 66.8%

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

   5. Taylor expanded in k around 0 63.3%

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

   if 7.19999999999999962e-8 < l

   1. Initial program 49.7%

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

      1. associate-/r* 49.7%

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

      2. sqr-neg 49.7%

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

      3. associate-*l* 49.7%

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

      4. sqr-neg 49.7%

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

      5. associate-/r* 55.4%

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

      6. associate-+l+ 55.4%

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

      7. unpow2 55.4%

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

      8. times-frac 43.5%

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

      9. sqr-neg 43.5%

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

      10. times-frac 55.4%

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

      11. unpow2 55.4%

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

   3. Simplified 55.4%

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

   5. Taylor expanded in k around 0 54.5%

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

      1. *-commutative 54.5%

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

      2. associate-/r* 54.4%

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

   7. Simplified 54.4%

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

      1. add-cube-cbrt 54.4%

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

      2. pow2 54.4%

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

      3. cbrt-div 54.4%

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

      4. unpow2 54.4%

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

      5. cbrt-prod 54.4%

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

      6. unpow2 54.4%

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

      7. unpow3 54.4%

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

      8. add-cbrt-cube 54.4%

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

      9. cbrt-div 54.4%

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

      10. unpow2 54.4%

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

      11. cbrt-prod 57.4%

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

      12. unpow2 57.4%

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

      13. unpow3 57.4%

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

      14. add-cbrt-cube 63.1%

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

   9. Applied egg-rr 63.1%

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

       1. pow-plus 63.1%

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

       2. metadata-eval 63.1%

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

   11. Simplified 63.1%

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

4. Final simplification 63.3%

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

Alternative 14: 64.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.8 \cdot 10^{-133}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \frac{k}{\frac{\ell}{{t\_m}^{3}}}}{\ell}}\\ \end{array} \end{array} \]

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-133)
    (* 2.0 (/ (* (cos k) (pow l 2.0)) (* t_m (pow k 4.0))))
    (/
     2.0
     (/
      (* (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))) (/ k (/ l (pow t_m 3.0))))
      l)))))

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-133) {
		tmp = 2.0 * ((cos(k) * pow(l, 2.0)) / (t_m * pow(k, 4.0)));
	} else {
		tmp = 2.0 / (((tan(k) * (2.0 + pow((k / t_m), 2.0))) * (k / (l / pow(t_m, 3.0)))) / l);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3.8d-133) then
        tmp = 2.0d0 * ((cos(k) * (l ** 2.0d0)) / (t_m * (k ** 4.0d0)))
    else
        tmp = 2.0d0 / (((tan(k) * (2.0d0 + ((k / t_m) ** 2.0d0))) * (k / (l / (t_m ** 3.0d0)))) / l)
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.8e-133) {
		tmp = 2.0 * ((Math.cos(k) * Math.pow(l, 2.0)) / (t_m * Math.pow(k, 4.0)));
	} else {
		tmp = 2.0 / (((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * (k / (l / Math.pow(t_m, 3.0)))) / l);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3.8e-133:
		tmp = 2.0 * ((math.cos(k) * math.pow(l, 2.0)) / (t_m * math.pow(k, 4.0)))
	else:
		tmp = 2.0 / (((math.tan(k) * (2.0 + math.pow((k / t_m), 2.0))) * (k / (l / math.pow(t_m, 3.0)))) / l)
	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-133)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * (l ^ 2.0)) / Float64(t_m * (k ^ 4.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(k / Float64(l / (t_m ^ 3.0)))) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3.8e-133)
		tmp = 2.0 * ((cos(k) * (l ^ 2.0)) / (t_m * (k ^ 4.0)));
	else
		tmp = 2.0 / (((tan(k) * (2.0 + ((k / t_m) ^ 2.0))) * (k / (l / (t_m ^ 3.0)))) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.8e-133], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k / N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.8000000000000003e-133

   1. Initial program 49.7%

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

      1. associate-/r* 49.8%

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

      2. sqr-neg 49.8%

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

      3. associate-*l* 45.6%

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

      4. sqr-neg 45.6%

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

      5. associate-/r* 53.0%

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

      6. associate-+l+ 53.0%

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

      7. unpow2 53.0%

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

      8. times-frac 33.5%

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

      9. sqr-neg 33.5%

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

      10. times-frac 53.0%

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

      11. unpow2 53.0%

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

   3. Simplified 53.0%

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

   5. Taylor expanded in t around 0 63.6%

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

   6. Taylor expanded in k around 0 52.5%

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

   if 3.8000000000000003e-133 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/r* 74.8%

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

      2. +-commutative 74.8%

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

      3. associate-+r+ 74.8%

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

      4. metadata-eval 74.8%

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

      5. associate-*r* 74.8%

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

      6. *-commutative 74.8%

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

      7. associate-*l/ 77.0%

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

      8. associate-*r/ 76.0%

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

   4. Applied egg-rr 76.0%

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

   5. Taylor expanded in k around 0 69.5%

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

      1. associate-/l* 69.4%

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

   7. Simplified 69.4%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-133}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{k}{\frac{\ell}{{t}^{3}}}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.4 \cdot 10^{-130}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot \frac{k \cdot {t\_m}^{3}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

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-130)
    (* 2.0 (/ (* (cos k) (pow l 2.0)) (* t_m (pow k 4.0))))
    (/
     2.0
     (/
      (* (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))) (/ (* k (pow t_m 3.0)) 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) {
	double tmp;
	if (t_m <= 1.4e-130) {
		tmp = 2.0 * ((cos(k) * pow(l, 2.0)) / (t_m * pow(k, 4.0)));
	} else {
		tmp = 2.0 / (((tan(k) * (2.0 + pow((k / t_m), 2.0))) * ((k * pow(t_m, 3.0)) / l)) / l);
	}
	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 <= 1.4d-130) then
        tmp = 2.0d0 * ((cos(k) * (l ** 2.0d0)) / (t_m * (k ** 4.0d0)))
    else
        tmp = 2.0d0 / (((tan(k) * (2.0d0 + ((k / t_m) ** 2.0d0))) * ((k * (t_m ** 3.0d0)) / l)) / l)
    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 <= 1.4e-130) {
		tmp = 2.0 * ((Math.cos(k) * Math.pow(l, 2.0)) / (t_m * Math.pow(k, 4.0)));
	} else {
		tmp = 2.0 / (((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * ((k * Math.pow(t_m, 3.0)) / l)) / l);
	}
	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 <= 1.4e-130:
		tmp = 2.0 * ((math.cos(k) * math.pow(l, 2.0)) / (t_m * math.pow(k, 4.0)))
	else:
		tmp = 2.0 / (((math.tan(k) * (2.0 + math.pow((k / t_m), 2.0))) * ((k * math.pow(t_m, 3.0)) / l)) / l)
	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-130)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * (l ^ 2.0)) / Float64(t_m * (k ^ 4.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(k * (t_m ^ 3.0)) / l)) / l));
	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 <= 1.4e-130)
		tmp = 2.0 * ((cos(k) * (l ^ 2.0)) / (t_m * (k ^ 4.0)));
	else
		tmp = 2.0 / (((tan(k) * (2.0 + ((k / t_m) ^ 2.0))) * ((k * (t_m ^ 3.0)) / l)) / l);
	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, 1.4e-130], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.40000000000000008e-130

   1. Initial program 49.7%

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

      1. associate-/r* 49.8%

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

      2. sqr-neg 49.8%

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

      3. associate-*l* 45.6%

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

      4. sqr-neg 45.6%

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

      5. associate-/r* 53.0%

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

      6. associate-+l+ 53.0%

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

      7. unpow2 53.0%

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

      8. times-frac 33.5%

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

      9. sqr-neg 33.5%

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

      10. times-frac 53.0%

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

      11. unpow2 53.0%

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

   3. Simplified 53.0%

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

   5. Taylor expanded in t around 0 63.6%

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

   6. Taylor expanded in k around 0 52.5%

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

   if 1.40000000000000008e-130 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/r* 74.8%

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

      2. +-commutative 74.8%

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

      3. associate-+r+ 74.8%

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

      4. metadata-eval 74.8%

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

      5. associate-*r* 74.8%

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

      6. *-commutative 74.8%

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

      7. associate-*l/ 77.0%

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

      8. associate-*r/ 76.0%

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

   4. Applied egg-rr 76.0%

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

   5. Taylor expanded in k around 0 69.5%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-130}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{k \cdot {t}^{3}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.9 \cdot 10^{-21}:\\ \;\;\;\;\frac{2}{\frac{\frac{2 \cdot \left({k}^{2} \cdot {t\_m}^{3}\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 5.9e-21)
    (/ 2.0 (/ (/ (* 2.0 (* (pow k 2.0) (pow t_m 3.0))) l) l))
    (* 2.0 (/ (* (cos k) (pow l 2.0)) (* t_m (pow k 4.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5.9e-21) {
		tmp = 2.0 / (((2.0 * (pow(k, 2.0) * pow(t_m, 3.0))) / l) / l);
	} else {
		tmp = 2.0 * ((cos(k) * pow(l, 2.0)) / (t_m * pow(k, 4.0)));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.9d-21) then
        tmp = 2.0d0 / (((2.0d0 * ((k ** 2.0d0) * (t_m ** 3.0d0))) / l) / l)
    else
        tmp = 2.0d0 * ((cos(k) * (l ** 2.0d0)) / (t_m * (k ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5.9e-21) {
		tmp = 2.0 / (((2.0 * (Math.pow(k, 2.0) * Math.pow(t_m, 3.0))) / l) / l);
	} else {
		tmp = 2.0 * ((Math.cos(k) * Math.pow(l, 2.0)) / (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 5.9e-21:
		tmp = 2.0 / (((2.0 * (math.pow(k, 2.0) * math.pow(t_m, 3.0))) / l) / l)
	else:
		tmp = 2.0 * ((math.cos(k) * math.pow(l, 2.0)) / (t_m * math.pow(k, 4.0)))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 5.9e-21)
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * Float64((k ^ 2.0) * (t_m ^ 3.0))) / l) / l));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) * (l ^ 2.0)) / Float64(t_m * (k ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 5.9e-21)
		tmp = 2.0 / (((2.0 * ((k ^ 2.0) * (t_m ^ 3.0))) / l) / l);
	else
		tmp = 2.0 * ((cos(k) * (l ^ 2.0)) / (t_m * (k ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 5.9e-21], N[(2.0 / N[(N[(N[(2.0 * N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.9000000000000003e-21

   1. Initial program 59.5%

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

      1. associate-/r* 66.9%

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

      2. +-commutative 66.9%

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

      3. associate-+r+ 66.9%

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

      4. metadata-eval 66.9%

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

      5. associate-*r* 66.9%

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

      6. *-commutative 66.9%

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

      7. associate-*l/ 68.5%

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

      8. associate-*r/ 68.0%

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

   4. Applied egg-rr 68.0%

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

   5. Taylor expanded in k around 0 63.2%

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

      1. associate-*r/ 63.2%

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

   7. Simplified 63.2%

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

   if 5.9000000000000003e-21 < k

   1. Initial program 49.2%

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

      1. associate-/r* 49.2%

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

      2. sqr-neg 49.2%

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

      3. associate-*l* 49.2%

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

      4. sqr-neg 49.2%

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

      5. associate-/r* 54.3%

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

      6. associate-+l+ 54.3%

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

      7. unpow2 54.3%

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

      8. times-frac 41.6%

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

      9. sqr-neg 41.6%

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

      10. times-frac 54.3%

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

      11. unpow2 54.3%

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

   3. Simplified 54.3%

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

   5. Taylor expanded in t around 0 70.1%

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

   6. Taylor expanded in k around 0 55.5%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.9 \cdot 10^{-21}:\\ \;\;\;\;\frac{2}{\frac{\frac{2 \cdot \left({k}^{2} \cdot {t}^{3}\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\frac{2 \cdot \frac{{k}^{2}}{\frac{\ell}{{t\_m}^{3}}}}{\ell}} \end{array} \]

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 (/ (* 2.0 (/ (pow k 2.0) (/ l (pow t_m 3.0)))) 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 / ((2.0 * (pow(k, 2.0) / (l / pow(t_m, 3.0)))) / 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 / ((2.0d0 * ((k ** 2.0d0) / (l / (t_m ** 3.0d0)))) / 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 / ((2.0 * (Math.pow(k, 2.0) / (l / Math.pow(t_m, 3.0)))) / 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 / ((2.0 * (math.pow(k, 2.0) / (l / math.pow(t_m, 3.0)))) / 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(2.0 * Float64((k ^ 2.0) / Float64(l / (t_m ^ 3.0)))) / 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 / ((2.0 * ((k ^ 2.0) / (l / (t_m ^ 3.0)))) / 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[(2.0 * N[(N[Power[k, 2.0], $MachinePrecision] / N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Add Preprocessing
3. Step-by-step derivation

   1. associate-/r* 63.1%

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

   2. +-commutative 63.1%

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

   3. associate-+r+ 63.1%

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

   4. metadata-eval 63.1%

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

   5. associate-*r* 63.1%

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

   6. *-commutative 63.1%

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

   7. associate-*l/ 64.2%

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

   8. associate-*r/ 63.9%

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

4. Applied egg-rr 63.9%

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

5. Taylor expanded in k around 0 58.5%

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

   1. associate-/l* 58.3%

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

7. Simplified 58.3%

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

8. Final simplification 58.3%

   \[\leadsto \frac{2}{\frac{2 \cdot \frac{{k}^{2}}{\frac{\ell}{{t}^{3}}}}{\ell}} \]
9. Add Preprocessing

Alternative 18: 55.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\frac{\frac{2 \cdot \left({k}^{2} \cdot {t\_m}^{3}\right)}{\ell}}{\ell}} \end{array} \]

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 (/ (/ (* 2.0 (* (pow k 2.0) (pow t_m 3.0))) 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 / (((2.0 * (pow(k, 2.0) * pow(t_m, 3.0))) / 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 / (((2.0d0 * ((k ** 2.0d0) * (t_m ** 3.0d0))) / 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 / (((2.0 * (Math.pow(k, 2.0) * Math.pow(t_m, 3.0))) / 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 / (((2.0 * (math.pow(k, 2.0) * math.pow(t_m, 3.0))) / 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(2.0 * Float64((k ^ 2.0) * (t_m ^ 3.0))) / 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 / (((2.0 * ((k ^ 2.0) * (t_m ^ 3.0))) / 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[(2.0 * N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Add Preprocessing
3. Step-by-step derivation

   1. associate-/r* 63.1%

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

   2. +-commutative 63.1%

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

   3. associate-+r+ 63.1%

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

   4. metadata-eval 63.1%

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

   5. associate-*r* 63.1%

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

   6. *-commutative 63.1%

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

   7. associate-*l/ 64.2%

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

   8. associate-*r/ 63.9%

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

4. Applied egg-rr 63.9%

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

5. Taylor expanded in k around 0 58.5%

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

   1. associate-*r/ 58.5%

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

7. Simplified 58.5%

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

8. Final simplification 58.5%

   \[\leadsto \frac{2}{\frac{\frac{2 \cdot \left({k}^{2} \cdot {t}^{3}\right)}{\ell}}{\ell}} \]
9. Add Preprocessing

Alternative 19: 51.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \frac{1}{\frac{t\_m}{\frac{{\ell}^{2}}{{k}^{4}}}}\right) \end{array} \]

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 (/ 1.0 (/ t_m (/ (pow l 2.0) (pow k 4.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * (1.0 / (t_m / (pow(l, 2.0) / pow(k, 4.0)))));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * (1.0d0 / (t_m / ((l ** 2.0d0) / (k ** 4.0d0)))))
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * (1.0 / (t_m / (Math.pow(l, 2.0) / Math.pow(k, 4.0)))));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * (1.0 / (t_m / (math.pow(l, 2.0) / math.pow(k, 4.0)))))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64(1.0 / Float64(t_m / Float64((l ^ 2.0) / (k ^ 4.0))))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * (1.0 / (t_m / ((l ^ 2.0) / (k ^ 4.0)))));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 * N[(1.0 / N[(t$95$m / N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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-/r* 56.4%

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

   2. sqr-neg 56.4%

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

   3. associate-*l* 52.8%

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

   4. sqr-neg 52.8%

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

   5. associate-/r* 59.1%

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

   6. associate-+l+ 59.1%

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

   7. unpow2 59.1%

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

   8. times-frac 44.6%

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

   9. sqr-neg 44.6%

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

   10. times-frac 59.1%

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

   11. unpow2 59.1%

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

3. Simplified 59.1%

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

5. Taylor expanded in t around 0 64.1%

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

6. Taylor expanded in k around 0 54.3%

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

   1. clear-num 54.6%

      \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]

   2. inv-pow 54.6%

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

   3. *-commutative 54.6%

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

8. Applied egg-rr 54.6%

   \[\leadsto 2 \cdot \color{blue}{{\left(\frac{t \cdot {k}^{4}}{{\ell}^{2}}\right)}^{-1}} \]
9. Step-by-step derivation

   1. unpow-1 54.6%

      \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{{\ell}^{2}}}} \]

   2. associate-/l* 54.5%

      \[\leadsto 2 \cdot \frac{1}{\color{blue}{\frac{t}{\frac{{\ell}^{2}}{{k}^{4}}}}} \]

10. Simplified 54.5%

    \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{t}{\frac{{\ell}^{2}}{{k}^{4}}}}} \]

11. Final simplification 54.5%

    \[\leadsto 2 \cdot \frac{1}{\frac{t}{\frac{{\ell}^{2}}{{k}^{4}}}} \]
12. Add Preprocessing

Alternative 20: 51.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k}^{4}}\right) \end{array} \]

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 l 2.0) (* t_m (pow k 4.0))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * (pow(l, 2.0) / (t_m * pow(k, 4.0))));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * ((l ** 2.0d0) / (t_m * (k ** 4.0d0))))
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 4.0))));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k, 4.0))))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 4.0)))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * ((l ^ 2.0) / (t_m * (k ^ 4.0))));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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-/r* 56.4%

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

   2. sqr-neg 56.4%

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

   3. associate-*l* 52.8%

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

   4. sqr-neg 52.8%

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

   5. associate-/r* 59.1%

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

   6. associate-+l+ 59.1%

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

   7. unpow2 59.1%

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

   8. times-frac 44.6%

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

   9. sqr-neg 44.6%

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

   10. times-frac 59.1%

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

   11. unpow2 59.1%

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

3. Simplified 59.1%

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

5. Taylor expanded in t around 0 64.1%

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

6. Taylor expanded in k around 0 54.3%

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

7. Final simplification 54.3%

   \[\leadsto 2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(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))))