Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.6% → 86.2%

Time: 20.3s

Alternatives: 29

Speedup: 1.9×

• 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: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\ell}^{2}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.2e-125)
    (*
     (/ 2.0 (pow k 2.0))
     (* (/ (cos k) (pow (sin k) 2.0)) (/ (pow l 2.0) t_m)))
    (/
     2.0
     (pow
      (*
       (* (cbrt (sin k)) (* t_m (pow (cbrt l) -2.0)))
       (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))))
      3.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.2e-125) {
		tmp = (2.0 / pow(k, 2.0)) * ((cos(k) / pow(sin(k), 2.0)) * (pow(l, 2.0) / t_m));
	} else {
		tmp = 2.0 / pow(((cbrt(sin(k)) * (t_m * pow(cbrt(l), -2.0))) * cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.2e-125) {
		tmp = (2.0 / Math.pow(k, 2.0)) * ((Math.cos(k) / Math.pow(Math.sin(k), 2.0)) * (Math.pow(l, 2.0) / t_m));
	} else {
		tmp = 2.0 / Math.pow(((Math.cbrt(Math.sin(k)) * (t_m * Math.pow(Math.cbrt(l), -2.0))) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.2e-125)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64(cos(k) / (sin(k) ^ 2.0)) * Float64((l ^ 2.0) / t_m)));
	else
		tmp = Float64(2.0 / (Float64(Float64(cbrt(sin(k)) * Float64(t_m * (cbrt(l) ^ -2.0))) * cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.2e-125], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(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] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.1999999999999998e-125

   1. Initial program 50.2%

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

   3. Simplified 50.5%

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

   5. Taylor expanded in k around inf 60.7%

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

      1. associate-*r/ 60.7%

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

      2. times-frac 61.5%

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

      3. *-commutative 61.5%

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

      4. *-commutative 61.5%

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

      5. times-frac 63.8%

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

   7. Simplified 63.8%

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

   if 3.1999999999999998e-125 < t

   1. Initial program 75.2%

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

   3. Simplified 75.2%

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

      1. add-cube-cbrt 75.0%

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

      2. pow3 75.0%

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

      3. *-commutative 75.0%

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

      4. cbrt-prod 75.0%

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

      5. cbrt-div 76.3%

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

      6. rem-cbrt-cube 84.0%

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

      7. cbrt-prod 90.5%

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

      8. pow2 90.5%

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

   6. Applied egg-rr 90.5%

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

      1. add-sqr-sqrt 55.9%

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

      2. pow2 55.9%

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

      3. associate-+r+ 55.9%

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

      4. metadata-eval 55.9%

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

      5. sqrt-prod 55.9%

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

      6. metadata-eval 55.9%

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

      7. associate-+r+ 55.9%

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

      8. add-sqr-sqrt 55.9%

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

      9. hypot-1-def 55.9%

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

      10. unpow2 55.9%

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

      11. hypot-1-def 55.9%

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

   8. Applied egg-rr 55.9%

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

      1. unpow2 55.9%

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

      2. swap-sqr 55.9%

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

      3. rem-square-sqrt 90.4%

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

      4. unpow2 90.4%

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

   10. Simplified 90.4%

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

   12. Applied egg-rr 95.1%

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

Alternative 2: 82.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\ell}^{2}}{t\_m}\right)\\ \mathbf{elif}\;t\_m \leq 820000000:\\ \;\;\;\;t\_2 \cdot \left(\frac{\frac{2}{\tan k}}{\sin k \cdot {t\_m}^{3}} \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right)\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (hypot 1.0 (hypot 1.0 (/ k t_m))))))
   (*
    t_s
    (if (<= t_m 1.2e-114)
      (*
       (/ 2.0 (pow k 2.0))
       (* (/ (cos k) (pow (sin k) 2.0)) (/ (pow l 2.0) t_m)))
      (if (<= t_m 820000000.0)
        (* t_2 (* (/ (/ 2.0 (tan k)) (* (sin k) (pow t_m 3.0))) t_2))
        (/
         2.0
         (*
          (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0)
          (* (tan k) (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0))))))))))

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 = l / hypot(1.0, hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 1.2e-114) {
		tmp = (2.0 / pow(k, 2.0)) * ((cos(k) / pow(sin(k), 2.0)) * (pow(l, 2.0) / t_m));
	} else if (t_m <= 820000000.0) {
		tmp = t_2 * (((2.0 / tan(k)) / (sin(k) * pow(t_m, 3.0))) * t_2);
	} else {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0) * (tan(k) * (1.0 + (pow((k / t_m), 2.0) + 1.0))));
	}
	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 = l / Math.hypot(1.0, Math.hypot(1.0, (k / t_m)));
	double tmp;
	if (t_m <= 1.2e-114) {
		tmp = (2.0 / Math.pow(k, 2.0)) * ((Math.cos(k) / Math.pow(Math.sin(k), 2.0)) * (Math.pow(l, 2.0) / t_m));
	} else if (t_m <= 820000000.0) {
		tmp = t_2 * (((2.0 / Math.tan(k)) / (Math.sin(k) * Math.pow(t_m, 3.0))) * t_2);
	} else {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (Math.tan(k) * (1.0 + (Math.pow((k / t_m), 2.0) + 1.0))));
	}
	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(l / hypot(1.0, hypot(1.0, Float64(k / t_m))))
	tmp = 0.0
	if (t_m <= 1.2e-114)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64(cos(k) / (sin(k) ^ 2.0)) * Float64((l ^ 2.0) / t_m)));
	elseif (t_m <= 820000000.0)
		tmp = Float64(t_2 * Float64(Float64(Float64(2.0 / tan(k)) / Float64(sin(k) * (t_m ^ 3.0))) * t_2));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0)))));
	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[(l / N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.2e-114], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 820000000.0], N[(t$95$2 * N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.2000000000000001e-114

   1. Initial program 50.0%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in k around inf 61.4%

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

      1. associate-*r/ 61.4%

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

      2. times-frac 62.1%

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

      3. *-commutative 62.1%

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

      4. *-commutative 62.1%

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

      5. times-frac 64.4%

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

   7. Simplified 64.4%

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

   if 1.2000000000000001e-114 < t < 8.2e8

   1. Initial program 85.8%

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

   3. Simplified 81.5%

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

      1. associate-*r* 81.4%

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

      2. add-sqr-sqrt 81.4%

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

      3. times-frac 81.6%

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

   6. Applied egg-rr 95.2%

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

      1. associate-/l* 95.2%

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

      2. metadata-eval 95.2%

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

      3. associate-*r/ 95.2%

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

      4. associate-/l/ 95.3%

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

      5. associate-*r/ 95.3%

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

      6. associate-*r/ 95.3%

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

      7. metadata-eval 95.3%

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

      8. *-commutative 95.3%

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

   8. Simplified 95.3%

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

   if 8.2e8 < t

   1. Initial program 73.3%

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

   3. Simplified 73.3%

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

      1. add-cube-cbrt 73.1%

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

      2. pow3 73.1%

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

      3. *-commutative 73.1%

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

      4. cbrt-prod 73.1%

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

      5. cbrt-div 75.0%

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

      6. rem-cbrt-cube 82.4%

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

      7. cbrt-prod 92.0%

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

      8. pow2 92.0%

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

   6. Applied egg-rr 92.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\ell}^{2}}{t}\right)\\ \mathbf{elif}\;t \leq 820000000:\\ \;\;\;\;\frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)} \cdot \left(\frac{\frac{2}{\tan k}}{\sin k \cdot {t}^{3}} \cdot \frac{\ell}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.3 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\ell}^{2}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.3e-125)
    (*
     (/ 2.0 (pow k 2.0))
     (* (/ (cos k) (pow (sin k) 2.0)) (/ (pow l 2.0) t_m)))
    (/
     2.0
     (*
      (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0)
      (* (tan k) (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.3e-125) {
		tmp = (2.0 / pow(k, 2.0)) * ((cos(k) / pow(sin(k), 2.0)) * (pow(l, 2.0) / t_m));
	} else {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0) * (tan(k) * (1.0 + (pow((k / t_m), 2.0) + 1.0))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 4.3e-125) {
		tmp = (2.0 / Math.pow(k, 2.0)) * ((Math.cos(k) / Math.pow(Math.sin(k), 2.0)) * (Math.pow(l, 2.0) / t_m));
	} else {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (Math.tan(k) * (1.0 + (Math.pow((k / t_m), 2.0) + 1.0))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 4.3e-125)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64(cos(k) / (sin(k) ^ 2.0)) * Float64((l ^ 2.0) / t_m)));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 4.3e-125], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.3000000000000002e-125

   1. Initial program 50.2%

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

   3. Simplified 50.5%

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

   5. Taylor expanded in k around inf 60.7%

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

      1. associate-*r/ 60.7%

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

      2. times-frac 61.5%

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

      3. *-commutative 61.5%

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

      4. *-commutative 61.5%

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

      5. times-frac 63.8%

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

   7. Simplified 63.8%

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

   if 4.3000000000000002e-125 < t

   1. Initial program 75.2%

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

   3. Simplified 75.2%

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

      1. add-cube-cbrt 75.0%

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

      2. pow3 75.0%

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

      3. *-commutative 75.0%

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

      4. cbrt-prod 75.0%

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

      5. cbrt-div 76.3%

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

      6. rem-cbrt-cube 84.0%

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

      7. cbrt-prod 90.5%

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

      8. pow2 90.5%

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

   6. Applied egg-rr 90.5%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.3 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\ell}^{2}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.25 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\ell}^{2}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.25e-125)
    (*
     (/ 2.0 (pow k 2.0))
     (* (/ (cos k) (pow (sin k) 2.0)) (/ (pow l 2.0) t_m)))
    (/
     2.0
     (*
      (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0)))
      (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 <= 4.25e-125) {
		tmp = (2.0 / pow(k, 2.0)) * ((cos(k) / pow(sin(k), 2.0)) * (pow(l, 2.0) / t_m));
	} else {
		tmp = 2.0 / ((tan(k) * (2.0 + pow((k / t_m), 2.0))) * 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 <= 4.25e-125) {
		tmp = (2.0 / Math.pow(k, 2.0)) * ((Math.cos(k) / Math.pow(Math.sin(k), 2.0)) * (Math.pow(l, 2.0) / t_m));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) * 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 <= 4.25e-125)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64(cos(k) / (sin(k) ^ 2.0)) * Float64((l ^ 2.0) / t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * (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, 4.25e-125], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.2500000000000001e-125

   1. Initial program 50.2%

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

   3. Simplified 50.5%

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

   5. Taylor expanded in k around inf 60.7%

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

      1. associate-*r/ 60.7%

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

      2. times-frac 61.5%

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

      3. *-commutative 61.5%

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

      4. *-commutative 61.5%

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

      5. times-frac 63.8%

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

   7. Simplified 63.8%

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

   if 4.2500000000000001e-125 < t

   1. Initial program 75.2%

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

   3. Simplified 75.2%

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

      1. add-cube-cbrt 75.0%

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

      2. pow3 75.0%

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

      3. *-commutative 75.0%

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

      4. cbrt-prod 75.0%

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

      5. cbrt-div 76.3%

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

      6. rem-cbrt-cube 84.0%

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

      7. cbrt-prod 90.5%

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

      8. pow2 90.5%

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

   6. Applied egg-rr 90.5%

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

      1. add-sqr-sqrt 55.9%

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

      2. pow2 55.9%

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

      3. associate-+r+ 55.9%

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

      4. metadata-eval 55.9%

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

      5. sqrt-prod 55.9%

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

      6. metadata-eval 55.9%

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

      7. associate-+r+ 55.9%

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

      8. add-sqr-sqrt 55.9%

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

      9. hypot-1-def 55.9%

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

      10. unpow2 55.9%

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

      11. hypot-1-def 55.9%

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

   8. Applied egg-rr 55.9%

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

      1. unpow2 55.9%

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

      2. swap-sqr 55.9%

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

      3. rem-square-sqrt 90.4%

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

      4. unpow2 90.4%

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

   10. Simplified 90.4%

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

   12. Applied egg-rr 95.1%

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

       1. cube-prod 90.4%

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

       2. rem-cube-cbrt 90.4%

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

   14. Simplified 90.4%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.25 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\ell}^{2}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.044:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2}}{\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(\sqrt[3]{2} \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.044)
    (pow
     (/
      (cbrt 2.0)
      (* (* t_m (pow (cbrt l) -2.0)) (* (cbrt 2.0) (pow (cbrt k) 2.0))))
     3.0)
    (* (/ 2.0 (tan k)) (/ (pow l 2.0) (* (pow k 2.0) (* t_m (sin k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.044) {
		tmp = pow((cbrt(2.0) / ((t_m * pow(cbrt(l), -2.0)) * (cbrt(2.0) * pow(cbrt(k), 2.0)))), 3.0);
	} else {
		tmp = (2.0 / tan(k)) * (pow(l, 2.0) / (pow(k, 2.0) * (t_m * sin(k))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.044) {
		tmp = Math.pow((Math.cbrt(2.0) / ((t_m * Math.pow(Math.cbrt(l), -2.0)) * (Math.cbrt(2.0) * Math.pow(Math.cbrt(k), 2.0)))), 3.0);
	} else {
		tmp = (2.0 / Math.tan(k)) * (Math.pow(l, 2.0) / (Math.pow(k, 2.0) * (t_m * Math.sin(k))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.044)
		tmp = Float64(cbrt(2.0) / Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * Float64(cbrt(2.0) * (cbrt(k) ^ 2.0)))) ^ 3.0;
	else
		tmp = Float64(Float64(2.0 / tan(k)) * Float64((l ^ 2.0) / Float64((k ^ 2.0) * Float64(t_m * sin(k)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 0.044], N[Power[N[(N[Power[2.0, 1/3], $MachinePrecision] / N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[2.0, 1/3], $MachinePrecision] * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.043999999999999997

   1. Initial program 59.6%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in k around 0 56.4%

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

      1. add-cube-cbrt 56.3%

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

      2. pow3 56.3%

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

      3. associate-/l/ 53.9%

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

      4. cbrt-div 53.9%

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

      5. unpow3 53.9%

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

      6. add-cbrt-cube 59.3%

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

      7. cbrt-prod 63.8%

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

      8. unpow2 63.8%

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

      9. div-inv 63.8%

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

      10. unpow-prod-down 53.4%

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

      11. pow-flip 53.4%

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

      12. metadata-eval 53.4%

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

   7. Applied egg-rr 53.4%

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

      1. cube-prod 63.8%

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

   9. Simplified 63.8%

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

       1. add-cube-cbrt 63.8%

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

       2. pow2 63.8%

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

       3. cbrt-div 63.8%

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

       4. cbrt-prod 63.8%

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

       5. unpow3 63.8%

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

       6. add-cbrt-cube 63.8%

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

       7. cbrt-div 63.8%

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

       8. cbrt-prod 63.8%

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

   11. Applied egg-rr 68.0%

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

       1. unpow2 68.0%

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

       2. unpow3 68.0%

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

   13. Simplified 68.0%

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

       1. *-commutative 68.0%

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

       2. cbrt-prod 68.0%

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

       3. unpow2 68.0%

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

       4. cbrt-prod 81.0%

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

       5. pow2 81.0%

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

   15. Applied egg-rr 81.0%

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

   if 0.043999999999999997 < k

   1. Initial program 50.9%

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

   3. Simplified 49.3%

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

      1. associate-*r* 52.5%

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

      2. *-un-lft-identity 52.5%

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

      3. times-frac 55.8%

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

      4. associate-/l/ 55.8%

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

   6. Applied egg-rr 55.8%

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

      1. /-rgt-identity 55.8%

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

      2. associate-*l/ 55.8%

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

      3. *-commutative 55.8%

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

      4. times-frac 55.8%

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

      5. *-commutative 55.8%

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

   8. Simplified 55.8%

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

      1. associate-*r/ 52.5%

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

   10. Applied egg-rr 52.5%

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

       1. associate-/l* 55.8%

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

       2. associate-*r* 55.8%

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

       3. *-commutative 55.8%

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

   12. Simplified 55.8%

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

   13. Taylor expanded in t around 0 74.3%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.044:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2}}{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(\sqrt[3]{2} \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\ell}^{2}}{t\_m}\right)\\ \mathbf{elif}\;t\_m \leq 9.5 \cdot 10^{+174}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \frac{\sqrt[3]{\frac{\ell}{\sin k}}}{t\_m}\right)}^{3} \cdot \frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.1e-88)
    (*
     (/ 2.0 (pow k 2.0))
     (* (/ (cos k) (pow (sin k) 2.0)) (/ (pow l 2.0) t_m)))
    (if (<= t_m 9.5e+174)
      (*
       (pow (* (cbrt (/ 2.0 (tan k))) (/ (cbrt (/ l (sin k))) t_m)) 3.0)
       (/ l (+ 2.0 (pow (/ k t_m) 2.0))))
      (/
       2.0
       (*
        (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0)
        (* 2.0 k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.1e-88) {
		tmp = (2.0 / pow(k, 2.0)) * ((cos(k) / pow(sin(k), 2.0)) * (pow(l, 2.0) / t_m));
	} else if (t_m <= 9.5e+174) {
		tmp = pow((cbrt((2.0 / tan(k))) * (cbrt((l / sin(k))) / t_m)), 3.0) * (l / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.1e-88) {
		tmp = (2.0 / Math.pow(k, 2.0)) * ((Math.cos(k) / Math.pow(Math.sin(k), 2.0)) * (Math.pow(l, 2.0) / t_m));
	} else if (t_m <= 9.5e+174) {
		tmp = Math.pow((Math.cbrt((2.0 / Math.tan(k))) * (Math.cbrt((l / Math.sin(k))) / t_m)), 3.0) * (l / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.1e-88)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64(cos(k) / (sin(k) ^ 2.0)) * Float64((l ^ 2.0) / t_m)));
	elseif (t_m <= 9.5e+174)
		tmp = Float64((Float64(cbrt(Float64(2.0 / tan(k))) * Float64(cbrt(Float64(l / sin(k))) / t_m)) ^ 3.0) * Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.1e-88], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.5e+174], N[(N[Power[N[(N[Power[N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.1e-88

   1. Initial program 51.0%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in k around inf 62.6%

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

      1. associate-*r/ 62.6%

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

      2. times-frac 63.3%

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

      3. *-commutative 63.3%

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

      4. *-commutative 63.3%

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

      5. times-frac 65.5%

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

   7. Simplified 65.5%

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

   if 2.1e-88 < t < 9.4999999999999992e174

   1. Initial program 78.6%

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

   3. Simplified 76.5%

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

      1. associate-*r* 76.5%

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

      2. *-un-lft-identity 76.5%

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

      3. times-frac 76.5%

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

      4. associate-/l/ 76.5%

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

   6. Applied egg-rr 76.5%

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

      1. /-rgt-identity 76.5%

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

      2. associate-*l/ 76.6%

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

      3. *-commutative 76.6%

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

      4. times-frac 76.6%

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

      5. *-commutative 76.6%

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

   8. Simplified 76.6%

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

      1. add-cube-cbrt 76.4%

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

      2. pow3 76.3%

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

      3. cbrt-prod 76.3%

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

      4. associate-/r* 76.3%

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

      5. cbrt-div 76.3%

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

      6. unpow3 76.3%

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

      7. add-cbrt-cube 89.9%

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

   10. Applied egg-rr 89.9%

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

   if 9.4999999999999992e174 < t

   1. Initial program 72.6%

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

   3. Simplified 72.6%

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

      1. add-cube-cbrt 72.6%

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

      2. pow3 72.6%

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

      3. *-commutative 72.6%

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

      4. cbrt-prod 72.6%

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

      5. cbrt-div 72.6%

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

      6. rem-cbrt-cube 72.9%

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

      7. cbrt-prod 92.2%

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

      8. pow2 92.2%

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

   6. Applied egg-rr 92.2%

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

   7. Taylor expanded in k around 0 88.7%

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

      1. *-commutative 88.7%

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

   9. Simplified 88.7%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-88}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\ell}^{2}}{t}\right)\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+174}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \frac{\sqrt[3]{\frac{\ell}{\sin k}}}{t}\right)}^{3} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\ell}^{2}}{t\_m}\right)\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell \cdot \frac{\frac{\ell}{\sin k}}{{t\_m}^{3}}}{t\_2}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(\tan k \cdot t\_2\right) \cdot {\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 2.0 (pow (/ k t_m) 2.0))))
   (*
    t_s
    (if (<= t_m 2e-79)
      (*
       (/ 2.0 (pow k 2.0))
       (* (/ (cos k) (pow (sin k) 2.0)) (/ (pow l 2.0) t_m)))
      (if (<= t_m 5.6e+102)
        (* 2.0 (/ (/ (* l (/ (/ l (sin k)) (pow t_m 3.0))) t_2) (tan k)))
        (/
         2.0
         (*
          (sin k)
          (* (* (tan 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 = 2.0 + pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 2e-79) {
		tmp = (2.0 / pow(k, 2.0)) * ((cos(k) / pow(sin(k), 2.0)) * (pow(l, 2.0) / t_m));
	} else if (t_m <= 5.6e+102) {
		tmp = 2.0 * (((l * ((l / sin(k)) / pow(t_m, 3.0))) / t_2) / tan(k));
	} else {
		tmp = 2.0 / (sin(k) * ((tan(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 = 2.0 + Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 2e-79) {
		tmp = (2.0 / Math.pow(k, 2.0)) * ((Math.cos(k) / Math.pow(Math.sin(k), 2.0)) * (Math.pow(l, 2.0) / t_m));
	} else if (t_m <= 5.6e+102) {
		tmp = 2.0 * (((l * ((l / Math.sin(k)) / Math.pow(t_m, 3.0))) / t_2) / Math.tan(k));
	} else {
		tmp = 2.0 / (Math.sin(k) * ((Math.tan(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(2.0 + (Float64(k / t_m) ^ 2.0))
	tmp = 0.0
	if (t_m <= 2e-79)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64(cos(k) / (sin(k) ^ 2.0)) * Float64((l ^ 2.0) / t_m)));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * Float64(Float64(l / sin(k)) / (t_m ^ 3.0))) / t_2) / tan(k)));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(Float64(tan(k) * 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[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2e-79], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(2.0 * N[(N[(N[(l * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision] * N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2e-79

   1. Initial program 51.5%

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

   3. Simplified 51.8%

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

   5. Taylor expanded in k around inf 63.0%

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

      1. associate-*r/ 63.0%

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

      2. times-frac 63.7%

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

      3. *-commutative 63.7%

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

      4. *-commutative 63.7%

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

      5. times-frac 65.9%

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

   7. Simplified 65.9%

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

   if 2e-79 < t < 5.60000000000000037e102

   1. Initial program 89.1%

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

   3. Simplified 86.0%

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

      1. associate-*r* 85.9%

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

      2. *-un-lft-identity 85.9%

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

      3. times-frac 86.0%

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

      4. associate-/l/ 86.0%

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

   6. Applied egg-rr 86.0%

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

      1. /-rgt-identity 86.0%

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

      2. associate-*l/ 86.0%

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

      3. *-commutative 86.0%

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

      4. times-frac 86.1%

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

      5. *-commutative 86.1%

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

   8. Simplified 86.1%

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

      1. associate-*r/ 86.0%

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

   10. Applied egg-rr 86.0%

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

       1. associate-/l* 86.1%

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

       2. associate-*r* 89.5%

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

       3. associate-*l/ 89.5%

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

       4. *-lft-identity 89.5%

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

       5. times-frac 89.5%

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

       6. metadata-eval 89.5%

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

       7. associate-*r/ 89.4%

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

       8. associate-/r* 92.8%

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

   12. Simplified 92.8%

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

   if 5.60000000000000037e102 < t

   1. Initial program 65.1%

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

   3. Simplified 65.1%

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

      1. add-cube-cbrt 65.1%

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

      2. pow3 65.1%

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

      3. *-commutative 65.1%

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

      4. cbrt-prod 65.1%

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

      5. cbrt-div 65.1%

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

      6. rem-cbrt-cube 75.7%

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

      7. cbrt-prod 89.2%

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

      8. pow2 89.2%

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

   6. Applied egg-rr 89.2%

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

      1. add-sqr-sqrt 52.8%

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

      2. pow2 52.8%

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

      3. associate-+r+ 52.8%

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

      4. metadata-eval 52.8%

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

      5. sqrt-prod 52.8%

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

      6. metadata-eval 52.8%

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

      7. associate-+r+ 52.8%

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

      8. add-sqr-sqrt 52.8%

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

      9. hypot-1-def 52.8%

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

      10. unpow2 52.8%

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

      11. hypot-1-def 52.8%

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

   8. Applied egg-rr 52.8%

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

      1. unpow2 52.8%

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

      2. swap-sqr 52.8%

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

      3. rem-square-sqrt 89.1%

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

      4. unpow2 89.1%

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

   10. Simplified 89.1%

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

   12. Applied egg-rr 99.0%

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

       1. div-inv 99.0%

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

       2. *-commutative 99.0%

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

       3. unpow-prod-down 89.0%

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

       4. pow3 89.1%

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

       5. add-cube-cbrt 89.1%

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

       6. unpow-prod-down 89.2%

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

       7. pow3 89.2%

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

       8. add-cube-cbrt 89.2%

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

   14. Applied egg-rr 89.2%

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

       1. associate-*r/ 89.2%

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

       2. metadata-eval 89.2%

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

       3. *-commutative 89.2%

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

       4. associate-*l* 89.2%

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

   16. Simplified 89.2%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\ell}^{2}}{t}\right)\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell \cdot \frac{\frac{\ell}{\sin k}}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\ell}^{2}}{t\_m}\right)\\ \mathbf{elif}\;t\_m \leq 1.52 \cdot 10^{+137}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \left(\frac{2}{\tan k} \cdot {\left(\frac{\sqrt[3]{\frac{\ell}{\sin k}}}{t\_m}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.5e-87)
    (*
     (/ 2.0 (pow k 2.0))
     (* (/ (cos k) (pow (sin k) 2.0)) (/ (pow l 2.0) t_m)))
    (if (<= t_m 1.52e+137)
      (*
       (/ l (+ 2.0 (pow (/ k t_m) 2.0)))
       (* (/ 2.0 (tan k)) (pow (/ (cbrt (/ l (sin k))) t_m) 3.0)))
      (/
       2.0
       (*
        (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0)
        (* 2.0 k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.5e-87) {
		tmp = (2.0 / pow(k, 2.0)) * ((cos(k) / pow(sin(k), 2.0)) * (pow(l, 2.0) / t_m));
	} else if (t_m <= 1.52e+137) {
		tmp = (l / (2.0 + pow((k / t_m), 2.0))) * ((2.0 / tan(k)) * pow((cbrt((l / sin(k))) / t_m), 3.0));
	} else {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 7.5e-87) {
		tmp = (2.0 / Math.pow(k, 2.0)) * ((Math.cos(k) / Math.pow(Math.sin(k), 2.0)) * (Math.pow(l, 2.0) / t_m));
	} else if (t_m <= 1.52e+137) {
		tmp = (l / (2.0 + Math.pow((k / t_m), 2.0))) * ((2.0 / Math.tan(k)) * Math.pow((Math.cbrt((l / Math.sin(k))) / t_m), 3.0));
	} else {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 7.5e-87)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64(cos(k) / (sin(k) ^ 2.0)) * Float64((l ^ 2.0) / t_m)));
	elseif (t_m <= 1.52e+137)
		tmp = Float64(Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(2.0 / tan(k)) * (Float64(cbrt(Float64(l / sin(k))) / t_m) ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 7.5e-87], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.52e+137], N[(N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.5000000000000002e-87

   1. Initial program 51.0%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in k around inf 62.6%

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

      1. associate-*r/ 62.6%

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

      2. times-frac 63.3%

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

      3. *-commutative 63.3%

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

      4. *-commutative 63.3%

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

      5. times-frac 65.5%

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

   7. Simplified 65.5%

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

   if 7.5000000000000002e-87 < t < 1.52000000000000006e137

   1. Initial program 81.5%

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

   3. Simplified 79.1%

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

      1. associate-*r* 79.0%

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

      2. *-un-lft-identity 79.0%

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

      3. times-frac 79.1%

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

      4. associate-/l/ 79.1%

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

   6. Applied egg-rr 79.1%

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

      1. /-rgt-identity 79.1%

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

      2. associate-*l/ 79.2%

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

      3. *-commutative 79.2%

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

      4. times-frac 79.2%

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

      5. *-commutative 79.2%

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

   8. Simplified 79.2%

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

      1. add-cube-cbrt 79.0%

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

      2. pow3 79.1%

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

      3. associate-/r* 79.1%

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

      4. cbrt-div 79.1%

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

      5. unpow3 79.0%

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

      6. add-cbrt-cube 88.2%

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

   10. Applied egg-rr 88.2%

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

   if 1.52000000000000006e137 < t

   1. Initial program 69.7%

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

   3. Simplified 69.7%

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

      1. add-cube-cbrt 69.7%

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

      2. pow3 69.7%

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

      3. *-commutative 69.7%

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

      4. cbrt-prod 69.7%

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

      5. cbrt-div 69.7%

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

      6. rem-cbrt-cube 73.5%

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

      7. cbrt-prod 90.1%

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

      8. pow2 90.1%

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

   6. Applied egg-rr 90.1%

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

   7. Taylor expanded in k around 0 87.1%

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

      1. *-commutative 87.1%

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

   9. Simplified 87.1%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\ell}^{2}}{t}\right)\\ \mathbf{elif}\;t \leq 1.52 \cdot 10^{+137}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{2}{\tan k} \cdot {\left(\frac{\sqrt[3]{\frac{\ell}{\sin k}}}{t}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\ell}^{2}}{t\_m}\right)\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell \cdot \frac{\frac{\ell}{\sin k}}{{t\_m}^{3}}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.2e-80)
    (*
     (/ 2.0 (pow k 2.0))
     (* (/ (cos k) (pow (sin k) 2.0)) (/ (pow l 2.0) t_m)))
    (if (<= t_m 5.6e+102)
      (*
       2.0
       (/
        (/ (* l (/ (/ l (sin k)) (pow t_m 3.0))) (+ 2.0 (pow (/ k t_m) 2.0)))
        (tan k)))
      (/
       2.0
       (*
        (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0)
        (* 2.0 k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.2e-80) {
		tmp = (2.0 / pow(k, 2.0)) * ((cos(k) / pow(sin(k), 2.0)) * (pow(l, 2.0) / t_m));
	} else if (t_m <= 5.6e+102) {
		tmp = 2.0 * (((l * ((l / sin(k)) / pow(t_m, 3.0))) / (2.0 + pow((k / t_m), 2.0))) / tan(k));
	} else {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3.2e-80) {
		tmp = (2.0 / Math.pow(k, 2.0)) * ((Math.cos(k) / Math.pow(Math.sin(k), 2.0)) * (Math.pow(l, 2.0) / t_m));
	} else if (t_m <= 5.6e+102) {
		tmp = 2.0 * (((l * ((l / Math.sin(k)) / Math.pow(t_m, 3.0))) / (2.0 + Math.pow((k / t_m), 2.0))) / Math.tan(k));
	} else {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3.2e-80)
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64(cos(k) / (sin(k) ^ 2.0)) * Float64((l ^ 2.0) / t_m)));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * Float64(Float64(l / sin(k)) / (t_m ^ 3.0))) / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) / tan(k)));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3.2e-80], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(2.0 * N[(N[(N[(l * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.1999999999999999e-80

   1. Initial program 51.5%

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

   3. Simplified 51.8%

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

   5. Taylor expanded in k around inf 63.0%

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

      1. associate-*r/ 63.0%

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

      2. times-frac 63.7%

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

      3. *-commutative 63.7%

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

      4. *-commutative 63.7%

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

      5. times-frac 65.9%

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

   7. Simplified 65.9%

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

   if 3.1999999999999999e-80 < t < 5.60000000000000037e102

   1. Initial program 89.1%

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

   3. Simplified 86.0%

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

      1. associate-*r* 85.9%

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

      2. *-un-lft-identity 85.9%

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

      3. times-frac 86.0%

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

      4. associate-/l/ 86.0%

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

   6. Applied egg-rr 86.0%

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

      1. /-rgt-identity 86.0%

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

      2. associate-*l/ 86.0%

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

      3. *-commutative 86.0%

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

      4. times-frac 86.1%

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

      5. *-commutative 86.1%

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

   8. Simplified 86.1%

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

      1. associate-*r/ 86.0%

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

   10. Applied egg-rr 86.0%

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

       1. associate-/l* 86.1%

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

       2. associate-*r* 89.5%

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

       3. associate-*l/ 89.5%

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

       4. *-lft-identity 89.5%

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

       5. times-frac 89.5%

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

       6. metadata-eval 89.5%

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

       7. associate-*r/ 89.4%

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

       8. associate-/r* 92.8%

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

   12. Simplified 92.8%

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

   if 5.60000000000000037e102 < t

   1. Initial program 65.1%

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

   3. Simplified 65.1%

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

      1. add-cube-cbrt 65.1%

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

      2. pow3 65.1%

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

      3. *-commutative 65.1%

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

      4. cbrt-prod 65.1%

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

      5. cbrt-div 65.1%

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

      6. rem-cbrt-cube 75.7%

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

      7. cbrt-prod 89.2%

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

      8. pow2 89.2%

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

   6. Applied egg-rr 89.2%

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

   7. Taylor expanded in k around 0 86.3%

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

      1. *-commutative 86.3%

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

   9. Simplified 86.3%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \left(\frac{\cos k}{{\sin k}^{2}} \cdot \frac{{\ell}^{2}}{t}\right)\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell \cdot \frac{\frac{\ell}{\sin k}}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.12 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell \cdot \frac{\frac{\ell}{\sin k}}{{t\_m}^{3}}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.12e-79)
    (* (/ 2.0 (tan k)) (/ (pow l 2.0) (* (pow k 2.0) (* t_m (sin k)))))
    (if (<= t_m 5.6e+102)
      (*
       2.0
       (/
        (/ (* l (/ (/ l (sin k)) (pow t_m 3.0))) (+ 2.0 (pow (/ k t_m) 2.0)))
        (tan k)))
      (/
       2.0
       (*
        (pow (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) 3.0)
        (* 2.0 k)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.12e-79) {
		tmp = (2.0 / tan(k)) * (pow(l, 2.0) / (pow(k, 2.0) * (t_m * sin(k))));
	} else if (t_m <= 5.6e+102) {
		tmp = 2.0 * (((l * ((l / sin(k)) / pow(t_m, 3.0))) / (2.0 + pow((k / t_m), 2.0))) / tan(k));
	} else {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.12e-79) {
		tmp = (2.0 / Math.tan(k)) * (Math.pow(l, 2.0) / (Math.pow(k, 2.0) * (t_m * Math.sin(k))));
	} else if (t_m <= 5.6e+102) {
		tmp = 2.0 * (((l * ((l / Math.sin(k)) / Math.pow(t_m, 3.0))) / (2.0 + Math.pow((k / t_m), 2.0))) / Math.tan(k));
	} else {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0) * (2.0 * k));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.12e-79)
		tmp = Float64(Float64(2.0 / tan(k)) * Float64((l ^ 2.0) / Float64((k ^ 2.0) * Float64(t_m * sin(k)))));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(2.0 * Float64(Float64(Float64(l * Float64(Float64(l / sin(k)) / (t_m ^ 3.0))) / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) / tan(k)));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.12e-79], N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(2.0 * N[(N[(N[(l * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.11999999999999996e-79

   1. Initial program 51.5%

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

   3. Simplified 51.8%

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

      1. associate-*r* 56.3%

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

      2. *-un-lft-identity 56.3%

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

      3. times-frac 56.2%

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

      4. associate-/l/ 56.2%

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

   6. Applied egg-rr 56.2%

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

      1. /-rgt-identity 56.2%

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

      2. associate-*l/ 56.3%

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

      3. *-commutative 56.3%

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

      4. times-frac 56.7%

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

      5. *-commutative 56.7%

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

   8. Simplified 56.7%

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

      1. associate-*r/ 56.8%

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

   10. Applied egg-rr 56.8%

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

       1. associate-/l* 56.7%

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

       2. associate-*r* 56.3%

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

       3. *-commutative 56.3%

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

   12. Simplified 56.3%

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

   13. Taylor expanded in t around 0 63.0%

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

   if 1.11999999999999996e-79 < t < 5.60000000000000037e102

   1. Initial program 89.1%

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

   3. Simplified 86.0%

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

      1. associate-*r* 85.9%

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

      2. *-un-lft-identity 85.9%

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

      3. times-frac 86.0%

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

      4. associate-/l/ 86.0%

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

   6. Applied egg-rr 86.0%

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

      1. /-rgt-identity 86.0%

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

      2. associate-*l/ 86.0%

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

      3. *-commutative 86.0%

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

      4. times-frac 86.1%

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

      5. *-commutative 86.1%

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

   8. Simplified 86.1%

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

      1. associate-*r/ 86.0%

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

   10. Applied egg-rr 86.0%

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

       1. associate-/l* 86.1%

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

       2. associate-*r* 89.5%

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

       3. associate-*l/ 89.5%

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

       4. *-lft-identity 89.5%

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

       5. times-frac 89.5%

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

       6. metadata-eval 89.5%

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

       7. associate-*r/ 89.4%

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

       8. associate-/r* 92.8%

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

   12. Simplified 92.8%

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

   if 5.60000000000000037e102 < t

   1. Initial program 65.1%

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

   3. Simplified 65.1%

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

      1. add-cube-cbrt 65.1%

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

      2. pow3 65.1%

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

      3. *-commutative 65.1%

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

      4. cbrt-prod 65.1%

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

      5. cbrt-div 65.1%

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

      6. rem-cbrt-cube 75.7%

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

      7. cbrt-prod 89.2%

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

      8. pow2 89.2%

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

   6. Applied egg-rr 89.2%

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

   7. Taylor expanded in k around 0 86.3%

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

      1. *-commutative 86.3%

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

   9. Simplified 86.3%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.12 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell \cdot \frac{\frac{\ell}{\sin k}}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{elif}\;k \leq 12:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.1e-151)
    (/ 2.0 (* (* 2.0 k) (pow (* (/ (pow t_m 1.5) l) (sqrt (sin k))) 2.0)))
    (if (<= k 12.0)
      (/
       2.0
       (pow (* (* t_m (pow (cbrt l) -2.0)) (cbrt (* 2.0 (pow k 2.0)))) 3.0))
      (* (/ 2.0 (tan k)) (/ (pow l 2.0) (* (pow k 2.0) (* t_m (sin k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.1e-151) {
		tmp = 2.0 / ((2.0 * k) * pow(((pow(t_m, 1.5) / l) * sqrt(sin(k))), 2.0));
	} else if (k <= 12.0) {
		tmp = 2.0 / pow(((t_m * pow(cbrt(l), -2.0)) * cbrt((2.0 * pow(k, 2.0)))), 3.0);
	} else {
		tmp = (2.0 / tan(k)) * (pow(l, 2.0) / (pow(k, 2.0) * (t_m * sin(k))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.1e-151) {
		tmp = 2.0 / ((2.0 * k) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.sqrt(Math.sin(k))), 2.0));
	} else if (k <= 12.0) {
		tmp = 2.0 / Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt((2.0 * Math.pow(k, 2.0)))), 3.0);
	} else {
		tmp = (2.0 / Math.tan(k)) * (Math.pow(l, 2.0) / (Math.pow(k, 2.0) * (t_m * Math.sin(k))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.1e-151)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(Float64((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0)));
	elseif (k <= 12.0)
		tmp = Float64(2.0 / (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * cbrt(Float64(2.0 * (k ^ 2.0)))) ^ 3.0));
	else
		tmp = Float64(Float64(2.0 / tan(k)) * Float64((l ^ 2.0) / Float64((k ^ 2.0) * Float64(t_m * sin(k)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.1e-151], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 12.0], N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.0999999999999999e-151

   1. Initial program 59.8%

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

   3. Simplified 59.8%

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

      1. add-sqr-sqrt 28.5%

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

      2. pow2 28.5%

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

      3. *-commutative 28.5%

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

      4. sqrt-prod 10.3%

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

      5. sqrt-div 10.3%

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

      6. sqrt-pow1 10.9%

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

      7. metadata-eval 10.9%

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

      8. sqrt-prod 9.0%

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

      9. add-sqr-sqrt 12.7%

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

   6. Applied egg-rr 12.7%

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

      1. *-commutative 12.7%

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

   8. Simplified 12.7%

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

   9. Taylor expanded in k around 0 11.6%

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

       1. *-commutative 74.8%

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

   11. Simplified 11.6%

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

   if 2.0999999999999999e-151 < k < 12

   1. Initial program 58.4%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in k around 0 62.9%

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

      1. add-cube-cbrt 62.7%

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

      2. pow3 62.7%

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

      3. associate-/l/ 62.0%

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

      4. cbrt-div 62.0%

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

      5. unpow3 62.0%

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

      6. add-cbrt-cube 68.7%

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

      7. cbrt-prod 81.9%

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

      8. unpow2 81.9%

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

      9. div-inv 81.9%

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

      10. unpow-prod-down 61.9%

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

      11. pow-flip 61.9%

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

      12. metadata-eval 61.9%

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

   7. Applied egg-rr 61.9%

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

      1. cube-prod 81.9%

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

   9. Simplified 81.9%

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

       1. add-cube-cbrt 82.0%

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

       2. pow3 82.1%

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

       3. cbrt-prod 82.0%

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

       4. unpow3 82.0%

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

       5. add-cbrt-cube 91.4%

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

   11. Applied egg-rr 91.4%

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

   if 12 < k

   1. Initial program 50.9%

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

   3. Simplified 49.3%

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

      1. associate-*r* 52.5%

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

      2. *-un-lft-identity 52.5%

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

      3. times-frac 55.8%

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

      4. associate-/l/ 55.8%

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

   6. Applied egg-rr 55.8%

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

      1. /-rgt-identity 55.8%

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

      2. associate-*l/ 55.8%

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

      3. *-commutative 55.8%

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

      4. times-frac 55.8%

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

      5. *-commutative 55.8%

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

   8. Simplified 55.8%

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

      1. associate-*r/ 52.5%

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

   10. Applied egg-rr 52.5%

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

       1. associate-/l* 55.8%

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

       2. associate-*r* 55.8%

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

       3. *-commutative 55.8%

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

   12. Simplified 55.8%

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

   13. Taylor expanded in t around 0 74.3%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{elif}\;k \leq 12:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{elif}\;k \leq 0.00185:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.15e-151)
    (/ 2.0 (* (* 2.0 k) (pow (* (/ (pow t_m 1.5) l) (sqrt (sin k))) 2.0)))
    (if (<= k 0.00185)
      (/
       2.0
       (pow (* t_m (* (pow (cbrt l) -2.0) (cbrt (* 2.0 (pow k 2.0))))) 3.0))
      (* (/ 2.0 (tan k)) (/ (pow l 2.0) (* (pow k 2.0) (* t_m (sin k)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.15e-151) {
		tmp = 2.0 / ((2.0 * k) * pow(((pow(t_m, 1.5) / l) * sqrt(sin(k))), 2.0));
	} else if (k <= 0.00185) {
		tmp = 2.0 / pow((t_m * (pow(cbrt(l), -2.0) * cbrt((2.0 * pow(k, 2.0))))), 3.0);
	} else {
		tmp = (2.0 / tan(k)) * (pow(l, 2.0) / (pow(k, 2.0) * (t_m * sin(k))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.15e-151) {
		tmp = 2.0 / ((2.0 * k) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.sqrt(Math.sin(k))), 2.0));
	} else if (k <= 0.00185) {
		tmp = 2.0 / Math.pow((t_m * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt((2.0 * Math.pow(k, 2.0))))), 3.0);
	} else {
		tmp = (2.0 / Math.tan(k)) * (Math.pow(l, 2.0) / (Math.pow(k, 2.0) * (t_m * Math.sin(k))));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.15e-151)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(Float64((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0)));
	elseif (k <= 0.00185)
		tmp = Float64(2.0 / (Float64(t_m * Float64((cbrt(l) ^ -2.0) * cbrt(Float64(2.0 * (k ^ 2.0))))) ^ 3.0));
	else
		tmp = Float64(Float64(2.0 / tan(k)) * Float64((l ^ 2.0) / Float64((k ^ 2.0) * Float64(t_m * sin(k)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.15e-151], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.00185], N[(2.0 / N[Power[N[(t$95$m * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.15000000000000009e-151

   1. Initial program 59.8%

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

   3. Simplified 59.8%

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

      1. add-sqr-sqrt 28.5%

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

      2. pow2 28.5%

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

      3. *-commutative 28.5%

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

      4. sqrt-prod 10.3%

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

      5. sqrt-div 10.3%

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

      6. sqrt-pow1 10.9%

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

      7. metadata-eval 10.9%

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

      8. sqrt-prod 9.0%

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

      9. add-sqr-sqrt 12.7%

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

   6. Applied egg-rr 12.7%

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

      1. *-commutative 12.7%

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

   8. Simplified 12.7%

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

   9. Taylor expanded in k around 0 11.6%

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

       1. *-commutative 74.8%

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

   11. Simplified 11.6%

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

   if 2.15000000000000009e-151 < k < 0.0018500000000000001

   1. Initial program 58.4%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in k around 0 62.9%

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

      1. add-cube-cbrt 62.7%

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

      2. pow3 62.7%

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

      3. cbrt-prod 62.7%

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

      4. associate-/l/ 62.0%

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

      5. cbrt-div 62.0%

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

      6. unpow3 62.0%

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

      7. add-cbrt-cube 71.9%

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

      8. cbrt-prod 91.3%

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

      9. unpow2 91.3%

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

      10. div-inv 91.3%

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

      11. pow-flip 91.4%

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

      12. metadata-eval 91.4%

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

   7. Applied egg-rr 91.4%

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

      1. associate-*l* 91.4%

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

   9. Simplified 91.4%

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

   if 0.0018500000000000001 < k

   1. Initial program 50.9%

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

   3. Simplified 49.3%

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

      1. associate-*r* 52.5%

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

      2. *-un-lft-identity 52.5%

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

      3. times-frac 55.8%

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

      4. associate-/l/ 55.8%

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

   6. Applied egg-rr 55.8%

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

      1. /-rgt-identity 55.8%

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

      2. associate-*l/ 55.8%

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

      3. *-commutative 55.8%

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

      4. times-frac 55.8%

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

      5. *-commutative 55.8%

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

   8. Simplified 55.8%

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

      1. associate-*r/ 52.5%

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

   10. Applied egg-rr 52.5%

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

       1. associate-/l* 55.8%

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

       2. associate-*r* 55.8%

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

       3. *-commutative 55.8%

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

   12. Simplified 55.8%

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

   13. Taylor expanded in t around 0 74.3%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{elif}\;k \leq 0.00185:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{2}{\tan k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{-162}:\\ \;\;\;\;t\_2 \cdot \left(\frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \frac{\frac{\ell}{k}}{{t\_m}^{3}}\right)\\ \mathbf{elif}\;k \leq 0.018:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot {\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ 2.0 (tan k))))
   (*
    t_s
    (if (<= k 1.02e-162)
      (* t_2 (* (/ l (+ 2.0 (pow (/ k t_m) 2.0))) (/ (/ l k) (pow t_m 3.0))))
      (if (<= k 0.018)
        (/ 2.0 (* (* 2.0 (pow k 2.0)) (pow (/ t_m (pow (cbrt l) 2.0)) 3.0)))
        (* t_2 (/ (pow l 2.0) (* (pow k 2.0) (* t_m (sin k))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 / tan(k);
	double tmp;
	if (k <= 1.02e-162) {
		tmp = t_2 * ((l / (2.0 + pow((k / t_m), 2.0))) * ((l / k) / pow(t_m, 3.0)));
	} else if (k <= 0.018) {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * pow((t_m / pow(cbrt(l), 2.0)), 3.0));
	} else {
		tmp = t_2 * (pow(l, 2.0) / (pow(k, 2.0) * (t_m * sin(k))));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 2.0 / Math.tan(k);
	double tmp;
	if (k <= 1.02e-162) {
		tmp = t_2 * ((l / (2.0 + Math.pow((k / t_m), 2.0))) * ((l / k) / Math.pow(t_m, 3.0)));
	} else if (k <= 0.018) {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0));
	} else {
		tmp = t_2 * (Math.pow(l, 2.0) / (Math.pow(k, 2.0) * (t_m * Math.sin(k))));
	}
	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(2.0 / tan(k))
	tmp = 0.0
	if (k <= 1.02e-162)
		tmp = Float64(t_2 * Float64(Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(l / k) / (t_m ^ 3.0))));
	elseif (k <= 0.018)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * (Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0)));
	else
		tmp = Float64(t_2 * Float64((l ^ 2.0) / Float64((k ^ 2.0) * Float64(t_m * sin(k)))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.02e-162], N[(t$95$2 * N[(N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.018], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $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], N[(t$95$2 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.01999999999999998e-162

   1. Initial program 59.2%

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

   3. Simplified 59.6%

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

      1. associate-*r* 63.2%

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

      2. *-un-lft-identity 63.2%

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

      3. times-frac 62.5%

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

      4. associate-/l/ 62.5%

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

   6. Applied egg-rr 62.5%

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

      1. /-rgt-identity 62.5%

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

      2. associate-*l/ 62.6%

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

      3. *-commutative 62.6%

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

      4. times-frac 63.2%

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

      5. *-commutative 63.2%

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

   8. Simplified 63.2%

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

      1. associate-*r/ 63.8%

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

   10. Applied egg-rr 63.8%

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

       1. associate-/l* 63.2%

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

       2. associate-*r* 64.3%

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

       3. *-commutative 64.3%

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

   12. Simplified 64.3%

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

   13. Taylor expanded in k around 0 59.8%

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

       1. associate-/r* 61.9%

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

   15. Simplified 61.9%

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

   if 1.01999999999999998e-162 < k < 0.0179999999999999986

   1. Initial program 61.6%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in k around 0 68.5%

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

      1. add-cube-cbrt 68.4%

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

      2. pow3 68.4%

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

      3. associate-/l/ 64.7%

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

      4. cbrt-div 64.7%

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

      5. unpow3 64.7%

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

      6. add-cbrt-cube 70.5%

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

      7. cbrt-prod 84.6%

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

      8. unpow2 84.6%

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

      9. div-inv 84.6%

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

      10. unpow-prod-down 64.7%

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

      11. pow-flip 64.7%

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

      12. metadata-eval 64.7%

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

   7. Applied egg-rr 64.7%

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

      1. cube-prod 84.6%

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

   9. Simplified 84.6%

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

       1. metadata-eval 84.6%

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

       2. pow-flip 84.6%

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

       3. div-inv 84.6%

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

   11. Applied egg-rr 84.6%

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

   if 0.0179999999999999986 < k

   1. Initial program 50.9%

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

   3. Simplified 49.3%

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

      1. associate-*r* 52.5%

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

      2. *-un-lft-identity 52.5%

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

      3. times-frac 55.8%

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

      4. associate-/l/ 55.8%

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

   6. Applied egg-rr 55.8%

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

      1. /-rgt-identity 55.8%

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

      2. associate-*l/ 55.8%

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

      3. *-commutative 55.8%

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

      4. times-frac 55.8%

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

      5. *-commutative 55.8%

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

   8. Simplified 55.8%

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

      1. associate-*r/ 52.5%

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

   10. Applied egg-rr 52.5%

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

       1. associate-/l* 55.8%

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

       2. associate-*r* 55.8%

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

       3. *-commutative 55.8%

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

   12. Simplified 55.8%

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

   13. Taylor expanded in t around 0 74.3%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \left(\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\right)\\ \mathbf{elif}\;k \leq 0.018:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{\frac{t\_m}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \left(\frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \frac{\frac{\ell}{k}}{{t\_m}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot {\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.2e-114)
    (/ 1.0 (* (/ t_m 2.0) (/ (pow k 4.0) (pow l 2.0))))
    (if (<= t_m 1.05e+101)
      (*
       (/ 2.0 (tan k))
       (* (/ l (+ 2.0 (pow (/ k t_m) 2.0))) (/ (/ l k) (pow t_m 3.0))))
      (/ 2.0 (* (* 2.0 (pow k 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 <= 1.2e-114) {
		tmp = 1.0 / ((t_m / 2.0) * (pow(k, 4.0) / pow(l, 2.0)));
	} else if (t_m <= 1.05e+101) {
		tmp = (2.0 / tan(k)) * ((l / (2.0 + pow((k / t_m), 2.0))) * ((l / k) / pow(t_m, 3.0)));
	} else {
		tmp = 2.0 / ((2.0 * pow(k, 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 <= 1.2e-114) {
		tmp = 1.0 / ((t_m / 2.0) * (Math.pow(k, 4.0) / Math.pow(l, 2.0)));
	} else if (t_m <= 1.05e+101) {
		tmp = (2.0 / Math.tan(k)) * ((l / (2.0 + Math.pow((k / t_m), 2.0))) * ((l / k) / Math.pow(t_m, 3.0)));
	} else {
		tmp = 2.0 / ((2.0 * Math.pow(k, 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 <= 1.2e-114)
		tmp = Float64(1.0 / Float64(Float64(t_m / 2.0) * Float64((k ^ 4.0) / (l ^ 2.0))));
	elseif (t_m <= 1.05e+101)
		tmp = Float64(Float64(2.0 / tan(k)) * Float64(Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(l / k) / (t_m ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 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, 1.2e-114], N[(1.0 / N[(N[(t$95$m / 2.0), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+101], N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $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]

Derivation

1. Split input into 3 regimes

2. if t < 1.2000000000000001e-114

   1. Initial program 50.0%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in k around inf 61.4%

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

   6. Taylor expanded in k around 0 52.6%

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

      1. associate-*r/ 52.6%

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

      2. *-commutative 52.6%

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

   8. Simplified 52.6%

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

      1. clear-num 52.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}}} \]

      2. inv-pow 52.6%

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

   10. Applied egg-rr 52.6%

       \[\leadsto \color{blue}{{\left(\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 52.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}}} \]

       2. times-frac 53.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   12. Simplified 53.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   if 1.2000000000000001e-114 < t < 1.05e101

   1. Initial program 88.8%

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

   3. Simplified 86.4%

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

      1. associate-*r* 86.3%

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

      2. *-un-lft-identity 86.3%

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

      3. times-frac 86.4%

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

      4. associate-/l/ 86.4%

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

   6. Applied egg-rr 86.4%

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

      1. /-rgt-identity 86.4%

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

      2. associate-*l/ 86.4%

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

      3. *-commutative 86.4%

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

      4. times-frac 86.5%

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

      5. *-commutative 86.5%

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

   8. Simplified 86.5%

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

      1. associate-*r/ 86.4%

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

   10. Applied egg-rr 86.4%

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

       1. associate-/l* 86.5%

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

       2. associate-*r* 89.2%

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

       3. *-commutative 89.2%

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

   12. Simplified 89.2%

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

   13. Taylor expanded in k around 0 71.5%

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

       1. associate-/r* 74.1%

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

   15. Simplified 74.1%

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

   if 1.05e101 < t

   1. Initial program 65.1%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in k around 0 56.4%

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

      1. add-cube-cbrt 56.4%

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

      2. pow3 56.4%

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

      3. associate-/l/ 50.4%

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

      4. cbrt-div 50.4%

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

      5. unpow3 50.4%

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

      6. add-cbrt-cube 58.3%

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

      7. cbrt-prod 69.2%

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

      8. unpow2 69.2%

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

      9. div-inv 69.2%

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

      10. unpow-prod-down 50.4%

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

      11. pow-flip 50.4%

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

      12. metadata-eval 50.4%

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

   7. Applied egg-rr 50.4%

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

      1. cube-prod 69.2%

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

   9. Simplified 69.2%

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

       1. metadata-eval 69.2%

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

       2. pow-flip 69.2%

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

       3. div-inv 69.2%

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

   11. Applied egg-rr 69.2%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \left(\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 65.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{\frac{t\_m}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{elif}\;t\_m \leq 1.05 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \left(\frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \frac{\frac{\ell}{k}}{{t\_m}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.2e-114)
    (/ 1.0 (* (/ t_m 2.0) (/ (pow k 4.0) (pow l 2.0))))
    (if (<= t_m 1.05e+101)
      (*
       (/ 2.0 (tan k))
       (* (/ l (+ 2.0 (pow (/ k t_m) 2.0))) (/ (/ l k) (pow t_m 3.0))))
      (/ 2.0 (* (pow (* t_m (pow (cbrt l) -2.0)) 3.0) (* 2.0 (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 (t_m <= 1.2e-114) {
		tmp = 1.0 / ((t_m / 2.0) * (pow(k, 4.0) / pow(l, 2.0)));
	} else if (t_m <= 1.05e+101) {
		tmp = (2.0 / tan(k)) * ((l / (2.0 + pow((k / t_m), 2.0))) * ((l / k) / pow(t_m, 3.0)));
	} else {
		tmp = 2.0 / (pow((t_m * pow(cbrt(l), -2.0)), 3.0) * (2.0 * 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 (t_m <= 1.2e-114) {
		tmp = 1.0 / ((t_m / 2.0) * (Math.pow(k, 4.0) / Math.pow(l, 2.0)));
	} else if (t_m <= 1.05e+101) {
		tmp = (2.0 / Math.tan(k)) * ((l / (2.0 + Math.pow((k / t_m), 2.0))) * ((l / k) / Math.pow(t_m, 3.0)));
	} else {
		tmp = 2.0 / (Math.pow((t_m * Math.pow(Math.cbrt(l), -2.0)), 3.0) * (2.0 * 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 (t_m <= 1.2e-114)
		tmp = Float64(1.0 / Float64(Float64(t_m / 2.0) * Float64((k ^ 4.0) / (l ^ 2.0))));
	elseif (t_m <= 1.05e+101)
		tmp = Float64(Float64(2.0 / tan(k)) * Float64(Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(l / k) / (t_m ^ 3.0))));
	else
		tmp = Float64(2.0 / Float64((Float64(t_m * (cbrt(l) ^ -2.0)) ^ 3.0) * Float64(2.0 * (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[t$95$m, 1.2e-114], N[(1.0 / N[(N[(t$95$m / 2.0), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.05e+101], N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.2000000000000001e-114

   1. Initial program 50.0%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in k around inf 61.4%

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

   6. Taylor expanded in k around 0 52.6%

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

      1. associate-*r/ 52.6%

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

      2. *-commutative 52.6%

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

   8. Simplified 52.6%

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

      1. clear-num 52.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}}} \]

      2. inv-pow 52.6%

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

   10. Applied egg-rr 52.6%

       \[\leadsto \color{blue}{{\left(\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 52.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}}} \]

       2. times-frac 53.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   12. Simplified 53.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   if 1.2000000000000001e-114 < t < 1.05e101

   1. Initial program 88.8%

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

   3. Simplified 86.4%

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

      1. associate-*r* 86.3%

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

      2. *-un-lft-identity 86.3%

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

      3. times-frac 86.4%

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

      4. associate-/l/ 86.4%

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

   6. Applied egg-rr 86.4%

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

      1. /-rgt-identity 86.4%

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

      2. associate-*l/ 86.4%

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

      3. *-commutative 86.4%

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

      4. times-frac 86.5%

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

      5. *-commutative 86.5%

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

   8. Simplified 86.5%

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

      1. associate-*r/ 86.4%

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

   10. Applied egg-rr 86.4%

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

       1. associate-/l* 86.5%

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

       2. associate-*r* 89.2%

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

       3. *-commutative 89.2%

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

   12. Simplified 89.2%

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

   13. Taylor expanded in k around 0 71.5%

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

       1. associate-/r* 74.1%

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

   15. Simplified 74.1%

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

   if 1.05e101 < t

   1. Initial program 65.1%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in k around 0 56.4%

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

      1. add-cube-cbrt 56.4%

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

      2. pow3 56.4%

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

      3. associate-/l/ 50.4%

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

      4. cbrt-div 50.4%

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

      5. unpow3 50.4%

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

      6. add-cbrt-cube 58.3%

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

      7. cbrt-prod 69.2%

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

      8. unpow2 69.2%

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

      9. div-inv 69.2%

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

      10. unpow-prod-down 50.4%

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

      11. pow-flip 50.4%

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

      12. metadata-eval 50.4%

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

   7. Applied egg-rr 50.4%

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

      1. cube-prod 69.2%

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

   9. Simplified 69.2%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \left(\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(2 \cdot {k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.7:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 0.7)
    (/ 2.0 (* (* 2.0 k) (pow (* (/ (pow t_m 1.5) l) (sqrt (sin k))) 2.0)))
    (* (/ 2.0 (tan k)) (/ (pow l 2.0) (* (pow k 2.0) (* t_m (sin k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.7) {
		tmp = 2.0 / ((2.0 * k) * pow(((pow(t_m, 1.5) / l) * sqrt(sin(k))), 2.0));
	} else {
		tmp = (2.0 / tan(k)) * (pow(l, 2.0) / (pow(k, 2.0) * (t_m * sin(k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 0.7d0) then
        tmp = 2.0d0 / ((2.0d0 * k) * ((((t_m ** 1.5d0) / l) * sqrt(sin(k))) ** 2.0d0))
    else
        tmp = (2.0d0 / tan(k)) * ((l ** 2.0d0) / ((k ** 2.0d0) * (t_m * sin(k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 0.7) {
		tmp = 2.0 / ((2.0 * k) * Math.pow(((Math.pow(t_m, 1.5) / l) * Math.sqrt(Math.sin(k))), 2.0));
	} else {
		tmp = (2.0 / Math.tan(k)) * (Math.pow(l, 2.0) / (Math.pow(k, 2.0) * (t_m * Math.sin(k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 0.7:
		tmp = 2.0 / ((2.0 * k) * math.pow(((math.pow(t_m, 1.5) / l) * math.sqrt(math.sin(k))), 2.0))
	else:
		tmp = (2.0 / math.tan(k)) * (math.pow(l, 2.0) / (math.pow(k, 2.0) * (t_m * math.sin(k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 0.7)
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * (Float64(Float64((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0)));
	else
		tmp = Float64(Float64(2.0 / tan(k)) * Float64((l ^ 2.0) / Float64((k ^ 2.0) * Float64(t_m * sin(k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 0.7)
		tmp = 2.0 / ((2.0 * k) * ((((t_m ^ 1.5) / l) * sqrt(sin(k))) ^ 2.0));
	else
		tmp = (2.0 / tan(k)) * ((l ^ 2.0) / ((k ^ 2.0) * (t_m * sin(k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 0.7], N[(2.0 / N[(N[(2.0 * k), $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Sin[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.69999999999999996

   1. Initial program 59.6%

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

   3. Simplified 59.6%

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

      1. add-sqr-sqrt 29.6%

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

      2. pow2 29.6%

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

      3. *-commutative 29.6%

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

      4. sqrt-prod 14.0%

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

      5. sqrt-div 14.0%

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

      6. sqrt-pow1 15.5%

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

      7. metadata-eval 15.5%

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

      8. sqrt-prod 9.8%

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

      9. add-sqr-sqrt 17.5%

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

   6. Applied egg-rr 17.5%

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

      1. *-commutative 17.5%

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

   8. Simplified 17.5%

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

   9. Taylor expanded in k around 0 16.1%

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

       1. *-commutative 76.3%

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

   11. Simplified 16.1%

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

   if 0.69999999999999996 < k

   1. Initial program 50.9%

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

   3. Simplified 49.3%

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

      1. associate-*r* 52.5%

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

      2. *-un-lft-identity 52.5%

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

      3. times-frac 55.8%

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

      4. associate-/l/ 55.8%

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

   6. Applied egg-rr 55.8%

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

      1. /-rgt-identity 55.8%

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

      2. associate-*l/ 55.8%

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

      3. *-commutative 55.8%

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

      4. times-frac 55.8%

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

      5. *-commutative 55.8%

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

   8. Simplified 55.8%

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

      1. associate-*r/ 52.5%

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

   10. Applied egg-rr 52.5%

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

       1. associate-/l* 55.8%

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

       2. associate-*r* 55.8%

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

       3. *-commutative 55.8%

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

   12. Simplified 55.8%

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

   13. Taylor expanded in t around 0 74.3%

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

4. Final simplification 30.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.7:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 65.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{\frac{t\_m}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \left(\frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \frac{\frac{\ell}{k}}{{t\_m}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.2e-114)
    (/ 1.0 (* (/ t_m 2.0) (/ (pow k 4.0) (pow l 2.0))))
    (*
     (/ 2.0 (tan k))
     (* (/ l (+ 2.0 (pow (/ k t_m) 2.0))) (/ (/ l k) (pow t_m 3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.2e-114) {
		tmp = 1.0 / ((t_m / 2.0) * (pow(k, 4.0) / pow(l, 2.0)));
	} else {
		tmp = (2.0 / tan(k)) * ((l / (2.0 + pow((k / t_m), 2.0))) * ((l / k) / pow(t_m, 3.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.2d-114) then
        tmp = 1.0d0 / ((t_m / 2.0d0) * ((k ** 4.0d0) / (l ** 2.0d0)))
    else
        tmp = (2.0d0 / tan(k)) * ((l / (2.0d0 + ((k / t_m) ** 2.0d0))) * ((l / k) / (t_m ** 3.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.2e-114) {
		tmp = 1.0 / ((t_m / 2.0) * (Math.pow(k, 4.0) / Math.pow(l, 2.0)));
	} else {
		tmp = (2.0 / Math.tan(k)) * ((l / (2.0 + Math.pow((k / t_m), 2.0))) * ((l / k) / Math.pow(t_m, 3.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.2e-114:
		tmp = 1.0 / ((t_m / 2.0) * (math.pow(k, 4.0) / math.pow(l, 2.0)))
	else:
		tmp = (2.0 / math.tan(k)) * ((l / (2.0 + math.pow((k / t_m), 2.0))) * ((l / k) / math.pow(t_m, 3.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.2e-114)
		tmp = Float64(1.0 / Float64(Float64(t_m / 2.0) * Float64((k ^ 4.0) / (l ^ 2.0))));
	else
		tmp = Float64(Float64(2.0 / tan(k)) * Float64(Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(l / k) / (t_m ^ 3.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.2e-114)
		tmp = 1.0 / ((t_m / 2.0) * ((k ^ 4.0) / (l ^ 2.0)));
	else
		tmp = (2.0 / tan(k)) * ((l / (2.0 + ((k / t_m) ^ 2.0))) * ((l / k) / (t_m ^ 3.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.2e-114], N[(1.0 / N[(N[(t$95$m / 2.0), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.2000000000000001e-114

   1. Initial program 50.0%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in k around inf 61.4%

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

   6. Taylor expanded in k around 0 52.6%

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

      1. associate-*r/ 52.6%

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

      2. *-commutative 52.6%

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

   8. Simplified 52.6%

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

      1. clear-num 52.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}}} \]

      2. inv-pow 52.6%

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

   10. Applied egg-rr 52.6%

       \[\leadsto \color{blue}{{\left(\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 52.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}}} \]

       2. times-frac 53.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   12. Simplified 53.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   if 1.2000000000000001e-114 < t

   1. Initial program 77.0%

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

   3. Simplified 75.7%

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

      1. associate-*r* 78.8%

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

      2. *-un-lft-identity 78.8%

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

      3. times-frac 78.8%

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

      4. associate-/l/ 78.8%

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

   6. Applied egg-rr 78.8%

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

      1. /-rgt-identity 78.8%

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

      2. associate-*l/ 78.8%

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

      3. *-commutative 78.8%

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

      4. times-frac 78.8%

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

      5. *-commutative 78.8%

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

   8. Simplified 78.8%

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

      1. associate-*r/ 78.8%

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

   10. Applied egg-rr 78.8%

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

       1. associate-/l* 78.8%

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

       2. associate-*r* 80.2%

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

       3. *-commutative 80.2%

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{{t}^{3} \cdot \sin k}} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\right) \]

   12. Simplified 80.2%

       \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \left(\frac{\ell}{{t}^{3} \cdot \sin k} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\right)} \]

   13. Taylor expanded in k around 0 71.4%

       \[\leadsto \frac{2}{\tan k} \cdot \left(\color{blue}{\frac{\ell}{k \cdot {t}^{3}}} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\right) \]
   14. Step-by-step derivation

       1. associate-/r* 72.6%

          \[\leadsto \frac{2}{\tan k} \cdot \left(\color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\right) \]

   15. Simplified 72.6%

       \[\leadsto \frac{2}{\tan k} \cdot \left(\color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \left(\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 64.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t\_m}\right)}^{2}} \cdot \left(\frac{2}{k} \cdot \frac{\ell}{\sin k \cdot {t\_m}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.3e-14)
    (* 2.0 (/ (* (cos k) (pow l 2.0)) (* t_m (pow k 4.0))))
    (*
     (/ l (+ 2.0 (pow (/ k t_m) 2.0)))
     (* (/ 2.0 k) (/ l (* (sin k) (pow t_m 3.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e-14) {
		tmp = 2.0 * ((cos(k) * pow(l, 2.0)) / (t_m * pow(k, 4.0)));
	} else {
		tmp = (l / (2.0 + pow((k / t_m), 2.0))) * ((2.0 / k) * (l / (sin(k) * pow(t_m, 3.0))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.3d-14) then
        tmp = 2.0d0 * ((cos(k) * (l ** 2.0d0)) / (t_m * (k ** 4.0d0)))
    else
        tmp = (l / (2.0d0 + ((k / t_m) ** 2.0d0))) * ((2.0d0 / k) * (l / (sin(k) * (t_m ** 3.0d0))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e-14) {
		tmp = 2.0 * ((Math.cos(k) * Math.pow(l, 2.0)) / (t_m * Math.pow(k, 4.0)));
	} else {
		tmp = (l / (2.0 + Math.pow((k / t_m), 2.0))) * ((2.0 / k) * (l / (Math.sin(k) * Math.pow(t_m, 3.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.3e-14:
		tmp = 2.0 * ((math.cos(k) * math.pow(l, 2.0)) / (t_m * math.pow(k, 4.0)))
	else:
		tmp = (l / (2.0 + math.pow((k / t_m), 2.0))) * ((2.0 / k) * (l / (math.sin(k) * math.pow(t_m, 3.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.3e-14)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * (l ^ 2.0)) / Float64(t_m * (k ^ 4.0))));
	else
		tmp = Float64(Float64(l / Float64(2.0 + (Float64(k / t_m) ^ 2.0))) * Float64(Float64(2.0 / k) * Float64(l / Float64(sin(k) * (t_m ^ 3.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.3e-14)
		tmp = 2.0 * ((cos(k) * (l ^ 2.0)) / (t_m * (k ^ 4.0)));
	else
		tmp = (l / (2.0 + ((k / t_m) ^ 2.0))) * ((2.0 / k) * (l / (sin(k) * (t_m ^ 3.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.3e-14], 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[(N[(l / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] * N[(l / N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.29999999999999998e-14

   1. Initial program 52.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf 63.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 54.6%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{4} \cdot t}} \]

   if 2.29999999999999998e-14 < t

   1. Initial program 75.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 79.6%

         \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. *-un-lft-identity 79.6%

         \[\leadsto \frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}{\color{blue}{1 \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      3. times-frac 79.6%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

      4. associate-/l/ 79.6%

         \[\leadsto \frac{\color{blue}{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 79.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. /-rgt-identity 79.6%

         \[\leadsto \color{blue}{\left(\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-*l/ 79.7%

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. *-commutative 79.7%

         \[\leadsto \frac{2 \cdot \ell}{\color{blue}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. times-frac 79.7%

         \[\leadsto \color{blue}{\left(\frac{2}{\tan k} \cdot \frac{\ell}{{t}^{3} \cdot \sin k}\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      5. *-commutative 79.7%

         \[\leadsto \left(\frac{2}{\tan k} \cdot \frac{\ell}{\color{blue}{\sin k \cdot {t}^{3}}}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   8. Simplified 79.7%

      \[\leadsto \color{blue}{\left(\frac{2}{\tan k} \cdot \frac{\ell}{\sin k \cdot {t}^{3}}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   9. Taylor expanded in k around 0 71.4%

      \[\leadsto \left(\frac{2}{\color{blue}{k}} \cdot \frac{\ell}{\sin k \cdot {t}^{3}}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{2}{k} \cdot \frac{\ell}{\sin k \cdot {t}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 61.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \left(0.5 \cdot \frac{{\ell}^{2}}{k \cdot {t\_m}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.3e-14)
    (* 2.0 (/ (* (cos k) (pow l 2.0)) (* t_m (pow k 4.0))))
    (* (/ 2.0 (tan k)) (* 0.5 (/ (pow l 2.0) (* k (pow t_m 3.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e-14) {
		tmp = 2.0 * ((cos(k) * pow(l, 2.0)) / (t_m * pow(k, 4.0)));
	} else {
		tmp = (2.0 / tan(k)) * (0.5 * (pow(l, 2.0) / (k * pow(t_m, 3.0))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.3d-14) then
        tmp = 2.0d0 * ((cos(k) * (l ** 2.0d0)) / (t_m * (k ** 4.0d0)))
    else
        tmp = (2.0d0 / tan(k)) * (0.5d0 * ((l ** 2.0d0) / (k * (t_m ** 3.0d0))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e-14) {
		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)) * (0.5 * (Math.pow(l, 2.0) / (k * Math.pow(t_m, 3.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.3e-14:
		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)) * (0.5 * (math.pow(l, 2.0) / (k * math.pow(t_m, 3.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.3e-14)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * (l ^ 2.0)) / Float64(t_m * (k ^ 4.0))));
	else
		tmp = Float64(Float64(2.0 / tan(k)) * Float64(0.5 * Float64((l ^ 2.0) / Float64(k * (t_m ^ 3.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.3e-14)
		tmp = 2.0 * ((cos(k) * (l ^ 2.0)) / (t_m * (k ^ 4.0)));
	else
		tmp = (2.0 / tan(k)) * (0.5 * ((l ^ 2.0) / (k * (t_m ^ 3.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.3e-14], 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[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.29999999999999998e-14

   1. Initial program 52.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf 63.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 54.6%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{4} \cdot t}} \]

   if 2.29999999999999998e-14 < t

   1. Initial program 75.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* 79.6%

         \[\leadsto \frac{\color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. *-un-lft-identity 79.6%

         \[\leadsto \frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell\right) \cdot \ell}{\color{blue}{1 \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

      3. times-frac 79.6%

         \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

      4. associate-/l/ 79.6%

         \[\leadsto \frac{\color{blue}{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Applied egg-rr 79.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell}{1} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. /-rgt-identity 79.6%

         \[\leadsto \color{blue}{\left(\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \ell\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      2. associate-*l/ 79.7%

         \[\leadsto \color{blue}{\frac{2 \cdot \ell}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. *-commutative 79.7%

         \[\leadsto \frac{2 \cdot \ell}{\color{blue}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. times-frac 79.7%

         \[\leadsto \color{blue}{\left(\frac{2}{\tan k} \cdot \frac{\ell}{{t}^{3} \cdot \sin k}\right)} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      5. *-commutative 79.7%

         \[\leadsto \left(\frac{2}{\tan k} \cdot \frac{\ell}{\color{blue}{\sin k \cdot {t}^{3}}}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   8. Simplified 79.7%

      \[\leadsto \color{blue}{\left(\frac{2}{\tan k} \cdot \frac{\ell}{\sin k \cdot {t}^{3}}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 79.7%

         \[\leadsto \color{blue}{\frac{\left(\frac{2}{\tan k} \cdot \frac{\ell}{\sin k \cdot {t}^{3}}\right) \cdot \ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   10. Applied egg-rr 79.7%

       \[\leadsto \color{blue}{\frac{\left(\frac{2}{\tan k} \cdot \frac{\ell}{\sin k \cdot {t}^{3}}\right) \cdot \ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   11. Step-by-step derivation

       1. associate-/l* 79.7%

          \[\leadsto \color{blue}{\left(\frac{2}{\tan k} \cdot \frac{\ell}{\sin k \cdot {t}^{3}}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

       2. associate-*r* 81.4%

          \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \left(\frac{\ell}{\sin k \cdot {t}^{3}} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\right)} \]

       3. *-commutative 81.4%

          \[\leadsto \frac{2}{\tan k} \cdot \left(\frac{\ell}{\color{blue}{{t}^{3} \cdot \sin k}} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\right) \]

   12. Simplified 81.4%

       \[\leadsto \color{blue}{\frac{2}{\tan k} \cdot \left(\frac{\ell}{{t}^{3} \cdot \sin k} \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\right)} \]

   13. Taylor expanded in k around 0 70.9%

       \[\leadsto \frac{2}{\tan k} \cdot \color{blue}{\left(0.5 \cdot \frac{{\ell}^{2}}{k \cdot {t}^{3}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \left(0.5 \cdot \frac{{\ell}^{2}}{k \cdot {t}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 61.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.3 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{k \cdot {t\_m}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.3e-14)
    (* 2.0 (/ (* (cos k) (pow l 2.0)) (* t_m (pow k 4.0))))
    (/
     (* (/ (/ 2.0 k) (* k (pow t_m 3.0))) (* l l))
     (+ 2.0 (pow (/ k t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e-14) {
		tmp = 2.0 * ((cos(k) * pow(l, 2.0)) / (t_m * pow(k, 4.0)));
	} else {
		tmp = (((2.0 / k) / (k * pow(t_m, 3.0))) * (l * l)) / (2.0 + pow((k / t_m), 2.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.3d-14) then
        tmp = 2.0d0 * ((cos(k) * (l ** 2.0d0)) / (t_m * (k ** 4.0d0)))
    else
        tmp = (((2.0d0 / k) / (k * (t_m ** 3.0d0))) * (l * l)) / (2.0d0 + ((k / t_m) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.3e-14) {
		tmp = 2.0 * ((Math.cos(k) * Math.pow(l, 2.0)) / (t_m * Math.pow(k, 4.0)));
	} else {
		tmp = (((2.0 / k) / (k * Math.pow(t_m, 3.0))) * (l * l)) / (2.0 + Math.pow((k / t_m), 2.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.3e-14:
		tmp = 2.0 * ((math.cos(k) * math.pow(l, 2.0)) / (t_m * math.pow(k, 4.0)))
	else:
		tmp = (((2.0 / k) / (k * math.pow(t_m, 3.0))) * (l * l)) / (2.0 + math.pow((k / t_m), 2.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.3e-14)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * (l ^ 2.0)) / Float64(t_m * (k ^ 4.0))));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 / k) / Float64(k * (t_m ^ 3.0))) * Float64(l * l)) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.3e-14)
		tmp = 2.0 * ((cos(k) * (l ^ 2.0)) / (t_m * (k ^ 4.0)));
	else
		tmp = (((2.0 / k) / (k * (t_m ^ 3.0))) * (l * l)) / (2.0 + ((k / t_m) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.3e-14], 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[(N[(N[(N[(2.0 / k), $MachinePrecision] / N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.29999999999999998e-14

   1. Initial program 52.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf 63.0%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 54.6%

      \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{{k}^{4} \cdot t}} \]

   if 2.29999999999999998e-14 < t

   1. Initial program 75.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 69.3%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{k \cdot {t}^{3}}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Taylor expanded in k around 0 69.0%

      \[\leadsto \frac{\frac{\color{blue}{\frac{2}{k}}}{k \cdot {t}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{k \cdot {t}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 60.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-150}:\\ \;\;\;\;\frac{1}{\frac{t\_m}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{elif}\;t\_m \leq 2.1 \cdot 10^{+25}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \frac{{t\_m}^{2}}{\ell}\right) \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e-150)
    (/ 1.0 (* (/ t_m 2.0) (/ (pow k 4.0) (pow l 2.0))))
    (if (<= t_m 2.1e+25)
      (/ 2.0 (* (* t_m (/ (pow t_m 2.0) l)) (/ (* 2.0 (pow k 2.0)) l)))
      (/ 2.0 (* (* 2.0 k) (* (sin 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 <= 8e-150) {
		tmp = 1.0 / ((t_m / 2.0) * (pow(k, 4.0) / pow(l, 2.0)));
	} else if (t_m <= 2.1e+25) {
		tmp = 2.0 / ((t_m * (pow(t_m, 2.0) / l)) * ((2.0 * pow(k, 2.0)) / l));
	} else {
		tmp = 2.0 / ((2.0 * k) * (sin(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 <= 8d-150) then
        tmp = 1.0d0 / ((t_m / 2.0d0) * ((k ** 4.0d0) / (l ** 2.0d0)))
    else if (t_m <= 2.1d+25) then
        tmp = 2.0d0 / ((t_m * ((t_m ** 2.0d0) / l)) * ((2.0d0 * (k ** 2.0d0)) / l))
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (sin(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 <= 8e-150) {
		tmp = 1.0 / ((t_m / 2.0) * (Math.pow(k, 4.0) / Math.pow(l, 2.0)));
	} else if (t_m <= 2.1e+25) {
		tmp = 2.0 / ((t_m * (Math.pow(t_m, 2.0) / l)) * ((2.0 * Math.pow(k, 2.0)) / l));
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(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 <= 8e-150:
		tmp = 1.0 / ((t_m / 2.0) * (math.pow(k, 4.0) / math.pow(l, 2.0)))
	elif t_m <= 2.1e+25:
		tmp = 2.0 / ((t_m * (math.pow(t_m, 2.0) / l)) * ((2.0 * math.pow(k, 2.0)) / l))
	else:
		tmp = 2.0 / ((2.0 * k) * (math.sin(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 <= 8e-150)
		tmp = Float64(1.0 / Float64(Float64(t_m / 2.0) * Float64((k ^ 4.0) / (l ^ 2.0))));
	elseif (t_m <= 2.1e+25)
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) * Float64(Float64(2.0 * (k ^ 2.0)) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(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 <= 8e-150)
		tmp = 1.0 / ((t_m / 2.0) * ((k ^ 4.0) / (l ^ 2.0)));
	elseif (t_m <= 2.1e+25)
		tmp = 2.0 / ((t_m * ((t_m ^ 2.0) / l)) * ((2.0 * (k ^ 2.0)) / l));
	else
		tmp = 2.0 / ((2.0 * k) * (sin(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, 8e-150], N[(1.0 / N[(N[(t$95$m / 2.0), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.1e+25], N[(2.0 / N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * 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]

Derivation

1. Split input into 3 regimes

2. if t < 8.00000000000000005e-150

   1. Initial program 50.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 51.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf 60.5%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 51.3%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   7. Step-by-step derivation

      1. associate-*r/ 51.3%

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. *-commutative 51.3%

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

   8. Simplified 51.3%

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}} \]
   9. Step-by-step derivation

      1. clear-num 51.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}}} \]

      2. inv-pow 51.3%

         \[\leadsto \color{blue}{{\left(\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}\right)}^{-1}} \]

   10. Applied egg-rr 51.3%

       \[\leadsto \color{blue}{{\left(\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 51.3%

          \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}}} \]

       2. times-frac 52.4%

          \[\leadsto \frac{1}{\color{blue}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   12. Simplified 52.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   if 8.00000000000000005e-150 < t < 2.0999999999999999e25

   1. Initial program 71.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 74.4%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 57.8%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 55.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   7. Applied egg-rr 55.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/l* 58.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

   9. Simplified 58.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]
   10. Step-by-step derivation

       1. cube-mult 58.0%

          \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}} \]

       2. *-un-lft-identity 58.0%

          \[\leadsto \frac{2}{\frac{t \cdot \left(t \cdot t\right)}{\color{blue}{1 \cdot \ell}} \cdot \frac{2 \cdot {k}^{2}}{\ell}} \]

       3. times-frac 69.4%

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{1} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \frac{2 \cdot {k}^{2}}{\ell}} \]

       4. pow2 69.4%

          \[\leadsto \frac{2}{\left(\frac{t}{1} \cdot \frac{\color{blue}{{t}^{2}}}{\ell}\right) \cdot \frac{2 \cdot {k}^{2}}{\ell}} \]

   11. Applied egg-rr 69.4%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{1} \cdot \frac{{t}^{2}}{\ell}\right)} \cdot \frac{2 \cdot {k}^{2}}{\ell}} \]

   if 2.0999999999999999e25 < t

   1. Initial program 72.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 72.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 70.6%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 87.1%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   7. Simplified 70.6%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-150}:\\ \;\;\;\;\frac{1}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+25}:\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{{t}^{2}}{\ell}\right) \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 60.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{\frac{t\_m}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{elif}\;t\_m \leq 8 \cdot 10^{+25}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell} \cdot \left({t\_m}^{3} \cdot \frac{1}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.2e-114)
    (/ 1.0 (* (/ t_m 2.0) (/ (pow k 4.0) (pow l 2.0))))
    (if (<= t_m 8e+25)
      (/ 2.0 (* (/ (* 2.0 (pow k 2.0)) l) (* (pow t_m 3.0) (/ 1.0 l))))
      (/ 2.0 (* (* 2.0 k) (* (sin 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 <= 1.2e-114) {
		tmp = 1.0 / ((t_m / 2.0) * (pow(k, 4.0) / pow(l, 2.0)));
	} else if (t_m <= 8e+25) {
		tmp = 2.0 / (((2.0 * pow(k, 2.0)) / l) * (pow(t_m, 3.0) * (1.0 / l)));
	} else {
		tmp = 2.0 / ((2.0 * k) * (sin(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 <= 1.2d-114) then
        tmp = 1.0d0 / ((t_m / 2.0d0) * ((k ** 4.0d0) / (l ** 2.0d0)))
    else if (t_m <= 8d+25) then
        tmp = 2.0d0 / (((2.0d0 * (k ** 2.0d0)) / l) * ((t_m ** 3.0d0) * (1.0d0 / l)))
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (sin(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 <= 1.2e-114) {
		tmp = 1.0 / ((t_m / 2.0) * (Math.pow(k, 4.0) / Math.pow(l, 2.0)));
	} else if (t_m <= 8e+25) {
		tmp = 2.0 / (((2.0 * Math.pow(k, 2.0)) / l) * (Math.pow(t_m, 3.0) * (1.0 / l)));
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(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 <= 1.2e-114:
		tmp = 1.0 / ((t_m / 2.0) * (math.pow(k, 4.0) / math.pow(l, 2.0)))
	elif t_m <= 8e+25:
		tmp = 2.0 / (((2.0 * math.pow(k, 2.0)) / l) * (math.pow(t_m, 3.0) * (1.0 / l)))
	else:
		tmp = 2.0 / ((2.0 * k) * (math.sin(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 <= 1.2e-114)
		tmp = Float64(1.0 / Float64(Float64(t_m / 2.0) * Float64((k ^ 4.0) / (l ^ 2.0))));
	elseif (t_m <= 8e+25)
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k ^ 2.0)) / l) * Float64((t_m ^ 3.0) * Float64(1.0 / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(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 <= 1.2e-114)
		tmp = 1.0 / ((t_m / 2.0) * ((k ^ 4.0) / (l ^ 2.0)));
	elseif (t_m <= 8e+25)
		tmp = 2.0 / (((2.0 * (k ^ 2.0)) / l) * ((t_m ^ 3.0) * (1.0 / l)));
	else
		tmp = 2.0 / ((2.0 * k) * (sin(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, 1.2e-114], N[(1.0 / N[(N[(t$95$m / 2.0), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8e+25], N[(2.0 / N[(N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * 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]

Derivation

1. Split input into 3 regimes

2. if t < 1.2000000000000001e-114

   1. Initial program 50.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 50.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf 61.4%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 52.6%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   7. Step-by-step derivation

      1. associate-*r/ 52.6%

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. *-commutative 52.6%

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

   8. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}} \]
   9. Step-by-step derivation

      1. clear-num 52.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}}} \]

      2. inv-pow 52.6%

         \[\leadsto \color{blue}{{\left(\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}\right)}^{-1}} \]

   10. Applied egg-rr 52.6%

       \[\leadsto \color{blue}{{\left(\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 52.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}}} \]

       2. times-frac 53.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   12. Simplified 53.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   if 1.2000000000000001e-114 < t < 8.00000000000000072e25

   1. Initial program 84.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 84.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 62.3%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 59.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   7. Applied egg-rr 59.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/l* 62.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

   9. Simplified 62.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]
   10. Step-by-step derivation

       1. div-inv 62.7%

          \[\leadsto \frac{2}{\color{blue}{\left({t}^{3} \cdot \frac{1}{\ell}\right)} \cdot \frac{2 \cdot {k}^{2}}{\ell}} \]

   11. Applied egg-rr 62.7%

       \[\leadsto \frac{2}{\color{blue}{\left({t}^{3} \cdot \frac{1}{\ell}\right)} \cdot \frac{2 \cdot {k}^{2}}{\ell}} \]

   if 8.00000000000000072e25 < t

   1. Initial program 72.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 72.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 70.6%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 87.1%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   7. Simplified 70.6%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+25}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell} \cdot \left({t}^{3} \cdot \frac{1}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 60.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{\frac{t\_m}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{elif}\;t\_m \leq 9 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell} \cdot \frac{{t\_m}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.2e-114)
    (/ 1.0 (* (/ t_m 2.0) (/ (pow k 4.0) (pow l 2.0))))
    (if (<= t_m 9e+24)
      (/ 2.0 (* (/ (* 2.0 (pow k 2.0)) l) (/ (pow t_m 3.0) l)))
      (/ 2.0 (* (* 2.0 k) (* (sin 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 <= 1.2e-114) {
		tmp = 1.0 / ((t_m / 2.0) * (pow(k, 4.0) / pow(l, 2.0)));
	} else if (t_m <= 9e+24) {
		tmp = 2.0 / (((2.0 * pow(k, 2.0)) / l) * (pow(t_m, 3.0) / l));
	} else {
		tmp = 2.0 / ((2.0 * k) * (sin(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 <= 1.2d-114) then
        tmp = 1.0d0 / ((t_m / 2.0d0) * ((k ** 4.0d0) / (l ** 2.0d0)))
    else if (t_m <= 9d+24) then
        tmp = 2.0d0 / (((2.0d0 * (k ** 2.0d0)) / l) * ((t_m ** 3.0d0) / l))
    else
        tmp = 2.0d0 / ((2.0d0 * k) * (sin(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 <= 1.2e-114) {
		tmp = 1.0 / ((t_m / 2.0) * (Math.pow(k, 4.0) / Math.pow(l, 2.0)));
	} else if (t_m <= 9e+24) {
		tmp = 2.0 / (((2.0 * Math.pow(k, 2.0)) / l) * (Math.pow(t_m, 3.0) / l));
	} else {
		tmp = 2.0 / ((2.0 * k) * (Math.sin(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 <= 1.2e-114:
		tmp = 1.0 / ((t_m / 2.0) * (math.pow(k, 4.0) / math.pow(l, 2.0)))
	elif t_m <= 9e+24:
		tmp = 2.0 / (((2.0 * math.pow(k, 2.0)) / l) * (math.pow(t_m, 3.0) / l))
	else:
		tmp = 2.0 / ((2.0 * k) * (math.sin(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 <= 1.2e-114)
		tmp = Float64(1.0 / Float64(Float64(t_m / 2.0) * Float64((k ^ 4.0) / (l ^ 2.0))));
	elseif (t_m <= 9e+24)
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k ^ 2.0)) / l) * Float64((t_m ^ 3.0) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(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 <= 1.2e-114)
		tmp = 1.0 / ((t_m / 2.0) * ((k ^ 4.0) / (l ^ 2.0)));
	elseif (t_m <= 9e+24)
		tmp = 2.0 / (((2.0 * (k ^ 2.0)) / l) * ((t_m ^ 3.0) / l));
	else
		tmp = 2.0 / ((2.0 * k) * (sin(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, 1.2e-114], N[(1.0 / N[(N[(t$95$m / 2.0), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9e+24], N[(2.0 / N[(N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * 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]

Derivation

1. Split input into 3 regimes

2. if t < 1.2000000000000001e-114

   1. Initial program 50.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 50.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf 61.4%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 52.6%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   7. Step-by-step derivation

      1. associate-*r/ 52.6%

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. *-commutative 52.6%

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

   8. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}} \]
   9. Step-by-step derivation

      1. clear-num 52.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}}} \]

      2. inv-pow 52.6%

         \[\leadsto \color{blue}{{\left(\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}\right)}^{-1}} \]

   10. Applied egg-rr 52.6%

       \[\leadsto \color{blue}{{\left(\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 52.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}}} \]

       2. times-frac 53.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   12. Simplified 53.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   if 1.2000000000000001e-114 < t < 9.00000000000000039e24

   1. Initial program 84.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 84.7%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 62.3%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 59.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   7. Applied egg-rr 59.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/l* 62.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

   9. Simplified 62.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

   if 9.00000000000000039e24 < t

   1. Initial program 72.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 72.6%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 70.6%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(2 \cdot k\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 87.1%

         \[\leadsto \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \color{blue}{\left(k \cdot 2\right)}} \]

   7. Simplified 70.6%

      \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(k \cdot 2\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+24}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell} \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot k\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 60.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{\frac{t\_m}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{k \cdot {t\_m}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.2e-114)
    (/ 1.0 (* (/ t_m 2.0) (/ (pow k 4.0) (pow l 2.0))))
    (/
     (* (/ (/ 2.0 k) (* k (pow t_m 3.0))) (* l l))
     (+ 2.0 (pow (/ k t_m) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.2e-114) {
		tmp = 1.0 / ((t_m / 2.0) * (pow(k, 4.0) / pow(l, 2.0)));
	} else {
		tmp = (((2.0 / k) / (k * pow(t_m, 3.0))) * (l * l)) / (2.0 + pow((k / t_m), 2.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.2d-114) then
        tmp = 1.0d0 / ((t_m / 2.0d0) * ((k ** 4.0d0) / (l ** 2.0d0)))
    else
        tmp = (((2.0d0 / k) / (k * (t_m ** 3.0d0))) * (l * l)) / (2.0d0 + ((k / t_m) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.2e-114) {
		tmp = 1.0 / ((t_m / 2.0) * (Math.pow(k, 4.0) / Math.pow(l, 2.0)));
	} else {
		tmp = (((2.0 / k) / (k * Math.pow(t_m, 3.0))) * (l * l)) / (2.0 + Math.pow((k / t_m), 2.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.2e-114:
		tmp = 1.0 / ((t_m / 2.0) * (math.pow(k, 4.0) / math.pow(l, 2.0)))
	else:
		tmp = (((2.0 / k) / (k * math.pow(t_m, 3.0))) * (l * l)) / (2.0 + math.pow((k / t_m), 2.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.2e-114)
		tmp = Float64(1.0 / Float64(Float64(t_m / 2.0) * Float64((k ^ 4.0) / (l ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 / k) / Float64(k * (t_m ^ 3.0))) * Float64(l * l)) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.2e-114)
		tmp = 1.0 / ((t_m / 2.0) * ((k ^ 4.0) / (l ^ 2.0)));
	else
		tmp = (((2.0 / k) / (k * (t_m ^ 3.0))) * (l * l)) / (2.0 + ((k / t_m) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.2e-114], N[(1.0 / N[(N[(t$95$m / 2.0), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 / k), $MachinePrecision] / N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.2000000000000001e-114

   1. Initial program 50.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 50.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf 61.4%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 52.6%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   7. Step-by-step derivation

      1. associate-*r/ 52.6%

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. *-commutative 52.6%

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

   8. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}} \]
   9. Step-by-step derivation

      1. clear-num 52.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}}} \]

      2. inv-pow 52.6%

         \[\leadsto \color{blue}{{\left(\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}\right)}^{-1}} \]

   10. Applied egg-rr 52.6%

       \[\leadsto \color{blue}{{\left(\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 52.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}}} \]

       2. times-frac 53.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   12. Simplified 53.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   if 1.2000000000000001e-114 < t

   1. Initial program 77.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 75.7%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 68.2%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{k \cdot {t}^{3}}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Taylor expanded in k around 0 66.4%

      \[\leadsto \frac{\frac{\color{blue}{\frac{2}{k}}}{k \cdot {t}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 58.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{\frac{t\_m}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell} \cdot \frac{{t\_m}^{3}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.2e-114)
    (/ 1.0 (* (/ t_m 2.0) (/ (pow k 4.0) (pow l 2.0))))
    (/ 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) {
	double tmp;
	if (t_m <= 1.2e-114) {
		tmp = 1.0 / ((t_m / 2.0) * (pow(k, 4.0) / pow(l, 2.0)));
	} else {
		tmp = 2.0 / (((2.0 * pow(k, 2.0)) / 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 <= 1.2d-114) then
        tmp = 1.0d0 / ((t_m / 2.0d0) * ((k ** 4.0d0) / (l ** 2.0d0)))
    else
        tmp = 2.0d0 / (((2.0d0 * (k ** 2.0d0)) / 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 <= 1.2e-114) {
		tmp = 1.0 / ((t_m / 2.0) * (Math.pow(k, 4.0) / Math.pow(l, 2.0)));
	} else {
		tmp = 2.0 / (((2.0 * Math.pow(k, 2.0)) / 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 <= 1.2e-114:
		tmp = 1.0 / ((t_m / 2.0) * (math.pow(k, 4.0) / math.pow(l, 2.0)))
	else:
		tmp = 2.0 / (((2.0 * math.pow(k, 2.0)) / 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 <= 1.2e-114)
		tmp = Float64(1.0 / Float64(Float64(t_m / 2.0) * Float64((k ^ 4.0) / (l ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k ^ 2.0)) / l) * Float64((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 <= 1.2e-114)
		tmp = 1.0 / ((t_m / 2.0) * ((k ^ 4.0) / (l ^ 2.0)));
	else
		tmp = 2.0 / (((2.0 * (k ^ 2.0)) / 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, 1.2e-114], N[(1.0 / N[(N[(t$95$m / 2.0), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.2000000000000001e-114

   1. Initial program 50.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 50.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf 61.4%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 52.6%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   7. Step-by-step derivation

      1. associate-*r/ 52.6%

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. *-commutative 52.6%

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

   8. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}} \]
   9. Step-by-step derivation

      1. clear-num 52.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}}} \]

      2. inv-pow 52.6%

         \[\leadsto \color{blue}{{\left(\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}\right)}^{-1}} \]

   10. Applied egg-rr 52.6%

       \[\leadsto \color{blue}{{\left(\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 52.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}}} \]

       2. times-frac 53.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   12. Simplified 53.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   if 1.2000000000000001e-114 < t

   1. Initial program 77.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 70.0%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 60.7%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 58.2%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   7. Applied egg-rr 58.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/l* 59.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

   9. Simplified 59.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell} \cdot \frac{{t}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 57.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.6 \cdot 10^{-116}:\\ \;\;\;\;\frac{1}{\frac{t\_m}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.6e-116)
    (/ 1.0 (* (/ t_m 2.0) (/ (pow k 4.0) (pow l 2.0))))
    (/ 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) {
	double tmp;
	if (t_m <= 4.6e-116) {
		tmp = 1.0 / ((t_m / 2.0) * (pow(k, 4.0) / pow(l, 2.0)));
	} else {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * ((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 <= 4.6d-116) then
        tmp = 1.0d0 / ((t_m / 2.0d0) * ((k ** 4.0d0) / (l ** 2.0d0)))
    else
        tmp = 2.0d0 / ((2.0d0 * (k ** 2.0d0)) * (((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 <= 4.6e-116) {
		tmp = 1.0 / ((t_m / 2.0) * (Math.pow(k, 4.0) / Math.pow(l, 2.0)));
	} else {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * ((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 <= 4.6e-116:
		tmp = 1.0 / ((t_m / 2.0) * (math.pow(k, 4.0) / math.pow(l, 2.0)))
	else:
		tmp = 2.0 / ((2.0 * math.pow(k, 2.0)) * ((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 <= 4.6e-116)
		tmp = Float64(1.0 / Float64(Float64(t_m / 2.0) * Float64((k ^ 4.0) / (l ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64(Float64((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 <= 4.6e-116)
		tmp = 1.0 / ((t_m / 2.0) * ((k ^ 4.0) / (l ^ 2.0)));
	else
		tmp = 2.0 / ((2.0 * (k ^ 2.0)) * (((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, 4.6e-116], N[(1.0 / N[(N[(t$95$m / 2.0), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.60000000000000003e-116

   1. Initial program 50.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 50.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf 61.4%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   6. Taylor expanded in k around 0 52.6%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   7. Step-by-step derivation

      1. associate-*r/ 52.6%

         \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

      2. *-commutative 52.6%

         \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

   8. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}} \]
   9. Step-by-step derivation

      1. clear-num 52.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}}} \]

      2. inv-pow 52.6%

         \[\leadsto \color{blue}{{\left(\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}\right)}^{-1}} \]

   10. Applied egg-rr 52.6%

       \[\leadsto \color{blue}{{\left(\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 52.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}}} \]

       2. times-frac 53.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   12. Simplified 53.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

   if 4.60000000000000003e-116 < t

   1. Initial program 77.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 70.0%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 60.7%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-116}:\\ \;\;\;\;\frac{1}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 51.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{1}{\frac{t\_m}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 1.0 (* (/ t_m 2.0) (/ (pow k 4.0) (pow l 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (1.0 / ((t_m / 2.0) * (pow(k, 4.0) / pow(l, 2.0))));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (1.0d0 / ((t_m / 2.0d0) * ((k ** 4.0d0) / (l ** 2.0d0))))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (1.0 / ((t_m / 2.0) * (Math.pow(k, 4.0) / Math.pow(l, 2.0))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (1.0 / ((t_m / 2.0) * (math.pow(k, 4.0) / math.pow(l, 2.0))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(1.0 / Float64(Float64(t_m / 2.0) * Float64((k ^ 4.0) / (l ^ 2.0)))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (1.0 / ((t_m / 2.0) * ((k ^ 4.0) / (l ^ 2.0))));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(1.0 / N[(N[(t$95$m / 2.0), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 57.4%

   \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
4. Add Preprocessing

5. Taylor expanded in k around inf 59.1%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

6. Taylor expanded in k around 0 50.3%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
7. Step-by-step derivation

   1. associate-*r/ 50.3%

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. *-commutative 50.3%

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

8. Simplified 50.3%

   \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}} \]
9. Step-by-step derivation

   1. clear-num 50.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}}} \]

   2. inv-pow 50.3%

      \[\leadsto \color{blue}{{\left(\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}\right)}^{-1}} \]

10. Applied egg-rr 50.3%

    \[\leadsto \color{blue}{{\left(\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}\right)}^{-1}} \]
11. Step-by-step derivation

    1. unpow-1 50.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot {k}^{4}}{2 \cdot {\ell}^{2}}}} \]

    2. times-frac 51.1%

       \[\leadsto \frac{1}{\color{blue}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]

12. Simplified 51.1%

    \[\leadsto \color{blue}{\frac{1}{\frac{t}{2} \cdot \frac{{k}^{4}}{{\ell}^{2}}}} \]
13. Add Preprocessing

Alternative 28: 51.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{2}{t\_m} \cdot \frac{{\ell}^{2}}{{k}^{4}}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ 2.0 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 / 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 / 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 / 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 / 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(Float64(2.0 / 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 / 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[(N[(2.0 / t$95$m), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 57.4%

   \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
4. Add Preprocessing

5. Taylor expanded in k around inf 59.1%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

6. Taylor expanded in k around 0 50.3%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
7. Step-by-step derivation

   1. associate-*r/ 50.3%

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. *-commutative 50.3%

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

8. Simplified 50.3%

   \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}} \]
9. Step-by-step derivation

   1. pow2 50.3%

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{t \cdot {k}^{4}} \]

   2. times-frac 51.1%

      \[\leadsto \color{blue}{\frac{2}{t} \cdot \frac{\ell \cdot \ell}{{k}^{4}}} \]

   3. pow2 51.1%

      \[\leadsto \frac{2}{t} \cdot \frac{\color{blue}{{\ell}^{2}}}{{k}^{4}} \]

10. Applied egg-rr 51.1%

    \[\leadsto \color{blue}{\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]
11. Add Preprocessing

Alternative 29: 51.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2 \cdot \left(\ell \cdot \ell\right)}{t\_m \cdot {k}^{4}} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (* 2.0 (* l l)) (* 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 * (l * l)) / (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 * l)) / (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 * (l * l)) / (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 * (l * l)) / (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(Float64(2.0 * Float64(l * l)) / 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 * l)) / (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[(N[(2.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

3. Simplified 57.4%

   \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
4. Add Preprocessing

5. Taylor expanded in k around inf 59.1%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

6. Taylor expanded in k around 0 50.3%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
7. Step-by-step derivation

   1. associate-*r/ 50.3%

      \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]

   2. *-commutative 50.3%

      \[\leadsto \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]

8. Simplified 50.3%

   \[\leadsto \color{blue}{\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}} \]
9. Step-by-step derivation

   1. unpow2 50.3%

      \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{t \cdot {k}^{4}} \]

10. Applied egg-rr 50.3%

    \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{t \cdot {k}^{4}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(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))))