Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.9% → 88.7%

Time: 18.5s

Alternatives: 21

Speedup: 1.9×

• 27.1% 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: 55.9% 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: 88.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\frac{1}{\ell}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.7 \cdot 10^{-193}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{{t\_m}^{1.5} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}{\ell}\right)}^{2}}}{\sin k \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot \left(t\_2 \cdot t\_2\right)\right) \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}\right)\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

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 (cbrt (/ 1.0 l))))
   (*
    t_s
    (if (<= t_m 3.7e-193)
      (/
       2.0
       (/ (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)) (* (pow l 2.0) (cos k))))
      (if (<= t_m 2e-71)
        (/
         (/
          2.0
          (pow (/ (* (pow t_m 1.5) (hypot 1.0 (hypot 1.0 (/ k t_m)))) l) 2.0))
         (* (sin k) (tan k)))
        (/
         2.0
         (pow
          (*
           (* t_m (* t_2 t_2))
           (* (cbrt (sin k)) (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))))
          3.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = cbrt((1.0 / l));
	double tmp;
	if (t_m <= 3.7e-193) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)) / (pow(l, 2.0) * cos(k)));
	} else if (t_m <= 2e-71) {
		tmp = (2.0 / pow(((pow(t_m, 1.5) * hypot(1.0, hypot(1.0, (k / t_m)))) / l), 2.0)) / (sin(k) * tan(k));
	} else {
		tmp = 2.0 / pow(((t_m * (t_2 * t_2)) * (cbrt(sin(k)) * cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.cbrt((1.0 / l));
	double tmp;
	if (t_m <= 3.7e-193) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)) / (Math.pow(l, 2.0) * Math.cos(k)));
	} else if (t_m <= 2e-71) {
		tmp = (2.0 / Math.pow(((Math.pow(t_m, 1.5) * Math.hypot(1.0, Math.hypot(1.0, (k / t_m)))) / l), 2.0)) / (Math.sin(k) * Math.tan(k));
	} else {
		tmp = 2.0 / Math.pow(((t_m * (t_2 * t_2)) * (Math.cbrt(Math.sin(k)) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = cbrt(Float64(1.0 / l))
	tmp = 0.0
	if (t_m <= 3.7e-193)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)) / Float64((l ^ 2.0) * cos(k))));
	elseif (t_m <= 2e-71)
		tmp = Float64(Float64(2.0 / (Float64(Float64((t_m ^ 1.5) * hypot(1.0, hypot(1.0, Float64(k / t_m)))) / l) ^ 2.0)) / Float64(sin(k) * tan(k)));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m * Float64(t_2 * t_2)) * Float64(cbrt(sin(k)) * cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(1.0 / l), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.7e-193], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2e-71], N[(N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.7000000000000002e-193

   1. Initial program 48.9%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in t around 0 67.1%

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

      1. associate-*r* 67.1%

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

   7. Simplified 67.1%

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

   if 3.7000000000000002e-193 < t < 1.9999999999999998e-71

   1. Initial program 56.3%

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

   3. Simplified 56.3%

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

   6. Applied egg-rr 70.3%

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

      1. *-un-lft-identity 70.3%

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

      2. associate-*r* 70.3%

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

      3. unpow-prod-down 70.4%

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

      4. pow2 70.4%

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

      5. add-sqr-sqrt 96.0%

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

   8. Applied egg-rr 96.0%

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

      1. *-lft-identity 96.0%

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

      2. associate-/r* 96.0%

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

      3. associate-*l/ 96.1%

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

   10. Simplified 96.1%

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

   if 1.9999999999999998e-71 < t

   1. Initial program 70.4%

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

   3. Simplified 70.4%

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

      1. associate-*l* 60.6%

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

      2. associate-/r* 63.4%

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

      3. associate-+r+ 63.4%

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

      4. metadata-eval 63.4%

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

      5. associate-*l* 63.4%

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

      6. add-cube-cbrt 63.3%

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

      7. pow3 63.3%

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

   6. Applied egg-rr 65.5%

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

      1. pow1/3 64.6%

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

      2. sqr-pow 64.6%

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

      3. metadata-eval 64.6%

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

      4. inv-pow 64.6%

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

      5. metadata-eval 64.6%

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

      6. inv-pow 64.6%

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

      7. unpow-prod-down 32.9%

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

   8. Applied egg-rr 32.9%

      \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{\ell}\right)}^{0.3333333333333333}\right)}\right) \cdot \sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}} \]
   9. Step-by-step derivation

      1. unpow1/3 33.0%

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

      2. unpow1/3 74.0%

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

   10. Simplified 74.0%

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

       1. associate-*l* 74.1%

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

       2. cbrt-prod 95.0%

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

   12. Applied egg-rr 95.0%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-193}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{{t}^{1.5} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\ell}\right)}^{2}}}{\sin k \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot \left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right)\right) \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\frac{1}{\ell}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot \left(t\_2 \cdot t\_2\right)\right) \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{{t\_m}^{1.5} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right)}{\ell}\right)}^{2}}}{\sin k \cdot \tan k}\\ \end{array} \end{array} \end{array} \]

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 (cbrt (/ 1.0 l))))
   (*
    t_s
    (if (<= k 2.1e-84)
      (/
       2.0
       (pow (* (* t_m (* t_2 t_2)) (* (pow (cbrt k) 2.0) (cbrt 2.0))) 3.0))
      (/
       (/
        2.0
        (pow (/ (* (pow t_m 1.5) (hypot 1.0 (hypot 1.0 (/ k t_m)))) l) 2.0))
       (* (sin k) (tan 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 = cbrt((1.0 / l));
	double tmp;
	if (k <= 2.1e-84) {
		tmp = 2.0 / pow(((t_m * (t_2 * t_2)) * (pow(cbrt(k), 2.0) * cbrt(2.0))), 3.0);
	} else {
		tmp = (2.0 / pow(((pow(t_m, 1.5) * hypot(1.0, hypot(1.0, (k / t_m)))) / l), 2.0)) / (sin(k) * tan(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 = Math.cbrt((1.0 / l));
	double tmp;
	if (k <= 2.1e-84) {
		tmp = 2.0 / Math.pow(((t_m * (t_2 * t_2)) * (Math.pow(Math.cbrt(k), 2.0) * Math.cbrt(2.0))), 3.0);
	} else {
		tmp = (2.0 / Math.pow(((Math.pow(t_m, 1.5) * Math.hypot(1.0, Math.hypot(1.0, (k / t_m)))) / l), 2.0)) / (Math.sin(k) * Math.tan(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 = cbrt(Float64(1.0 / l))
	tmp = 0.0
	if (k <= 2.1e-84)
		tmp = Float64(2.0 / (Float64(Float64(t_m * Float64(t_2 * t_2)) * Float64((cbrt(k) ^ 2.0) * cbrt(2.0))) ^ 3.0));
	else
		tmp = Float64(Float64(2.0 / (Float64(Float64((t_m ^ 1.5) * hypot(1.0, hypot(1.0, Float64(k / t_m)))) / l) ^ 2.0)) / Float64(sin(k) * tan(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[Power[N[(1.0 / l), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 2.1e-84], N[(2.0 / N[Power[N[(N[(t$95$m * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.09999999999999998e-84

   1. Initial program 56.1%

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

   3. Simplified 56.1%

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

      1. associate-*l* 50.8%

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

      2. associate-/r* 55.4%

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

      3. associate-+r+ 55.4%

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

      4. metadata-eval 55.4%

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

      5. associate-*l* 55.4%

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

      6. add-cube-cbrt 55.4%

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

      7. pow3 55.4%

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

   6. Applied egg-rr 62.4%

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

      1. pow1/3 61.9%

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

      2. sqr-pow 61.9%

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

      3. metadata-eval 61.9%

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

      4. inv-pow 61.9%

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

      5. metadata-eval 61.9%

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

      6. inv-pow 61.9%

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

      7. unpow-prod-down 36.8%

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

   8. Applied egg-rr 36.8%

      \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{\ell}\right)}^{0.3333333333333333}\right)}\right) \cdot \sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}} \]
   9. Step-by-step derivation

      1. unpow1/3 36.9%

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

      2. unpow1/3 72.2%

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

   10. Simplified 72.2%

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

   11. Taylor expanded in k around 0 67.0%

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

       1. *-un-lft-identity 67.0%

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

       2. unpow2 67.0%

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

       3. cbrt-prod 79.9%

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

       4. pow2 79.9%

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

   13. Applied egg-rr 79.9%

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

       1. *-lft-identity 79.9%

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

   15. Simplified 79.9%

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

   if 2.09999999999999998e-84 < k

   1. Initial program 54.9%

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

   3. Simplified 54.9%

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

   6. Applied egg-rr 31.6%

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

      1. *-un-lft-identity 31.6%

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

      2. associate-*r* 31.5%

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

      3. unpow-prod-down 31.5%

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

      4. pow2 31.5%

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

      5. add-sqr-sqrt 43.6%

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

   8. Applied egg-rr 43.6%

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

      1. *-lft-identity 43.6%

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

      2. associate-/r* 43.6%

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

      3. associate-*l/ 43.7%

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

   10. Simplified 43.7%

       \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{{t}^{1.5} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}{\ell}\right)}^{2}}}{\sin k \cdot \tan k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\frac{1}{\ell}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{-84}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot \left(t\_2 \cdot t\_2\right)\right) \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right) \cdot \frac{{t\_m}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

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 (cbrt (/ 1.0 l))))
   (*
    t_s
    (if (<= k 1e-84)
      (/
       2.0
       (pow (* (* t_m (* t_2 t_2)) (* (pow (cbrt k) 2.0) (cbrt 2.0))) 3.0))
      (/
       2.0
       (*
        (* (sin k) (tan k))
        (pow
         (* (hypot 1.0 (hypot 1.0 (/ k t_m))) (/ (pow t_m 1.5) l))
         2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = cbrt((1.0 / l));
	double tmp;
	if (k <= 1e-84) {
		tmp = 2.0 / pow(((t_m * (t_2 * t_2)) * (pow(cbrt(k), 2.0) * cbrt(2.0))), 3.0);
	} else {
		tmp = 2.0 / ((sin(k) * tan(k)) * pow((hypot(1.0, hypot(1.0, (k / t_m))) * (pow(t_m, 1.5) / l)), 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 t_2 = Math.cbrt((1.0 / l));
	double tmp;
	if (k <= 1e-84) {
		tmp = 2.0 / Math.pow(((t_m * (t_2 * t_2)) * (Math.pow(Math.cbrt(k), 2.0) * Math.cbrt(2.0))), 3.0);
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.tan(k)) * Math.pow((Math.hypot(1.0, Math.hypot(1.0, (k / t_m))) * (Math.pow(t_m, 1.5) / l)), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = cbrt(Float64(1.0 / l))
	tmp = 0.0
	if (k <= 1e-84)
		tmp = Float64(2.0 / (Float64(Float64(t_m * Float64(t_2 * t_2)) * Float64((cbrt(k) ^ 2.0) * cbrt(2.0))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * tan(k)) * (Float64(hypot(1.0, hypot(1.0, Float64(k / t_m))) * Float64((t_m ^ 1.5) / l)) ^ 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_] := Block[{t$95$2 = N[Power[N[(1.0 / l), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1e-84], N[(2.0 / N[Power[N[(N[(t$95$m * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1e-84

   1. Initial program 56.1%

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

   3. Simplified 56.1%

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

      1. associate-*l* 50.8%

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

      2. associate-/r* 55.4%

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

      3. associate-+r+ 55.4%

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

      4. metadata-eval 55.4%

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

      5. associate-*l* 55.4%

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

      6. add-cube-cbrt 55.4%

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

      7. pow3 55.4%

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

   6. Applied egg-rr 62.4%

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

      1. pow1/3 61.9%

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

      2. sqr-pow 61.9%

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

      3. metadata-eval 61.9%

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

      4. inv-pow 61.9%

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

      5. metadata-eval 61.9%

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

      6. inv-pow 61.9%

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

      7. unpow-prod-down 36.8%

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

   8. Applied egg-rr 36.8%

      \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{\ell}\right)}^{0.3333333333333333}\right)}\right) \cdot \sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}} \]
   9. Step-by-step derivation

      1. unpow1/3 36.9%

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

      2. unpow1/3 72.2%

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

   10. Simplified 72.2%

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

   11. Taylor expanded in k around 0 67.0%

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

       1. *-un-lft-identity 67.0%

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

       2. unpow2 67.0%

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

       3. cbrt-prod 79.9%

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

       4. pow2 79.9%

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

   13. Applied egg-rr 79.9%

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

       1. *-lft-identity 79.9%

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

   15. Simplified 79.9%

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

   if 1e-84 < k

   1. Initial program 54.9%

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

   3. Simplified 54.9%

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

   6. Applied egg-rr 31.6%

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

      1. associate-*r* 31.5%

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

      2. unpow-prod-down 31.5%

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

      3. pow2 31.5%

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

      4. add-sqr-sqrt 43.6%

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

   8. Applied egg-rr 43.6%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-84}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot \left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right)\right) \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\frac{1}{\ell}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot \left(t\_2 \cdot t\_2\right)\right) \cdot \left(\sqrt[3]{2} \cdot {k}^{0.6666666666666666}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right) \cdot \frac{{t\_m}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

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 (cbrt (/ 1.0 l))))
   (*
    t_s
    (if (<= k 6.5e-85)
      (/
       2.0
       (pow
        (* (* t_m (* t_2 t_2)) (* (cbrt 2.0) (pow k 0.6666666666666666)))
        3.0))
      (/
       2.0
       (*
        (* (sin k) (tan k))
        (pow
         (* (hypot 1.0 (hypot 1.0 (/ k t_m))) (/ (pow t_m 1.5) l))
         2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = cbrt((1.0 / l));
	double tmp;
	if (k <= 6.5e-85) {
		tmp = 2.0 / pow(((t_m * (t_2 * t_2)) * (cbrt(2.0) * pow(k, 0.6666666666666666))), 3.0);
	} else {
		tmp = 2.0 / ((sin(k) * tan(k)) * pow((hypot(1.0, hypot(1.0, (k / t_m))) * (pow(t_m, 1.5) / l)), 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 t_2 = Math.cbrt((1.0 / l));
	double tmp;
	if (k <= 6.5e-85) {
		tmp = 2.0 / Math.pow(((t_m * (t_2 * t_2)) * (Math.cbrt(2.0) * Math.pow(k, 0.6666666666666666))), 3.0);
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.tan(k)) * Math.pow((Math.hypot(1.0, Math.hypot(1.0, (k / t_m))) * (Math.pow(t_m, 1.5) / l)), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = cbrt(Float64(1.0 / l))
	tmp = 0.0
	if (k <= 6.5e-85)
		tmp = Float64(2.0 / (Float64(Float64(t_m * Float64(t_2 * t_2)) * Float64(cbrt(2.0) * (k ^ 0.6666666666666666))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * tan(k)) * (Float64(hypot(1.0, hypot(1.0, Float64(k / t_m))) * Float64((t_m ^ 1.5) / l)) ^ 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_] := Block[{t$95$2 = N[Power[N[(1.0 / l), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 6.5e-85], N[(2.0 / N[Power[N[(N[(t$95$m * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[Power[2.0, 1/3], $MachinePrecision] * N[Power[k, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.5e-85

   1. Initial program 56.1%

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

   3. Simplified 56.1%

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

      1. associate-*l* 50.8%

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

      2. associate-/r* 55.4%

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

      3. associate-+r+ 55.4%

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

      4. metadata-eval 55.4%

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

      5. associate-*l* 55.4%

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

      6. add-cube-cbrt 55.4%

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

      7. pow3 55.4%

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

   6. Applied egg-rr 62.4%

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

      1. pow1/3 61.9%

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

      2. sqr-pow 61.9%

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

      3. metadata-eval 61.9%

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

      4. inv-pow 61.9%

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

      5. metadata-eval 61.9%

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

      6. inv-pow 61.9%

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

      7. unpow-prod-down 36.8%

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

   8. Applied egg-rr 36.8%

      \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{\ell}\right)}^{0.3333333333333333}\right)}\right) \cdot \sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}} \]
   9. Step-by-step derivation

      1. unpow1/3 36.9%

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

      2. unpow1/3 72.2%

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

   10. Simplified 72.2%

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

   11. Taylor expanded in k around 0 67.0%

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

       1. pow1/3 66.4%

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

       2. pow-pow 26.1%

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

       3. metadata-eval 26.1%

          \[\leadsto \frac{2}{{\left(\left(t \cdot \left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right)\right) \cdot \left({k}^{\color{blue}{0.6666666666666666}} \cdot \sqrt[3]{2}\right)\right)}^{3}} \]

   13. Applied egg-rr 26.1%

       \[\leadsto \frac{2}{{\left(\left(t \cdot \left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right)\right) \cdot \left(\color{blue}{{k}^{0.6666666666666666}} \cdot \sqrt[3]{2}\right)\right)}^{3}} \]

   if 6.5e-85 < k

   1. Initial program 54.9%

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

   3. Simplified 54.9%

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

   6. Applied egg-rr 31.6%

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

      1. associate-*r* 31.5%

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

      2. unpow-prod-down 31.5%

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

      3. pow2 31.5%

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

      4. add-sqr-sqrt 43.6%

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

   8. Applied egg-rr 43.6%

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2} \cdot \left(\sin k \cdot \tan k\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot \left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right)\right) \cdot \left(\sqrt[3]{2} \cdot {k}^{0.6666666666666666}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.65 \cdot 10^{-137}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t\_m}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 3.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + t\_2\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 9.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \left(\sin k \cdot \frac{\tan k \cdot \left(2 + t\_2\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot {\left(t\_m \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot \sqrt[3]{{\ell}^{-2}}\right)\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

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 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 1.65e-137)
      (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt (/ t_m (cos k)))) 2.0))
      (if (<= t_m 3.4e-94)
        (/
         2.0
         (*
          (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))
          (* (tan k) (+ 1.0 (+ 1.0 t_2)))))
        (if (<= t_m 9.8e+89)
          (/
           2.0
           (* (/ (pow t_m 3.0) l) (* (sin k) (/ (* (tan k) (+ 2.0 t_2)) l))))
          (/
           2.0
           (*
            2.0
            (pow
             (* t_m (* (pow (cbrt k) 2.0) (cbrt (pow 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 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.65e-137) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt((t_m / cos(k)))), 2.0);
	} else if (t_m <= 3.4e-94) {
		tmp = 2.0 / ((sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))) * (tan(k) * (1.0 + (1.0 + t_2))));
	} else if (t_m <= 9.8e+89) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * (sin(k) * ((tan(k) * (2.0 + t_2)) / l)));
	} else {
		tmp = 2.0 / (2.0 * pow((t_m * (pow(cbrt(k), 2.0) * cbrt(pow(l, -2.0)))), 3.0));
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.65e-137) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt((t_m / Math.cos(k)))), 2.0);
	} else if (t_m <= 3.4e-94) {
		tmp = 2.0 / ((Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))) * (Math.tan(k) * (1.0 + (1.0 + t_2))));
	} else if (t_m <= 9.8e+89) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * (Math.sin(k) * ((Math.tan(k) * (2.0 + t_2)) / l)));
	} else {
		tmp = 2.0 / (2.0 * Math.pow((t_m * (Math.pow(Math.cbrt(k), 2.0) * Math.cbrt(Math.pow(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(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 1.65e-137)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(Float64(t_m / cos(k)))) ^ 2.0));
	elseif (t_m <= 3.4e-94)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l))) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + t_2)))));
	elseif (t_m <= 9.8e+89)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(sin(k) * Float64(Float64(tan(k) * Float64(2.0 + t_2)) / l))));
	else
		tmp = Float64(2.0 / Float64(2.0 * (Float64(t_m * Float64((cbrt(k) ^ 2.0) * 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[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.65e-137], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.4e-94], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.8e+89], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[Power[N[(t$95$m * N[(N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Power[l, -2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 1.6500000000000001e-137

   1. Initial program 48.7%

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

   3. Simplified 48.7%

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

   6. Applied egg-rr 10.7%

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

   7. Taylor expanded in t around 0 29.2%

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

   if 1.6500000000000001e-137 < t < 3.3999999999999998e-94

   1. Initial program 38.9%

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

   3. Simplified 38.9%

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

      1. unpow3 38.9%

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

      2. times-frac 74.8%

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

      3. pow2 74.8%

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

   6. Applied egg-rr 74.8%

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

   if 3.3999999999999998e-94 < t < 9.79999999999999992e89

   1. Initial program 77.3%

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

   3. Simplified 77.3%

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

      1. associate-*l* 75.0%

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

      2. associate-/r* 77.2%

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

      3. associate-+r+ 77.2%

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

      4. metadata-eval 77.2%

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

      5. associate-*l* 77.2%

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

      6. associate-*l/ 79.8%

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

   6. Applied egg-rr 79.8%

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

      1. associate-/l* 79.8%

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

      2. associate-*l* 79.8%

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

   8. Simplified 79.8%

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

      1. associate-/l* 86.6%

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

   10. Applied egg-rr 86.6%

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

   if 9.79999999999999992e89 < t

   1. Initial program 66.4%

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

   3. Simplified 66.4%

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

      1. associate-*l* 50.3%

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

      2. associate-/r* 53.2%

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

      3. associate-+r+ 53.2%

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

      4. metadata-eval 53.2%

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

      5. associate-*l* 53.2%

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

      6. add-cube-cbrt 53.2%

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

      7. pow3 53.2%

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

   6. Applied egg-rr 59.7%

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

      1. pow1/3 59.2%

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

      2. sqr-pow 59.2%

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

      3. metadata-eval 59.2%

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

      4. inv-pow 59.2%

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

      5. metadata-eval 59.2%

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

      6. inv-pow 59.2%

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

      7. unpow-prod-down 39.1%

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

   8. Applied egg-rr 39.1%

      \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{\ell}\right)}^{0.3333333333333333}\right)}\right) \cdot \sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}} \]
   9. Step-by-step derivation

      1. unpow1/3 39.2%

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

      2. unpow1/3 69.6%

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

   10. Simplified 69.6%

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

   11. Taylor expanded in k around 0 65.0%

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

       1. associate-*r* 65.0%

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

       2. unpow-prod-down 65.0%

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

       3. cbrt-unprod 59.6%

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

       4. inv-pow 59.6%

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

       5. inv-pow 59.6%

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

       6. pow-prod-up 59.6%

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

       7. metadata-eval 59.6%

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

       8. unpow2 59.6%

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

       9. cbrt-prod 84.8%

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

       10. pow2 84.8%

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

   13. Applied egg-rr 84.8%

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

       1. rem-cube-cbrt 84.8%

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

       2. *-commutative 84.8%

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

       3. associate-*l* 84.9%

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

   15. Simplified 84.9%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{-137}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot {\left(t \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot \sqrt[3]{{\ell}^{-2}}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.7 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t\_m \leq 9.5 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \left(\sin k \cdot \frac{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot {\left(t\_m \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot \sqrt[3]{{\ell}^{-2}}\right)\right)}^{3}}\\ \end{array} \end{array} \]

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 5.7e-73)
    (/
     2.0
     (/ (* (* t_m (pow k 2.0)) (pow (sin k) 2.0)) (* (pow l 2.0) (cos k))))
    (if (<= t_m 9.5e+89)
      (/
       2.0
       (*
        (/ (pow t_m 3.0) l)
        (* (sin k) (/ (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))) l))))
      (/
       2.0
       (*
        2.0
        (pow (* t_m (* (pow (cbrt k) 2.0) (cbrt (pow 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 <= 5.7e-73) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)) / (pow(l, 2.0) * cos(k)));
	} else if (t_m <= 9.5e+89) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * (sin(k) * ((tan(k) * (2.0 + pow((k / t_m), 2.0))) / l)));
	} else {
		tmp = 2.0 / (2.0 * pow((t_m * (pow(cbrt(k), 2.0) * cbrt(pow(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 <= 5.7e-73) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)) / (Math.pow(l, 2.0) * Math.cos(k)));
	} else if (t_m <= 9.5e+89) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * (Math.sin(k) * ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) / l)));
	} else {
		tmp = 2.0 / (2.0 * Math.pow((t_m * (Math.pow(Math.cbrt(k), 2.0) * Math.cbrt(Math.pow(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 <= 5.7e-73)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)) / Float64((l ^ 2.0) * cos(k))));
	elseif (t_m <= 9.5e+89)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(sin(k) * Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) / l))));
	else
		tmp = Float64(2.0 / Float64(2.0 * (Float64(t_m * Float64((cbrt(k) ^ 2.0) * 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, 5.7e-73], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9.5e+89], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[Power[N[(t$95$m * N[(N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Power[l, -2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.6999999999999998e-73

   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 50.0%

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

   5. Taylor expanded in t around 0 69.8%

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

      1. associate-*r* 69.8%

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

   7. Simplified 69.8%

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

   if 5.6999999999999998e-73 < t < 9.5000000000000003e89

   1. Initial program 74.9%

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

   3. Simplified 75.0%

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

      1. associate-*l* 72.1%

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

      2. associate-/r* 74.8%

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

      3. associate-+r+ 74.8%

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

      4. metadata-eval 74.8%

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

      5. associate-*l* 74.8%

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

      6. associate-*l/ 78.0%

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

   6. Applied egg-rr 78.0%

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

      1. associate-/l* 78.0%

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

      2. associate-*l* 78.0%

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

   8. Simplified 78.0%

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

      1. associate-/l* 86.4%

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

   10. Applied egg-rr 86.4%

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

   if 9.5000000000000003e89 < t

   1. Initial program 66.4%

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

   3. Simplified 66.4%

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

      1. associate-*l* 50.3%

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

      2. associate-/r* 53.2%

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

      3. associate-+r+ 53.2%

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

      4. metadata-eval 53.2%

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

      5. associate-*l* 53.2%

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

      6. add-cube-cbrt 53.2%

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

      7. pow3 53.2%

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

   6. Applied egg-rr 59.7%

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

      1. pow1/3 59.2%

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

      2. sqr-pow 59.2%

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

      3. metadata-eval 59.2%

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

      4. inv-pow 59.2%

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

      5. metadata-eval 59.2%

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

      6. inv-pow 59.2%

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

      7. unpow-prod-down 39.1%

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

   8. Applied egg-rr 39.1%

      \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{\ell}\right)}^{0.3333333333333333}\right)}\right) \cdot \sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}} \]
   9. Step-by-step derivation

      1. unpow1/3 39.2%

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

      2. unpow1/3 69.6%

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

   10. Simplified 69.6%

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

   11. Taylor expanded in k around 0 65.0%

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

       1. associate-*r* 65.0%

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

       2. unpow-prod-down 65.0%

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

       3. cbrt-unprod 59.6%

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

       4. inv-pow 59.6%

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

       5. inv-pow 59.6%

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

       6. pow-prod-up 59.6%

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

       7. metadata-eval 59.6%

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

       8. unpow2 59.6%

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

       9. cbrt-prod 84.8%

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

       10. pow2 84.8%

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

   13. Applied egg-rr 84.8%

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

       1. rem-cube-cbrt 84.8%

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

       2. *-commutative 84.8%

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

       3. associate-*l* 84.9%

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

   15. Simplified 84.9%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.7 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot {\left(t \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot \sqrt[3]{{\ell}^{-2}}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{elif}\;t\_m \leq 7 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \left(\sin k \cdot \frac{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot {\left(t\_m \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot \sqrt[3]{{\ell}^{-2}}\right)\right)}^{3}}\\ \end{array} \end{array} \]

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 8e-73)
    (/
     2.0
     (* (/ (* t_m (pow k 2.0)) (pow l 2.0)) (/ (pow (sin k) 2.0) (cos k))))
    (if (<= t_m 7e+89)
      (/
       2.0
       (*
        (/ (pow t_m 3.0) l)
        (* (sin k) (/ (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))) l))))
      (/
       2.0
       (*
        2.0
        (pow (* t_m (* (pow (cbrt k) 2.0) (cbrt (pow 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 <= 8e-73) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / pow(l, 2.0)) * (pow(sin(k), 2.0) / cos(k)));
	} else if (t_m <= 7e+89) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * (sin(k) * ((tan(k) * (2.0 + pow((k / t_m), 2.0))) / l)));
	} else {
		tmp = 2.0 / (2.0 * pow((t_m * (pow(cbrt(k), 2.0) * cbrt(pow(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 <= 8e-73) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.pow(l, 2.0)) * (Math.pow(Math.sin(k), 2.0) / Math.cos(k)));
	} else if (t_m <= 7e+89) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * (Math.sin(k) * ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) / l)));
	} else {
		tmp = 2.0 / (2.0 * Math.pow((t_m * (Math.pow(Math.cbrt(k), 2.0) * Math.cbrt(Math.pow(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 <= 8e-73)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / (l ^ 2.0)) * Float64((sin(k) ^ 2.0) / cos(k))));
	elseif (t_m <= 7e+89)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(sin(k) * Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) / l))));
	else
		tmp = Float64(2.0 / Float64(2.0 * (Float64(t_m * Float64((cbrt(k) ^ 2.0) * 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, 8e-73], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7e+89], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[Power[N[(t$95$m * N[(N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Power[l, -2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.99999999999999998e-73

   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 50.0%

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

   5. Taylor expanded in t around 0 69.8%

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

      1. associate-*r* 69.8%

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

      2. times-frac 71.4%

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

   7. Simplified 71.4%

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

   if 7.99999999999999998e-73 < t < 7.0000000000000001e89

   1. Initial program 74.9%

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

   3. Simplified 75.0%

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

      1. associate-*l* 72.1%

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

      2. associate-/r* 74.8%

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

      3. associate-+r+ 74.8%

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

      4. metadata-eval 74.8%

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

      5. associate-*l* 74.8%

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

      6. associate-*l/ 78.0%

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

   6. Applied egg-rr 78.0%

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

      1. associate-/l* 78.0%

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

      2. associate-*l* 78.0%

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

   8. Simplified 78.0%

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

      1. associate-/l* 86.4%

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

   10. Applied egg-rr 86.4%

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

   if 7.0000000000000001e89 < t

   1. Initial program 66.4%

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

   3. Simplified 66.4%

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

      1. associate-*l* 50.3%

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

      2. associate-/r* 53.2%

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

      3. associate-+r+ 53.2%

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

      4. metadata-eval 53.2%

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

      5. associate-*l* 53.2%

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

      6. add-cube-cbrt 53.2%

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

      7. pow3 53.2%

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

   6. Applied egg-rr 59.7%

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

      1. pow1/3 59.2%

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

      2. sqr-pow 59.2%

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

      3. metadata-eval 59.2%

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

      4. inv-pow 59.2%

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

      5. metadata-eval 59.2%

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

      6. inv-pow 59.2%

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

      7. unpow-prod-down 39.1%

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

   8. Applied egg-rr 39.1%

      \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{\ell}\right)}^{0.3333333333333333}\right)}\right) \cdot \sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}} \]
   9. Step-by-step derivation

      1. unpow1/3 39.2%

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

      2. unpow1/3 69.6%

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

   10. Simplified 69.6%

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

   11. Taylor expanded in k around 0 65.0%

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

       1. associate-*r* 65.0%

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

       2. unpow-prod-down 65.0%

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

       3. cbrt-unprod 59.6%

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

       4. inv-pow 59.6%

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

       5. inv-pow 59.6%

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

       6. pow-prod-up 59.6%

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

       7. metadata-eval 59.6%

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

       8. unpow2 59.6%

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

       9. cbrt-prod 84.8%

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

       10. pow2 84.8%

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

   13. Applied egg-rr 84.8%

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

       1. rem-cube-cbrt 84.8%

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

       2. *-commutative 84.8%

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

       3. associate-*l* 84.9%

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

   15. Simplified 84.9%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+89}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot {\left(t \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot \sqrt[3]{{\ell}^{-2}}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\frac{1}{\ell}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot \left(t\_2 \cdot t\_2\right)\right) \cdot \left(\sqrt[3]{2} \cdot {k}^{0.6666666666666666}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_m \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \end{array} \end{array} \]

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 (cbrt (/ 1.0 l))))
   (*
    t_s
    (if (<= k 1.5e-17)
      (/
       2.0
       (pow
        (* (* t_m (* t_2 t_2)) (* (cbrt 2.0) (pow k 0.6666666666666666)))
        3.0))
      (/
       2.0
       (/
        (* (* t_m (pow k 2.0)) (pow (sin k) 2.0))
        (* (pow l 2.0) (cos 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 = cbrt((1.0 / l));
	double tmp;
	if (k <= 1.5e-17) {
		tmp = 2.0 / pow(((t_m * (t_2 * t_2)) * (cbrt(2.0) * pow(k, 0.6666666666666666))), 3.0);
	} else {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) * pow(sin(k), 2.0)) / (pow(l, 2.0) * cos(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 = Math.cbrt((1.0 / l));
	double tmp;
	if (k <= 1.5e-17) {
		tmp = 2.0 / Math.pow(((t_m * (t_2 * t_2)) * (Math.cbrt(2.0) * Math.pow(k, 0.6666666666666666))), 3.0);
	} else {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) * Math.pow(Math.sin(k), 2.0)) / (Math.pow(l, 2.0) * Math.cos(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 = cbrt(Float64(1.0 / l))
	tmp = 0.0
	if (k <= 1.5e-17)
		tmp = Float64(2.0 / (Float64(Float64(t_m * Float64(t_2 * t_2)) * Float64(cbrt(2.0) * (k ^ 0.6666666666666666))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) * (sin(k) ^ 2.0)) / Float64((l ^ 2.0) * cos(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[Power[N[(1.0 / l), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.5e-17], N[(2.0 / N[Power[N[(N[(t$95$m * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[Power[2.0, 1/3], $MachinePrecision] * N[Power[k, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.50000000000000003e-17

   1. Initial program 57.1%

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

   3. Simplified 57.1%

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

      1. associate-*l* 52.1%

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

      2. associate-/r* 57.0%

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

      3. associate-+r+ 57.0%

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

      4. metadata-eval 57.0%

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

      5. associate-*l* 57.0%

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

      6. add-cube-cbrt 56.9%

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

      7. pow3 56.9%

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

   6. Applied egg-rr 64.0%

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

      1. pow1/3 63.4%

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

      2. sqr-pow 63.4%

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

      3. metadata-eval 63.4%

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

      4. inv-pow 63.4%

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

      5. metadata-eval 63.4%

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

      6. inv-pow 63.4%

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

      7. unpow-prod-down 37.3%

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

   8. Applied egg-rr 37.3%

      \[\leadsto \frac{2}{{\left(\left(t \cdot \color{blue}{\left({\left(\frac{1}{\ell}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{\ell}\right)}^{0.3333333333333333}\right)}\right) \cdot \sqrt[3]{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}} \]
   9. Step-by-step derivation

      1. unpow1/3 37.4%

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

      2. unpow1/3 73.8%

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

   10. Simplified 73.8%

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

   11. Taylor expanded in k around 0 67.4%

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

       1. pow1/3 66.9%

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

       2. pow-pow 28.8%

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

       3. metadata-eval 28.8%

          \[\leadsto \frac{2}{{\left(\left(t \cdot \left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right)\right) \cdot \left({k}^{\color{blue}{0.6666666666666666}} \cdot \sqrt[3]{2}\right)\right)}^{3}} \]

   13. Applied egg-rr 28.8%

       \[\leadsto \frac{2}{{\left(\left(t \cdot \left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right)\right) \cdot \left(\color{blue}{{k}^{0.6666666666666666}} \cdot \sqrt[3]{2}\right)\right)}^{3}} \]

   if 1.50000000000000003e-17 < k

   1. Initial program 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in t around 0 74.8%

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

      1. associate-*r* 74.8%

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

   7. Simplified 74.8%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot \left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right)\right) \cdot \left(\sqrt[3]{2} \cdot {k}^{0.6666666666666666}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot {k}^{2}\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.2 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t\_m}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 1.5 \cdot 10^{-92}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + t\_2\right)\right)\right)}\\ \mathbf{elif}\;t\_m \leq 2.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \left(\sin k \cdot \frac{\tan k \cdot \left(2 + t\_2\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

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 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 4.2e-138)
      (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt (/ t_m (cos k)))) 2.0))
      (if (<= t_m 1.5e-92)
        (/
         2.0
         (*
          (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))
          (* (tan k) (+ 1.0 (+ 1.0 t_2)))))
        (if (<= t_m 2.4e+55)
          (/
           2.0
           (* (/ (pow t_m 3.0) l) (* (sin k) (/ (* (tan k) (+ 2.0 t_2)) l))))
          (/ 2.0 (pow (* (/ (pow t_m 1.5) l) (* k (sqrt 2.0))) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 4.2e-138) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt((t_m / cos(k)))), 2.0);
	} else if (t_m <= 1.5e-92) {
		tmp = 2.0 / ((sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l))) * (tan(k) * (1.0 + (1.0 + t_2))));
	} else if (t_m <= 2.4e+55) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * (sin(k) * ((tan(k) * (2.0 + t_2)) / l)));
	} else {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (k * sqrt(2.0))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k / t_m) ** 2.0d0
    if (t_m <= 4.2d-138) then
        tmp = 2.0d0 / ((((k * sin(k)) / l) * sqrt((t_m / cos(k)))) ** 2.0d0)
    else if (t_m <= 1.5d-92) then
        tmp = 2.0d0 / ((sin(k) * (((t_m ** 2.0d0) / l) * (t_m / l))) * (tan(k) * (1.0d0 + (1.0d0 + t_2))))
    else if (t_m <= 2.4d+55) then
        tmp = 2.0d0 / (((t_m ** 3.0d0) / l) * (sin(k) * ((tan(k) * (2.0d0 + t_2)) / l)))
    else
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) * (k * sqrt(2.0d0))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 4.2e-138) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt((t_m / Math.cos(k)))), 2.0);
	} else if (t_m <= 1.5e-92) {
		tmp = 2.0 / ((Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))) * (Math.tan(k) * (1.0 + (1.0 + t_2))));
	} else if (t_m <= 2.4e+55) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * (Math.sin(k) * ((Math.tan(k) * (2.0 + t_2)) / l)));
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (k * Math.sqrt(2.0))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow((k / t_m), 2.0)
	tmp = 0
	if t_m <= 4.2e-138:
		tmp = 2.0 / math.pow((((k * math.sin(k)) / l) * math.sqrt((t_m / math.cos(k)))), 2.0)
	elif t_m <= 1.5e-92:
		tmp = 2.0 / ((math.sin(k) * ((math.pow(t_m, 2.0) / l) * (t_m / l))) * (math.tan(k) * (1.0 + (1.0 + t_2))))
	elif t_m <= 2.4e+55:
		tmp = 2.0 / ((math.pow(t_m, 3.0) / l) * (math.sin(k) * ((math.tan(k) * (2.0 + t_2)) / l)))
	else:
		tmp = 2.0 / math.pow(((math.pow(t_m, 1.5) / l) * (k * math.sqrt(2.0))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 4.2e-138)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(Float64(t_m / cos(k)))) ^ 2.0));
	elseif (t_m <= 1.5e-92)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l))) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + t_2)))));
	elseif (t_m <= 2.4e+55)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(sin(k) * Float64(Float64(tan(k) * Float64(2.0 + t_2)) / l))));
	else
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(k * sqrt(2.0))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (k / t_m) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 4.2e-138)
		tmp = 2.0 / ((((k * sin(k)) / l) * sqrt((t_m / cos(k)))) ^ 2.0);
	elseif (t_m <= 1.5e-92)
		tmp = 2.0 / ((sin(k) * (((t_m ^ 2.0) / l) * (t_m / l))) * (tan(k) * (1.0 + (1.0 + t_2))));
	elseif (t_m <= 2.4e+55)
		tmp = 2.0 / (((t_m ^ 3.0) / l) * (sin(k) * ((tan(k) * (2.0 + t_2)) / l)));
	else
		tmp = 2.0 / ((((t_m ^ 1.5) / l) * (k * sqrt(2.0))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.2e-138], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.5e-92], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.4e+55], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 4.19999999999999972e-138

   1. Initial program 48.7%

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

   3. Simplified 48.7%

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

   6. Applied egg-rr 10.7%

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

   7. Taylor expanded in t around 0 29.2%

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

   if 4.19999999999999972e-138 < t < 1.50000000000000007e-92

   1. Initial program 38.9%

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

   3. Simplified 38.9%

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

      1. unpow3 38.9%

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

      2. times-frac 74.8%

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

      3. pow2 74.8%

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

   6. Applied egg-rr 74.8%

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

   if 1.50000000000000007e-92 < t < 2.3999999999999999e55

   1. Initial program 81.2%

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

   3. Simplified 81.2%

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

      1. associate-*l* 78.7%

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

      2. associate-/r* 81.2%

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

      3. associate-+r+ 81.2%

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

      4. metadata-eval 81.2%

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

      5. associate-*l* 81.2%

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

      6. associate-*l/ 84.2%

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

   6. Applied egg-rr 84.2%

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

      1. associate-/l* 84.2%

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

      2. associate-*l* 84.1%

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

   8. Simplified 84.1%

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

      1. associate-/l* 91.8%

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

   10. Applied egg-rr 91.8%

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

   if 2.3999999999999999e55 < t

   1. Initial program 64.3%

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

   3. Simplified 64.4%

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

   6. Applied egg-rr 59.0%

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

   7. Taylor expanded in k around 0 78.3%

      \[\leadsto \frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{\left(k \cdot \sqrt{2}\right)}\right)}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{-138}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-92}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-92}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t\_m}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 3.5 \cdot 10^{+57}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \left(\sin k \cdot \frac{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \end{array} \end{array} \]

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 1.6e-92)
    (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt (/ t_m (cos k)))) 2.0))
    (if (<= t_m 3.5e+57)
      (/
       2.0
       (*
        (/ (pow t_m 3.0) l)
        (* (sin k) (/ (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))) l))))
      (/ 2.0 (pow (* (/ (pow t_m 1.5) l) (* k (sqrt 2.0))) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.6e-92) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt((t_m / cos(k)))), 2.0);
	} else if (t_m <= 3.5e+57) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * (sin(k) * ((tan(k) * (2.0 + pow((k / t_m), 2.0))) / l)));
	} else {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (k * sqrt(2.0))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.6d-92) then
        tmp = 2.0d0 / ((((k * sin(k)) / l) * sqrt((t_m / cos(k)))) ** 2.0d0)
    else if (t_m <= 3.5d+57) then
        tmp = 2.0d0 / (((t_m ** 3.0d0) / l) * (sin(k) * ((tan(k) * (2.0d0 + ((k / t_m) ** 2.0d0))) / l)))
    else
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) * (k * sqrt(2.0d0))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.6e-92) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt((t_m / Math.cos(k)))), 2.0);
	} else if (t_m <= 3.5e+57) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * (Math.sin(k) * ((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0))) / l)));
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (k * Math.sqrt(2.0))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.6e-92:
		tmp = 2.0 / math.pow((((k * math.sin(k)) / l) * math.sqrt((t_m / math.cos(k)))), 2.0)
	elif t_m <= 3.5e+57:
		tmp = 2.0 / ((math.pow(t_m, 3.0) / l) * (math.sin(k) * ((math.tan(k) * (2.0 + math.pow((k / t_m), 2.0))) / l)))
	else:
		tmp = 2.0 / math.pow(((math.pow(t_m, 1.5) / l) * (k * math.sqrt(2.0))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.6e-92)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(Float64(t_m / cos(k)))) ^ 2.0));
	elseif (t_m <= 3.5e+57)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(sin(k) * Float64(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))) / l))));
	else
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(k * sqrt(2.0))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.6e-92)
		tmp = 2.0 / ((((k * sin(k)) / l) * sqrt((t_m / cos(k)))) ^ 2.0);
	elseif (t_m <= 3.5e+57)
		tmp = 2.0 / (((t_m ^ 3.0) / l) * (sin(k) * ((tan(k) * (2.0 + ((k / t_m) ^ 2.0))) / l)));
	else
		tmp = 2.0 / ((((t_m ^ 1.5) / l) * (k * sqrt(2.0))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.6e-92], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.5e+57], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.5999999999999998e-92

   1. Initial program 48.3%

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

   3. Simplified 48.3%

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

   6. Applied egg-rr 12.5%

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

   7. Taylor expanded in t around 0 30.7%

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

   if 1.5999999999999998e-92 < t < 3.4999999999999997e57

   1. Initial program 81.2%

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

   3. Simplified 81.2%

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

      1. associate-*l* 78.7%

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

      2. associate-/r* 81.2%

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

      3. associate-+r+ 81.2%

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

      4. metadata-eval 81.2%

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

      5. associate-*l* 81.2%

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

      6. associate-*l/ 84.2%

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

   6. Applied egg-rr 84.2%

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

      1. associate-/l* 84.2%

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

      2. associate-*l* 84.1%

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

   8. Simplified 84.1%

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

      1. associate-/l* 91.8%

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

   10. Applied egg-rr 91.8%

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

   if 3.4999999999999997e57 < t

   1. Initial program 64.3%

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

   3. Simplified 64.4%

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

   6. Applied egg-rr 59.0%

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

   7. Taylor expanded in k around 0 78.3%

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

Alternative 11: 69.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{k}^{2}}{\cos k}}\\ \end{array} \end{array} \]

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 1.15e-19)
    (/ 2.0 (pow (* (/ (pow t_m 1.5) l) (* k (sqrt 2.0))) 2.0))
    (/ 2.0 (* (/ (* t_m (pow k 2.0)) (pow l 2.0)) (/ (pow k 2.0) (cos 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 <= 1.15e-19) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (k * sqrt(2.0))), 2.0);
	} else {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / pow(l, 2.0)) * (pow(k, 2.0) / cos(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 <= 1.15d-19) then
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) * (k * sqrt(2.0d0))) ** 2.0d0)
    else
        tmp = 2.0d0 / (((t_m * (k ** 2.0d0)) / (l ** 2.0d0)) * ((k ** 2.0d0) / cos(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 <= 1.15e-19) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (k * Math.sqrt(2.0))), 2.0);
	} else {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.pow(l, 2.0)) * (Math.pow(k, 2.0) / Math.cos(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 <= 1.15e-19:
		tmp = 2.0 / math.pow(((math.pow(t_m, 1.5) / l) * (k * math.sqrt(2.0))), 2.0)
	else:
		tmp = 2.0 / (((t_m * math.pow(k, 2.0)) / math.pow(l, 2.0)) * (math.pow(k, 2.0) / math.cos(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 <= 1.15e-19)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(k * sqrt(2.0))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / (l ^ 2.0)) * Float64((k ^ 2.0) / cos(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 <= 1.15e-19)
		tmp = 2.0 / ((((t_m ^ 1.5) / l) * (k * sqrt(2.0))) ^ 2.0);
	else
		tmp = 2.0 / (((t_m * (k ^ 2.0)) / (l ^ 2.0)) * ((k ^ 2.0) / cos(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, 1.15e-19], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.1499999999999999e-19

   1. Initial program 57.1%

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

   3. Simplified 57.1%

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

   6. Applied egg-rr 29.0%

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

   7. Taylor expanded in k around 0 33.0%

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

   if 1.1499999999999999e-19 < k

   1. Initial program 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in t around 0 74.8%

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

      1. associate-*r* 74.8%

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

      2. times-frac 74.8%

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

   7. Simplified 74.8%

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

   8. Taylor expanded in k around 0 64.5%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \frac{{k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t\_m}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 1.25 \cdot 10^{-75}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \frac{{t\_m}^{2}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t\_m \cdot \frac{t\_m}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \end{array} \end{array} \]

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 1.8e-137)
    (/ 2.0 (pow (* (/ (* k (sin k)) l) (sqrt (/ t_m (cos k)))) 2.0))
    (if (<= t_m 1.25e-75)
      (/
       2.0
       (*
        (/ (* t_m (/ (pow t_m 2.0) l)) l)
        (* (* (sin k) (tan k)) (+ 2.0 (/ k (* t_m (/ t_m k)))))))
      (/ 2.0 (pow (* (/ (pow t_m 1.5) l) (* k (sqrt 2.0))) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.8e-137) {
		tmp = 2.0 / pow((((k * sin(k)) / l) * sqrt((t_m / cos(k)))), 2.0);
	} else if (t_m <= 1.25e-75) {
		tmp = 2.0 / (((t_m * (pow(t_m, 2.0) / l)) / l) * ((sin(k) * tan(k)) * (2.0 + (k / (t_m * (t_m / k))))));
	} else {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (k * sqrt(2.0))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.8d-137) then
        tmp = 2.0d0 / ((((k * sin(k)) / l) * sqrt((t_m / cos(k)))) ** 2.0d0)
    else if (t_m <= 1.25d-75) then
        tmp = 2.0d0 / (((t_m * ((t_m ** 2.0d0) / l)) / l) * ((sin(k) * tan(k)) * (2.0d0 + (k / (t_m * (t_m / k))))))
    else
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) * (k * sqrt(2.0d0))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.8e-137) {
		tmp = 2.0 / Math.pow((((k * Math.sin(k)) / l) * Math.sqrt((t_m / Math.cos(k)))), 2.0);
	} else if (t_m <= 1.25e-75) {
		tmp = 2.0 / (((t_m * (Math.pow(t_m, 2.0) / l)) / l) * ((Math.sin(k) * Math.tan(k)) * (2.0 + (k / (t_m * (t_m / k))))));
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (k * Math.sqrt(2.0))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.8e-137:
		tmp = 2.0 / math.pow((((k * math.sin(k)) / l) * math.sqrt((t_m / math.cos(k)))), 2.0)
	elif t_m <= 1.25e-75:
		tmp = 2.0 / (((t_m * (math.pow(t_m, 2.0) / l)) / l) * ((math.sin(k) * math.tan(k)) * (2.0 + (k / (t_m * (t_m / k))))))
	else:
		tmp = 2.0 / math.pow(((math.pow(t_m, 1.5) / l) * (k * math.sqrt(2.0))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.8e-137)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k * sin(k)) / l) * sqrt(Float64(t_m / cos(k)))) ^ 2.0));
	elseif (t_m <= 1.25e-75)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) / l) * Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(k / Float64(t_m * Float64(t_m / k)))))));
	else
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(k * sqrt(2.0))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.8e-137)
		tmp = 2.0 / ((((k * sin(k)) / l) * sqrt((t_m / cos(k)))) ^ 2.0);
	elseif (t_m <= 1.25e-75)
		tmp = 2.0 / (((t_m * ((t_m ^ 2.0) / l)) / l) * ((sin(k) * tan(k)) * (2.0 + (k / (t_m * (t_m / k))))));
	else
		tmp = 2.0 / ((((t_m ^ 1.5) / l) * (k * sqrt(2.0))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.8e-137], N[(2.0 / N[Power[N[(N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.25e-75], N[(2.0 / N[(N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(k / N[(t$95$m * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.80000000000000003e-137

   1. Initial program 48.7%

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

   3. Simplified 48.7%

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

   6. Applied egg-rr 10.7%

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

   7. Taylor expanded in t around 0 29.2%

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

   if 1.80000000000000003e-137 < t < 1.24999999999999995e-75

   1. Initial program 67.4%

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

   3. Simplified 73.3%

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

      1. unpow2 73.3%

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

      2. clear-num 73.3%

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

      3. frac-times 73.3%

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

      4. *-un-lft-identity 73.3%

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

   6. Applied egg-rr 73.3%

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

      1. cube-mult 73.3%

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

      2. *-un-lft-identity 73.3%

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

      3. times-frac 86.5%

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

      4. pow2 86.5%

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

   8. Applied egg-rr 86.5%

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

   if 1.24999999999999995e-75 < t

   1. Initial program 69.5%

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

   3. Simplified 69.5%

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

   6. Applied egg-rr 60.8%

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

   7. Taylor expanded in k around 0 77.9%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sin k}{\ell} \cdot \sqrt{\frac{t}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-75}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{{t}^{2}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t\_m}{\cos k}} \cdot \left(k \cdot \frac{\sin k}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \end{array} \end{array} \]

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 1.6e-89)
    (/ 2.0 (pow (* (sqrt (/ t_m (cos k))) (* k (/ (sin k) l))) 2.0))
    (/ 2.0 (pow (* (/ (pow t_m 1.5) l) (* k (sqrt 2.0))) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.6e-89) {
		tmp = 2.0 / pow((sqrt((t_m / cos(k))) * (k * (sin(k) / l))), 2.0);
	} else {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (k * sqrt(2.0))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.6d-89) then
        tmp = 2.0d0 / ((sqrt((t_m / cos(k))) * (k * (sin(k) / l))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) * (k * sqrt(2.0d0))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.6e-89) {
		tmp = 2.0 / Math.pow((Math.sqrt((t_m / Math.cos(k))) * (k * (Math.sin(k) / l))), 2.0);
	} else {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (k * Math.sqrt(2.0))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.6e-89:
		tmp = 2.0 / math.pow((math.sqrt((t_m / math.cos(k))) * (k * (math.sin(k) / l))), 2.0)
	else:
		tmp = 2.0 / math.pow(((math.pow(t_m, 1.5) / l) * (k * math.sqrt(2.0))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.6e-89)
		tmp = Float64(2.0 / (Float64(sqrt(Float64(t_m / cos(k))) * Float64(k * Float64(sin(k) / l))) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(k * sqrt(2.0))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.6e-89)
		tmp = 2.0 / ((sqrt((t_m / cos(k))) * (k * (sin(k) / l))) ^ 2.0);
	else
		tmp = 2.0 / ((((t_m ^ 1.5) / l) * (k * sqrt(2.0))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.6e-89], N[(2.0 / N[Power[N[(N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.59999999999999999e-89

   1. Initial program 48.8%

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

   3. Simplified 48.8%

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

   6. Applied egg-rr 13.5%

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

   7. Taylor expanded in t around 0 31.4%

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

      1. associate-/l* 31.4%

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

   9. Simplified 31.4%

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

   if 1.59999999999999999e-89 < t

   1. Initial program 71.4%

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

   3. Simplified 71.4%

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

   6. Applied egg-rr 62.1%

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

   7. Taylor expanded in k around 0 78.2%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t}{\cos k}} \cdot \left(k \cdot \frac{\sin k}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+123}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot \frac{{t\_m}^{2}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t\_m \cdot \frac{t\_m}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\ell}^{-2} \cdot \left(t\_m \cdot {k}^{4}\right)}\\ \end{array} \end{array} \]

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 3.2e-18)
    (/ 2.0 (pow (* (/ (pow t_m 1.5) l) (* k (sqrt 2.0))) 2.0))
    (if (<= k 3e+123)
      (/
       2.0
       (*
        (/ (* t_m (/ (pow t_m 2.0) l)) l)
        (* (* (sin k) (tan k)) (+ 2.0 (/ k (* t_m (/ t_m k)))))))
      (/ 2.0 (* (pow l -2.0) (* t_m (pow k 4.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.2e-18) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (k * sqrt(2.0))), 2.0);
	} else if (k <= 3e+123) {
		tmp = 2.0 / (((t_m * (pow(t_m, 2.0) / l)) / l) * ((sin(k) * tan(k)) * (2.0 + (k / (t_m * (t_m / k))))));
	} else {
		tmp = 2.0 / (pow(l, -2.0) * (t_m * pow(k, 4.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.2d-18) then
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) * (k * sqrt(2.0d0))) ** 2.0d0)
    else if (k <= 3d+123) then
        tmp = 2.0d0 / (((t_m * ((t_m ** 2.0d0) / l)) / l) * ((sin(k) * tan(k)) * (2.0d0 + (k / (t_m * (t_m / k))))))
    else
        tmp = 2.0d0 / ((l ** (-2.0d0)) * (t_m * (k ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.2e-18) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (k * Math.sqrt(2.0))), 2.0);
	} else if (k <= 3e+123) {
		tmp = 2.0 / (((t_m * (Math.pow(t_m, 2.0) / l)) / l) * ((Math.sin(k) * Math.tan(k)) * (2.0 + (k / (t_m * (t_m / k))))));
	} else {
		tmp = 2.0 / (Math.pow(l, -2.0) * (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3.2e-18:
		tmp = 2.0 / math.pow(((math.pow(t_m, 1.5) / l) * (k * math.sqrt(2.0))), 2.0)
	elif k <= 3e+123:
		tmp = 2.0 / (((t_m * (math.pow(t_m, 2.0) / l)) / l) * ((math.sin(k) * math.tan(k)) * (2.0 + (k / (t_m * (t_m / k))))))
	else:
		tmp = 2.0 / (math.pow(l, -2.0) * (t_m * math.pow(k, 4.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.2e-18)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(k * sqrt(2.0))) ^ 2.0));
	elseif (k <= 3e+123)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) / l) * Float64(Float64(sin(k) * tan(k)) * Float64(2.0 + Float64(k / Float64(t_m * Float64(t_m / k)))))));
	else
		tmp = Float64(2.0 / Float64((l ^ -2.0) * Float64(t_m * (k ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3.2e-18)
		tmp = 2.0 / ((((t_m ^ 1.5) / l) * (k * sqrt(2.0))) ^ 2.0);
	elseif (k <= 3e+123)
		tmp = 2.0 / (((t_m * ((t_m ^ 2.0) / l)) / l) * ((sin(k) * tan(k)) * (2.0 + (k / (t_m * (t_m / k))))));
	else
		tmp = 2.0 / ((l ^ -2.0) * (t_m * (k ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.2e-18], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3e+123], N[(2.0 / N[(N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(k / N[(t$95$m * N[(t$95$m / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[l, -2.0], $MachinePrecision] * N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.1999999999999999e-18

   1. Initial program 57.1%

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

   3. Simplified 57.1%

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

   6. Applied egg-rr 29.0%

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

   7. Taylor expanded in k around 0 33.0%

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

   if 3.1999999999999999e-18 < k < 3.00000000000000008e123

   1. Initial program 51.7%

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

   3. Simplified 58.2%

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

      1. unpow2 58.2%

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

      2. clear-num 58.2%

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

      3. frac-times 58.2%

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

      4. *-un-lft-identity 58.2%

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

   6. Applied egg-rr 58.2%

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

      1. cube-mult 58.2%

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

      2. *-un-lft-identity 58.2%

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

      3. times-frac 58.2%

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

      4. pow2 58.2%

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

   8. Applied egg-rr 58.2%

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

   if 3.00000000000000008e123 < k

   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 51.5%

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

   5. Taylor expanded in t around 0 76.1%

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

      1. associate-*r* 76.1%

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

      2. times-frac 76.0%

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

   7. Simplified 76.0%

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

   8. Taylor expanded in k around 0 67.3%

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

      1. associate-/l* 67.1%

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

   10. Simplified 67.1%

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

       1. pow1 67.1%

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

       2. div-inv 67.1%

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

       3. pow-flip 67.1%

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

       4. metadata-eval 67.1%

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

   12. Applied egg-rr 67.1%

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

       1. unpow1 67.1%

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

       2. associate-*r* 67.3%

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

       3. *-commutative 67.3%

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

   14. Simplified 67.3%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+123}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{{t}^{2}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\ell}^{-2} \cdot \left(t \cdot {k}^{4}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\ell}^{-2} \cdot \left(t\_m \cdot {k}^{4}\right)}\\ \end{array} \end{array} \]

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 3.7e-17)
    (/ 2.0 (pow (* (/ (pow t_m 1.5) l) (* k (sqrt 2.0))) 2.0))
    (/ 2.0 (* (pow l -2.0) (* t_m (pow k 4.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.7e-17) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (k * sqrt(2.0))), 2.0);
	} else {
		tmp = 2.0 / (pow(l, -2.0) * (t_m * pow(k, 4.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.7d-17) then
        tmp = 2.0d0 / ((((t_m ** 1.5d0) / l) * (k * sqrt(2.0d0))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((l ** (-2.0d0)) * (t_m * (k ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 3.7e-17) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (k * Math.sqrt(2.0))), 2.0);
	} else {
		tmp = 2.0 / (Math.pow(l, -2.0) * (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 3.7e-17:
		tmp = 2.0 / math.pow(((math.pow(t_m, 1.5) / l) * (k * math.sqrt(2.0))), 2.0)
	else:
		tmp = 2.0 / (math.pow(l, -2.0) * (t_m * math.pow(k, 4.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 3.7e-17)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(k * sqrt(2.0))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((l ^ -2.0) * Float64(t_m * (k ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 3.7e-17)
		tmp = 2.0 / ((((t_m ^ 1.5) / l) * (k * sqrt(2.0))) ^ 2.0);
	else
		tmp = 2.0 / ((l ^ -2.0) * (t_m * (k ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 3.7e-17], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[l, -2.0], $MachinePrecision] * N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.6999999999999997e-17

   1. Initial program 57.1%

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

   3. Simplified 57.1%

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

   6. Applied egg-rr 29.0%

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

   7. Taylor expanded in k around 0 33.0%

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

   if 3.6999999999999997e-17 < k

   1. Initial program 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in t around 0 74.8%

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

      1. associate-*r* 74.8%

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

      2. times-frac 74.8%

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

   7. Simplified 74.8%

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

   8. Taylor expanded in k around 0 57.9%

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

      1. associate-/l* 57.8%

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

   10. Simplified 57.8%

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

       1. pow1 57.8%

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

       2. div-inv 57.8%

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

       3. pow-flip 57.8%

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

       4. metadata-eval 57.8%

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

   12. Applied egg-rr 57.8%

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

       1. unpow1 57.8%

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

       2. associate-*r* 57.9%

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

       3. *-commutative 57.9%

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

   14. Simplified 57.9%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \sqrt{2}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\ell}^{-2} \cdot \left(t \cdot {k}^{4}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{-76}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \mathbf{elif}\;t\_m \leq 1.65 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3} \cdot \left(2 \cdot \frac{{k}^{2}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

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 5e-76)
    (/ (* 2.0 (pow l 2.0)) (* t_m (pow k 4.0)))
    (if (<= t_m 1.65e+101)
      (/ 2.0 (/ (* (pow t_m 3.0) (* 2.0 (/ (pow k 2.0) l))) l))
      (/ 2.0 (* (* (sin k) (/ (pow t_m 3.0) (* l l))) (* 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 <= 5e-76) {
		tmp = (2.0 * pow(l, 2.0)) / (t_m * pow(k, 4.0));
	} else if (t_m <= 1.65e+101) {
		tmp = 2.0 / ((pow(t_m, 3.0) * (2.0 * (pow(k, 2.0) / l))) / l);
	} else {
		tmp = 2.0 / ((sin(k) * (pow(t_m, 3.0) / (l * l))) * (2.0 * 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 (t_m <= 5d-76) then
        tmp = (2.0d0 * (l ** 2.0d0)) / (t_m * (k ** 4.0d0))
    else if (t_m <= 1.65d+101) then
        tmp = 2.0d0 / (((t_m ** 3.0d0) * (2.0d0 * ((k ** 2.0d0) / l))) / l)
    else
        tmp = 2.0d0 / ((sin(k) * ((t_m ** 3.0d0) / (l * l))) * (2.0d0 * 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 (t_m <= 5e-76) {
		tmp = (2.0 * Math.pow(l, 2.0)) / (t_m * Math.pow(k, 4.0));
	} else if (t_m <= 1.65e+101) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) * (2.0 * (Math.pow(k, 2.0) / l))) / l);
	} else {
		tmp = 2.0 / ((Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l))) * (2.0 * 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 t_m <= 5e-76:
		tmp = (2.0 * math.pow(l, 2.0)) / (t_m * math.pow(k, 4.0))
	elif t_m <= 1.65e+101:
		tmp = 2.0 / ((math.pow(t_m, 3.0) * (2.0 * (math.pow(k, 2.0) / l))) / l)
	else:
		tmp = 2.0 / ((math.sin(k) * (math.pow(t_m, 3.0) / (l * l))) * (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 <= 5e-76)
		tmp = Float64(Float64(2.0 * (l ^ 2.0)) / Float64(t_m * (k ^ 4.0)));
	elseif (t_m <= 1.65e+101)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) * Float64(2.0 * Float64((k ^ 2.0) / l))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l))) * Float64(2.0 * 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 (t_m <= 5e-76)
		tmp = (2.0 * (l ^ 2.0)) / (t_m * (k ^ 4.0));
	elseif (t_m <= 1.65e+101)
		tmp = 2.0 / (((t_m ^ 3.0) * (2.0 * ((k ^ 2.0) / l))) / l);
	else
		tmp = 2.0 / ((sin(k) * ((t_m ^ 3.0) / (l * l))) * (2.0 * 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[t$95$m, 5e-76], N[(N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.65e+101], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(2.0 * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.9999999999999998e-76

   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 50.2%

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

   5. Taylor expanded in t around 0 70.2%

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

      1. associate-*r* 70.2%

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

      2. times-frac 71.7%

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

   7. Simplified 71.7%

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

   8. Taylor expanded in k around 0 59.9%

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

      1. associate-*r/ 59.9%

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

      2. *-commutative 59.9%

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

   10. Simplified 59.9%

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

   if 4.9999999999999998e-76 < t < 1.65000000000000006e101

   1. Initial program 71.7%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in k around 0 59.1%

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

      1. associate-*l/ 59.6%

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

   7. Applied egg-rr 59.6%

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

      1. associate-/l* 62.0%

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

   9. Simplified 62.0%

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

       1. associate-*l/ 66.1%

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

       2. associate-/l* 66.1%

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

   11. Applied egg-rr 66.1%

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

   if 1.65000000000000006e101 < t

   1. Initial program 67.1%

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

   3. Simplified 67.1%

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

   5. Taylor expanded in k around 0 67.1%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-76}:\\ \;\;\;\;\frac{2 \cdot {\ell}^{2}}{t \cdot {k}^{4}}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3} \cdot \left(2 \cdot \frac{{k}^{2}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.22 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3} \cdot \left(2 \cdot \frac{{k}^{2}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\ell}^{-2} \cdot \left(t\_m \cdot {k}^{4}\right)}\\ \end{array} \end{array} \]

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 1.22e-24)
    (/ 2.0 (/ (* (pow t_m 3.0) (* 2.0 (/ (pow k 2.0) l))) l))
    (/ 2.0 (* (pow l -2.0) (* t_m (pow k 4.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.22e-24) {
		tmp = 2.0 / ((pow(t_m, 3.0) * (2.0 * (pow(k, 2.0) / l))) / l);
	} else {
		tmp = 2.0 / (pow(l, -2.0) * (t_m * pow(k, 4.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.22d-24) then
        tmp = 2.0d0 / (((t_m ** 3.0d0) * (2.0d0 * ((k ** 2.0d0) / l))) / l)
    else
        tmp = 2.0d0 / ((l ** (-2.0d0)) * (t_m * (k ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.22e-24) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) * (2.0 * (Math.pow(k, 2.0) / l))) / l);
	} else {
		tmp = 2.0 / (Math.pow(l, -2.0) * (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.22e-24:
		tmp = 2.0 / ((math.pow(t_m, 3.0) * (2.0 * (math.pow(k, 2.0) / l))) / l)
	else:
		tmp = 2.0 / (math.pow(l, -2.0) * (t_m * math.pow(k, 4.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.22e-24)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) * Float64(2.0 * Float64((k ^ 2.0) / l))) / l));
	else
		tmp = Float64(2.0 / Float64((l ^ -2.0) * Float64(t_m * (k ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.22e-24)
		tmp = 2.0 / (((t_m ^ 3.0) * (2.0 * ((k ^ 2.0) / l))) / l);
	else
		tmp = 2.0 / ((l ^ -2.0) * (t_m * (k ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.22e-24], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(2.0 * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[l, -2.0], $MachinePrecision] * N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.22000000000000004e-24

   1. Initial program 56.7%

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

   3. Simplified 56.6%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   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. associate-*l/ 56.6%

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

   7. Applied egg-rr 56.6%

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

      1. associate-/l* 56.9%

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

   9. Simplified 56.9%

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

       1. associate-*l/ 57.8%

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

       2. associate-/l* 57.8%

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

   11. Applied egg-rr 57.8%

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

   if 1.22000000000000004e-24 < k

   1. Initial program 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in t around 0 74.8%

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

      1. associate-*r* 74.8%

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

      2. times-frac 74.8%

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

   7. Simplified 74.8%

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

   8. Taylor expanded in k around 0 59.0%

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

      1. associate-/l* 58.9%

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

   10. Simplified 58.9%

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

       1. pow1 58.9%

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

       2. div-inv 58.9%

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

       3. pow-flip 58.9%

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

       4. metadata-eval 58.9%

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

   12. Applied egg-rr 58.9%

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

       1. unpow1 58.9%

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

       2. associate-*r* 59.0%

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

       3. *-commutative 59.0%

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

   14. Simplified 59.0%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.22 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3} \cdot \left(2 \cdot \frac{{k}^{2}}{\ell}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\ell}^{-2} \cdot \left(t \cdot {k}^{4}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 57.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t\_m}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\ell}^{-2} \cdot \left(t\_m \cdot {k}^{4}\right)}\\ \end{array} \end{array} \]

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 8e-25)
    (/ 2.0 (/ (* (/ (pow t_m 3.0) l) (* 2.0 (pow k 2.0))) l))
    (/ 2.0 (* (pow l -2.0) (* t_m (pow k 4.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 8e-25) {
		tmp = 2.0 / (((pow(t_m, 3.0) / l) * (2.0 * pow(k, 2.0))) / l);
	} else {
		tmp = 2.0 / (pow(l, -2.0) * (t_m * pow(k, 4.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 8d-25) then
        tmp = 2.0d0 / ((((t_m ** 3.0d0) / l) * (2.0d0 * (k ** 2.0d0))) / l)
    else
        tmp = 2.0d0 / ((l ** (-2.0d0)) * (t_m * (k ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 8e-25) {
		tmp = 2.0 / (((Math.pow(t_m, 3.0) / l) * (2.0 * Math.pow(k, 2.0))) / l);
	} else {
		tmp = 2.0 / (Math.pow(l, -2.0) * (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 8e-25:
		tmp = 2.0 / (((math.pow(t_m, 3.0) / l) * (2.0 * math.pow(k, 2.0))) / l)
	else:
		tmp = 2.0 / (math.pow(l, -2.0) * (t_m * math.pow(k, 4.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 8e-25)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_m ^ 3.0) / l) * Float64(2.0 * (k ^ 2.0))) / l));
	else
		tmp = Float64(2.0 / Float64((l ^ -2.0) * Float64(t_m * (k ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 8e-25)
		tmp = 2.0 / ((((t_m ^ 3.0) / l) * (2.0 * (k ^ 2.0))) / l);
	else
		tmp = 2.0 / ((l ^ -2.0) * (t_m * (k ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 8e-25], N[(2.0 / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[l, -2.0], $MachinePrecision] * N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 8.00000000000000031e-25

   1. Initial program 56.7%

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

   3. Simplified 56.6%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   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. associate-*l/ 56.6%

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

   7. Applied egg-rr 56.6%

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

   if 8.00000000000000031e-25 < k

   1. Initial program 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in t around 0 74.8%

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

      1. associate-*r* 74.8%

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

      2. times-frac 74.8%

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

   7. Simplified 74.8%

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

   8. Taylor expanded in k around 0 59.0%

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

      1. associate-/l* 58.9%

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

   10. Simplified 58.9%

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

       1. pow1 58.9%

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

       2. div-inv 58.9%

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

       3. pow-flip 58.9%

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

       4. metadata-eval 58.9%

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

   12. Applied egg-rr 58.9%

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

       1. unpow1 58.9%

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

       2. associate-*r* 59.0%

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

       3. *-commutative 59.0%

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

   14. Simplified 59.0%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\ell}^{-2} \cdot \left(t \cdot {k}^{4}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 57.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\ell}^{-2} \cdot \left(t\_m \cdot {k}^{4}\right)}\\ \end{array} \end{array} \]

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 5.8e-25)
    (/ 2.0 (* (* 2.0 (pow k 2.0)) (/ (/ (pow t_m 3.0) l) l)))
    (/ 2.0 (* (pow l -2.0) (* t_m (pow k 4.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5.8e-25) {
		tmp = 2.0 / ((2.0 * pow(k, 2.0)) * ((pow(t_m, 3.0) / l) / l));
	} else {
		tmp = 2.0 / (pow(l, -2.0) * (t_m * pow(k, 4.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.8d-25) then
        tmp = 2.0d0 / ((2.0d0 * (k ** 2.0d0)) * (((t_m ** 3.0d0) / l) / l))
    else
        tmp = 2.0d0 / ((l ** (-2.0d0)) * (t_m * (k ** 4.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 5.8e-25) {
		tmp = 2.0 / ((2.0 * Math.pow(k, 2.0)) * ((Math.pow(t_m, 3.0) / l) / l));
	} else {
		tmp = 2.0 / (Math.pow(l, -2.0) * (t_m * Math.pow(k, 4.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 5.8e-25:
		tmp = 2.0 / ((2.0 * math.pow(k, 2.0)) * ((math.pow(t_m, 3.0) / l) / l))
	else:
		tmp = 2.0 / (math.pow(l, -2.0) * (t_m * math.pow(k, 4.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 5.8e-25)
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k ^ 2.0)) * Float64(Float64((t_m ^ 3.0) / l) / l)));
	else
		tmp = Float64(2.0 / Float64((l ^ -2.0) * Float64(t_m * (k ^ 4.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 5.8e-25)
		tmp = 2.0 / ((2.0 * (k ^ 2.0)) * (((t_m ^ 3.0) / l) / l));
	else
		tmp = 2.0 / ((l ^ -2.0) * (t_m * (k ^ 4.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 5.8e-25], 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], N[(2.0 / N[(N[Power[l, -2.0], $MachinePrecision] * N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.8000000000000001e-25

   1. Initial program 56.7%

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

   3. Simplified 56.6%

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

   5. Taylor expanded in k around 0 56.4%

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

   if 5.8000000000000001e-25 < k

   1. Initial program 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in t around 0 74.8%

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

      1. associate-*r* 74.8%

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

      2. times-frac 74.8%

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

   7. Simplified 74.8%

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

   8. Taylor expanded in k around 0 59.0%

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

      1. associate-/l* 58.9%

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

   10. Simplified 58.9%

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

       1. pow1 58.9%

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

       2. div-inv 58.9%

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

       3. pow-flip 58.9%

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

       4. metadata-eval 58.9%

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

   12. Applied egg-rr 58.9%

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

       1. unpow1 58.9%

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

       2. associate-*r* 59.0%

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

       3. *-commutative 59.0%

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

   14. Simplified 59.0%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\ell}^{-2} \cdot \left(t \cdot {k}^{4}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 52.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{{k}^{4} \cdot \frac{t\_m}{{\ell}^{2}}} \end{array} \]

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 (/ 2.0 (* (pow k 4.0) (/ t_m (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 * (2.0 / (pow(k, 4.0) * (t_m / 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 * (2.0d0 / ((k ** 4.0d0) * (t_m / (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 * (2.0 / (Math.pow(k, 4.0) * (t_m / 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 * (2.0 / (math.pow(k, 4.0) * (t_m / 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(2.0 / Float64((k ^ 4.0) * Float64(t_m / (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 * (2.0 / ((k ^ 4.0) * (t_m / (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[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.7%

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

3. Simplified 55.7%

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

5. Taylor expanded in t around 0 63.6%

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

   1. associate-*r* 63.6%

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

   2. times-frac 65.5%

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

7. Simplified 65.5%

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

8. Taylor expanded in k around 0 55.6%

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

   1. associate-/l* 55.9%

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

10. Simplified 55.9%

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

Alternative 21: 52.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{2}{{k}^{4}} \cdot \frac{{\ell}^{2}}{t\_m}\right) \end{array} \]

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 (pow k 4.0)) (/ (pow l 2.0) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * ((2.0 / pow(k, 4.0)) * (pow(l, 2.0) / t_m));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * ((2.0d0 / (k ** 4.0d0)) * ((l ** 2.0d0) / t_m))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * ((2.0 / Math.pow(k, 4.0)) * (Math.pow(l, 2.0) / t_m));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * ((2.0 / math.pow(k, 4.0)) * (math.pow(l, 2.0) / t_m))

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 / (k ^ 4.0)) * Float64((l ^ 2.0) / t_m)))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * ((2.0 / (k ^ 4.0)) * ((l ^ 2.0) / t_m));
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[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.7%

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

3. Simplified 55.7%

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

5. Taylor expanded in t around 0 63.6%

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

   1. associate-*r* 63.6%

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

   2. times-frac 65.5%

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

7. Simplified 65.5%

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

8. Taylor expanded in k around 0 55.6%

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

   1. associate-/l* 55.9%

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

10. Simplified 55.9%

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

11. Taylor expanded in k around 0 55.6%

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

    1. associate-*r/ 55.6%

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

    2. times-frac 55.8%

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

13. Simplified 55.8%

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

Reproduce

herbie shell --seed 2024137 
(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))))