Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.0% → 86.3%

Time: 18.9s

Alternatives: 25

Speedup: 24.8×

• 27.3% 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: 55.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left({\left(\sqrt[3]{k\_m}\right)}^{2} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{elif}\;k\_m \leq 0.00094:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t\_m}\right)\right)\right) \cdot \sqrt{\sin k\_m \cdot \tan k\_m}}\right)}^{2}\\ \mathbf{elif}\;k\_m \leq 4.4 \cdot 10^{+117}:\\ \;\;\;\;\frac{2}{t\_m \cdot \mathsf{fma}\left(2, \frac{t\_2}{\cos k\_m} \cdot {\left(\frac{t\_m}{\ell}\right)}^{2}, \frac{\frac{t\_2 \cdot {k\_m}^{2}}{{\ell}^{2}}}{\cos k\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k\_m \cdot \left(\tan k\_m \cdot \left(2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)\right)}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (pow (sin k_m) 2.0)))
   (*
    t_s
    (if (<= k_m 4.5e-125)
      (/
       2.0
       (pow
        (* (* t_m (pow (cbrt l) -2.0)) (* (pow (cbrt k_m) 2.0) (cbrt 2.0)))
        3.0))
      (if (<= k_m 0.00094)
        (pow
         (/
          (sqrt 2.0)
          (*
           (* (/ (pow t_m 1.5) l) (hypot 1.0 (hypot 1.0 (/ k_m t_m))))
           (sqrt (* (sin k_m) (tan k_m)))))
         2.0)
        (if (<= k_m 4.4e+117)
          (/
           2.0
           (*
            t_m
            (fma
             2.0
             (* (/ t_2 (cos k_m)) (pow (/ t_m l) 2.0))
             (/ (/ (* t_2 (pow k_m 2.0)) (pow l 2.0)) (cos k_m)))))
          (/
           2.0
           (pow
            (*
             (/ t_m (pow (cbrt l) 2.0))
             (cbrt (* (sin k_m) (* (tan k_m) (+ 2.0 (pow (/ k_m t_m) 2.0))))))
            3.0))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = pow(sin(k_m), 2.0);
	double tmp;
	if (k_m <= 4.5e-125) {
		tmp = 2.0 / pow(((t_m * pow(cbrt(l), -2.0)) * (pow(cbrt(k_m), 2.0) * cbrt(2.0))), 3.0);
	} else if (k_m <= 0.00094) {
		tmp = pow((sqrt(2.0) / (((pow(t_m, 1.5) / l) * hypot(1.0, hypot(1.0, (k_m / t_m)))) * sqrt((sin(k_m) * tan(k_m))))), 2.0);
	} else if (k_m <= 4.4e+117) {
		tmp = 2.0 / (t_m * fma(2.0, ((t_2 / cos(k_m)) * pow((t_m / l), 2.0)), (((t_2 * pow(k_m, 2.0)) / pow(l, 2.0)) / cos(k_m))));
	} else {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * cbrt((sin(k_m) * (tan(k_m) * (2.0 + pow((k_m / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = sin(k_m) ^ 2.0
	tmp = 0.0
	if (k_m <= 4.5e-125)
		tmp = Float64(2.0 / (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * Float64((cbrt(k_m) ^ 2.0) * cbrt(2.0))) ^ 3.0));
	elseif (k_m <= 0.00094)
		tmp = Float64(sqrt(2.0) / Float64(Float64(Float64((t_m ^ 1.5) / l) * hypot(1.0, hypot(1.0, Float64(k_m / t_m)))) * sqrt(Float64(sin(k_m) * tan(k_m))))) ^ 2.0;
	elseif (k_m <= 4.4e+117)
		tmp = Float64(2.0 / Float64(t_m * fma(2.0, Float64(Float64(t_2 / cos(k_m)) * (Float64(t_m / l) ^ 2.0)), Float64(Float64(Float64(t_2 * (k_m ^ 2.0)) / (l ^ 2.0)) / cos(k_m)))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(Float64(sin(k_m) * Float64(tan(k_m) * Float64(2.0 + (Float64(k_m / t_m) ^ 2.0)))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := Block[{t$95$2 = N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 4.5e-125], N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 0.00094], N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[k$95$m, 4.4e+117], N[(2.0 / N[(t$95$m * N[(2.0 * N[(N[(t$95$2 / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$m / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$2 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if k < 4.50000000000000012e-125

   1. Initial program 56.8%

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

   3. Simplified 57.8%

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

   5. Taylor expanded in k around 0 56.2%

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

      1. unpow2 56.2%

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

   7. Applied egg-rr 56.2%

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

      1. add-cube-cbrt 56.2%

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

      2. pow3 56.2%

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

   9. Applied egg-rr 65.6%

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

       1. *-commutative 65.6%

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

       2. cbrt-prod 65.6%

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

       3. pow2 65.6%

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

       4. cbrt-prod 76.8%

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

       5. pow2 76.8%

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

   11. Applied egg-rr 76.8%

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

   if 4.50000000000000012e-125 < k < 9.39999999999999972e-4

   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

   6. Applied egg-rr 57.0%

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

      1. unpow2 57.0%

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

      2. associate-*r* 57.0%

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

   8. Simplified 57.0%

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

   if 9.39999999999999972e-4 < k < 4.40000000000000028e117

   1. Initial program 47.1%

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

   3. Simplified 47.2%

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

      1. add-cube-cbrt 47.2%

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

      2. pow3 47.2%

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

      3. associate-/r* 51.3%

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

      4. *-commutative 51.3%

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

      5. cbrt-prod 51.1%

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

      6. associate-/r* 47.0%

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

      7. cbrt-div 47.0%

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

      8. rem-cbrt-cube 47.7%

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

      9. cbrt-prod 64.8%

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

      10. pow2 64.8%

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

   6. Applied egg-rr 64.8%

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

      1. add-sqr-sqrt 25.9%

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

      2. pow2 25.9%

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

      3. associate-+r+ 25.9%

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

      4. metadata-eval 25.9%

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

      5. sqrt-prod 25.9%

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

      6. metadata-eval 25.9%

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

      7. associate-+r+ 25.9%

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

      8. add-sqr-sqrt 25.9%

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

      9. hypot-1-def 25.9%

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

      10. unpow2 25.9%

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

      11. hypot-1-def 25.9%

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

   8. Applied egg-rr 25.9%

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

      1. unpow2 25.9%

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

      2. swap-sqr 25.9%

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

      3. rem-square-sqrt 65.0%

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

      4. unpow2 65.0%

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

   10. Simplified 65.0%

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

   11. Taylor expanded in t around 0 76.7%

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

       1. fma-define 76.7%

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

       2. *-commutative 76.7%

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

       3. *-commutative 76.7%

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

       4. times-frac 76.7%

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

       5. unpow2 76.7%

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

       6. unpow2 76.7%

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

       7. times-frac 91.4%

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

       8. unpow1 91.4%

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

       9. pow-plus 91.4%

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

       10. metadata-eval 91.4%

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

       11. associate-/r* 91.4%

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

   13. Simplified 91.4%

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

   if 4.40000000000000028e117 < k

   1. Initial program 54.5%

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

   3. Simplified 54.5%

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

      1. associate-*l* 54.5%

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

      2. associate-/r* 59.1%

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

      3. associate-+r+ 59.1%

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

      4. metadata-eval 59.1%

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

      5. associate-*l* 59.1%

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

      6. add-cube-cbrt 59.1%

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

      7. pow3 59.0%

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

   6. Applied egg-rr 81.9%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 0.00094:\\ \;\;\;\;{\left(\frac{\sqrt{2}}{\left(\frac{{t}^{1.5}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right) \cdot \sqrt{\sin k \cdot \tan k}}\right)}^{2}\\ \mathbf{elif}\;k \leq 4.4 \cdot 10^{+117}:\\ \;\;\;\;\frac{2}{t \cdot \mathsf{fma}\left(2, \frac{{\sin k}^{2}}{\cos k} \cdot {\left(\frac{t}{\ell}\right)}^{2}, \frac{\frac{{\sin k}^{2} \cdot {k}^{2}}{{\ell}^{2}}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{\cos k\_m \cdot \left(2 \cdot {\ell}^{2}\right)}{t\_m \cdot {k\_m}^{2}}}{{\sin k\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k\_m}\right)}^{3} \cdot \left(\tan k\_m \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t\_m}\right)\right)\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5e-64)
    (/
     (/ (* (cos k_m) (* 2.0 (pow l 2.0))) (* t_m (pow k_m 2.0)))
     (pow (sin k_m) 2.0))
    (/
     2.0
     (*
      (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (sin k_m))) 3.0)
      (* (tan k_m) (pow (hypot 1.0 (hypot 1.0 (/ k_m t_m))) 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 5e-64) {
		tmp = ((cos(k_m) * (2.0 * pow(l, 2.0))) / (t_m * pow(k_m, 2.0))) / pow(sin(k_m), 2.0);
	} else {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(sin(k_m))), 3.0) * (tan(k_m) * pow(hypot(1.0, hypot(1.0, (k_m / t_m))), 2.0)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (t_m <= 5e-64) {
		tmp = ((Math.cos(k_m) * (2.0 * Math.pow(l, 2.0))) / (t_m * Math.pow(k_m, 2.0))) / Math.pow(Math.sin(k_m), 2.0);
	} else {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k_m))), 3.0) * (Math.tan(k_m) * Math.pow(Math.hypot(1.0, Math.hypot(1.0, (k_m / t_m))), 2.0)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 5e-64)
		tmp = Float64(Float64(Float64(cos(k_m) * Float64(2.0 * (l ^ 2.0))) / Float64(t_m * (k_m ^ 2.0))) / (sin(k_m) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(sin(k_m))) ^ 3.0) * Float64(tan(k_m) * (hypot(1.0, hypot(1.0, Float64(k_m / t_m))) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 5e-64], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[Power[N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.00000000000000033e-64

   1. Initial program 54.1%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in t around 0 63.4%

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

      1. associate-*r/ 63.4%

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

      2. associate-*r* 63.5%

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

   7. Simplified 63.5%

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

      1. *-un-lft-identity 63.5%

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

      2. associate-/r* 65.2%

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

      3. associate-*r* 65.3%

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

      4. *-commutative 65.3%

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

   9. Applied egg-rr 65.3%

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

   if 5.00000000000000033e-64 < t

   1. Initial program 61.6%

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

   3. Simplified 61.6%

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

      1. add-cube-cbrt 61.6%

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

      2. pow3 61.6%

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

      3. associate-/r* 69.0%

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

      4. *-commutative 69.0%

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

      5. cbrt-prod 68.9%

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

      6. associate-/r* 61.6%

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

      7. cbrt-div 62.1%

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

      8. rem-cbrt-cube 67.9%

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

      9. cbrt-prod 90.4%

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

      10. pow2 90.4%

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

   6. Applied egg-rr 90.4%

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

      1. add-sqr-sqrt 42.7%

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

      2. pow2 42.7%

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

      3. associate-+r+ 42.7%

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

      4. metadata-eval 42.7%

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

      5. sqrt-prod 42.7%

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

      6. metadata-eval 42.7%

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

      7. associate-+r+ 42.7%

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

      8. add-sqr-sqrt 42.7%

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

      9. hypot-1-def 42.7%

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

      10. unpow2 42.7%

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

      11. hypot-1-def 42.7%

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

   8. Applied egg-rr 42.7%

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

      1. unpow2 42.7%

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

      2. swap-sqr 42.7%

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

      3. rem-square-sqrt 90.5%

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

      4. unpow2 90.5%

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

   10. Simplified 90.5%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \left(2 \cdot {\ell}^{2}\right)}{t \cdot {k}^{2}}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot {\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{-65}:\\ \;\;\;\;\frac{\frac{\cos k\_m \cdot \left(2 \cdot {\ell}^{2}\right)}{t\_m \cdot {k\_m}^{2}}}{{\sin k\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k\_m}\right)}^{3} \cdot \left(\tan k\_m \cdot \left(1 + \left(1 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2e-65)
    (/
     (/ (* (cos k_m) (* 2.0 (pow l 2.0))) (* t_m (pow k_m 2.0)))
     (pow (sin k_m) 2.0))
    (/
     2.0
     (*
      (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (sin k_m))) 3.0)
      (* (tan k_m) (+ 1.0 (+ 1.0 (pow (/ k_m t_m) 2.0)))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 2e-65) {
		tmp = ((cos(k_m) * (2.0 * pow(l, 2.0))) / (t_m * pow(k_m, 2.0))) / pow(sin(k_m), 2.0);
	} else {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(sin(k_m))), 3.0) * (tan(k_m) * (1.0 + (1.0 + pow((k_m / t_m), 2.0)))));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (t_m <= 2e-65) {
		tmp = ((Math.cos(k_m) * (2.0 * Math.pow(l, 2.0))) / (t_m * Math.pow(k_m, 2.0))) / Math.pow(Math.sin(k_m), 2.0);
	} else {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k_m))), 3.0) * (Math.tan(k_m) * (1.0 + (1.0 + Math.pow((k_m / t_m), 2.0)))));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 2e-65)
		tmp = Float64(Float64(Float64(cos(k_m) * Float64(2.0 * (l ^ 2.0))) / Float64(t_m * (k_m ^ 2.0))) / (sin(k_m) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(sin(k_m))) ^ 3.0) * Float64(tan(k_m) * Float64(1.0 + Float64(1.0 + (Float64(k_m / t_m) ^ 2.0))))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2e-65], N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * N[(2.0 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.99999999999999985e-65

   1. Initial program 54.1%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in t around 0 63.4%

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

      1. associate-*r/ 63.4%

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

      2. associate-*r* 63.5%

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

   7. Simplified 63.5%

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

      1. *-un-lft-identity 63.5%

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

      2. associate-/r* 65.2%

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

      3. associate-*r* 65.3%

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

      4. *-commutative 65.3%

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

   9. Applied egg-rr 65.3%

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

   if 1.99999999999999985e-65 < t

   1. Initial program 61.6%

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

   3. Simplified 61.6%

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

      1. add-cube-cbrt 61.6%

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

      2. pow3 61.6%

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

      3. associate-/r* 69.0%

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

      4. *-commutative 69.0%

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

      5. cbrt-prod 68.9%

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

      6. associate-/r* 61.6%

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

      7. cbrt-div 62.1%

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

      8. rem-cbrt-cube 67.9%

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

      9. cbrt-prod 90.4%

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

      10. pow2 90.4%

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

   6. Applied egg-rr 90.4%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-65}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \left(2 \cdot {\ell}^{2}\right)}{t \cdot {k}^{2}}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 7.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left({\left(\sqrt[3]{k\_m}\right)}^{2} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k\_m \cdot \left(\tan k\_m \cdot \left(2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 7.2e-69)
    (/
     2.0
     (pow
      (* (* t_m (pow (cbrt l) -2.0)) (* (pow (cbrt k_m) 2.0) (cbrt 2.0)))
      3.0))
    (/
     2.0
     (pow
      (*
       (/ t_m (pow (cbrt l) 2.0))
       (cbrt (* (sin k_m) (* (tan k_m) (+ 2.0 (pow (/ k_m t_m) 2.0))))))
      3.0)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 7.2e-69) {
		tmp = 2.0 / pow(((t_m * pow(cbrt(l), -2.0)) * (pow(cbrt(k_m), 2.0) * cbrt(2.0))), 3.0);
	} else {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * cbrt((sin(k_m) * (tan(k_m) * (2.0 + pow((k_m / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 7.2e-69) {
		tmp = 2.0 / Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * (Math.pow(Math.cbrt(k_m), 2.0) * Math.cbrt(2.0))), 3.0);
	} else {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt((Math.sin(k_m) * (Math.tan(k_m) * (2.0 + Math.pow((k_m / t_m), 2.0)))))), 3.0);
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 7.2e-69)
		tmp = Float64(2.0 / (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * Float64((cbrt(k_m) ^ 2.0) * cbrt(2.0))) ^ 3.0));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(Float64(sin(k_m) * Float64(tan(k_m) * Float64(2.0 + (Float64(k_m / t_m) ^ 2.0)))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 7.2e-69], N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Power[k$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 7.20000000000000035e-69

   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 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. Taylor expanded in k around 0 58.4%

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

      1. unpow2 58.4%

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

   7. Applied egg-rr 58.4%

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

      1. add-cube-cbrt 58.4%

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

      2. pow3 58.4%

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

   9. Applied egg-rr 67.2%

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

       1. *-commutative 67.2%

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

       2. cbrt-prod 67.2%

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

       3. pow2 67.2%

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

       4. cbrt-prod 77.7%

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

       5. pow2 77.7%

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

   11. Applied egg-rr 77.7%

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

   if 7.20000000000000035e-69 < k

   1. Initial program 55.3%

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

   3. Simplified 55.3%

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

      1. associate-*l* 55.3%

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

      2. associate-/r* 60.3%

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

      3. associate-+r+ 60.3%

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

      4. metadata-eval 60.3%

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

      5. associate-*l* 60.3%

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

      6. add-cube-cbrt 60.2%

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

      7. pow3 60.2%

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

   6. Applied egg-rr 78.6%

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

Alternative 5: 77.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{2}{{\left(t\_2 \cdot \left({\left(\sqrt[3]{k\_m}\right)}^{2} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)\right) \cdot {t\_2}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (* t_m (pow (cbrt l) -2.0))))
   (*
    t_s
    (if (<= k_m 2.4e-68)
      (/ 2.0 (pow (* t_2 (* (pow (cbrt k_m) 2.0) (cbrt 2.0))) 3.0))
      (/
       2.0
       (*
        (* (* (sin k_m) (tan k_m)) (+ 2.0 (pow (/ k_m t_m) 2.0)))
        (pow t_2 3.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = t_m * pow(cbrt(l), -2.0);
	double tmp;
	if (k_m <= 2.4e-68) {
		tmp = 2.0 / pow((t_2 * (pow(cbrt(k_m), 2.0) * cbrt(2.0))), 3.0);
	} else {
		tmp = 2.0 / (((sin(k_m) * tan(k_m)) * (2.0 + pow((k_m / t_m), 2.0))) * pow(t_2, 3.0));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = t_m * Math.pow(Math.cbrt(l), -2.0);
	double tmp;
	if (k_m <= 2.4e-68) {
		tmp = 2.0 / Math.pow((t_2 * (Math.pow(Math.cbrt(k_m), 2.0) * Math.cbrt(2.0))), 3.0);
	} else {
		tmp = 2.0 / (((Math.sin(k_m) * Math.tan(k_m)) * (2.0 + Math.pow((k_m / t_m), 2.0))) * Math.pow(t_2, 3.0));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(t_m * (cbrt(l) ^ -2.0))
	tmp = 0.0
	if (k_m <= 2.4e-68)
		tmp = Float64(2.0 / (Float64(t_2 * Float64((cbrt(k_m) ^ 2.0) * cbrt(2.0))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k_m) * tan(k_m)) * Float64(2.0 + (Float64(k_m / t_m) ^ 2.0))) * (t_2 ^ 3.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 2.4e-68], N[(2.0 / N[Power[N[(t$95$2 * N[(N[Power[N[Power[k$95$m, 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[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.39999999999999991e-68

   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 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. Taylor expanded in k around 0 58.4%

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

      1. unpow2 58.4%

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

   7. Applied egg-rr 58.4%

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

      1. add-cube-cbrt 58.4%

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

      2. pow3 58.4%

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

   9. Applied egg-rr 67.2%

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

       1. *-commutative 67.2%

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

       2. cbrt-prod 67.2%

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

       3. pow2 67.2%

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

       4. cbrt-prod 77.7%

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

       5. pow2 77.7%

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

   11. Applied egg-rr 77.7%

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

   if 2.39999999999999991e-68 < k

   1. Initial program 55.3%

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

   3. Simplified 60.3%

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

      1. associate-/r* 55.3%

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

      2. unpow3 55.3%

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

      3. times-frac 66.3%

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

      4. pow2 66.3%

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

   6. Applied egg-rr 66.3%

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

      1. add-cube-cbrt 66.1%

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

      2. pow3 66.1%

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

   8. Applied egg-rr 78.5%

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

      1. *-commutative 78.5%

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

      2. cube-prod 75.6%

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

      3. rem-cube-cbrt 75.7%

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

      4. associate-*r* 75.7%

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

   10. Simplified 75.7%

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

Alternative 6: 80.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 5.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left({\left(\sqrt[3]{k\_m}\right)}^{2} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t\_m \cdot {\sin k\_m}^{2}\right)}{\cos k\_m \cdot {\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 5.9e-6)
    (/
     2.0
     (pow
      (* (* t_m (pow (cbrt l) -2.0)) (* (pow (cbrt k_m) 2.0) (cbrt 2.0)))
      3.0))
    (/
     2.0
     (/
      (* (pow k_m 2.0) (* t_m (pow (sin k_m) 2.0)))
      (* (cos k_m) (pow l 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 5.9e-6) {
		tmp = 2.0 / pow(((t_m * pow(cbrt(l), -2.0)) * (pow(cbrt(k_m), 2.0) * cbrt(2.0))), 3.0);
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t_m * pow(sin(k_m), 2.0))) / (cos(k_m) * pow(l, 2.0)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 5.9e-6) {
		tmp = 2.0 / Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * (Math.pow(Math.cbrt(k_m), 2.0) * Math.cbrt(2.0))), 3.0);
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t_m * Math.pow(Math.sin(k_m), 2.0))) / (Math.cos(k_m) * Math.pow(l, 2.0)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 5.9e-6)
		tmp = Float64(2.0 / (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * Float64((cbrt(k_m) ^ 2.0) * cbrt(2.0))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t_m * (sin(k_m) ^ 2.0))) / Float64(cos(k_m) * (l ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 5.9e-6], N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Power[k$95$m, 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[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.90000000000000026e-6

   1. Initial program 58.0%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in k around 0 60.2%

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

      1. unpow2 60.2%

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

   7. Applied egg-rr 60.2%

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

      1. add-cube-cbrt 60.1%

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

      2. pow3 60.1%

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

   9. Applied egg-rr 68.9%

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

       1. *-commutative 68.9%

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

       2. cbrt-prod 68.9%

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

       3. pow2 68.9%

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

       4. cbrt-prod 78.8%

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

       5. pow2 78.8%

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

   11. Applied egg-rr 78.8%

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

   if 5.90000000000000026e-6 < k

   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 51.7%

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

   5. Taylor expanded in t around 0 72.4%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.4 \cdot 10^{-164}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k\_m \cdot \left(\sin k\_m \cdot {t\_m}^{3}\right)}\right) \cdot \frac{\ell}{2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{2 \cdot {k\_m}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k\_m \cdot {\ell}^{2}\right)}{{\sin k\_m}^{2} \cdot \left(t\_m \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.4e-164)
    (*
     (* l (/ 2.0 (* (tan k_m) (* (sin k_m) (pow t_m 3.0)))))
     (/ l (+ 2.0 (pow (/ k_m t_m) 2.0))))
    (if (<= k_m 5.5e-6)
      (/
       2.0
       (pow (* (* t_m (pow (cbrt l) -2.0)) (cbrt (* 2.0 (pow k_m 2.0)))) 3.0))
      (/
       (* 2.0 (* (cos k_m) (pow l 2.0)))
       (* (pow (sin k_m) 2.0) (* t_m (* k_m k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.4e-164) {
		tmp = (l * (2.0 / (tan(k_m) * (sin(k_m) * pow(t_m, 3.0))))) * (l / (2.0 + pow((k_m / t_m), 2.0)));
	} else if (k_m <= 5.5e-6) {
		tmp = 2.0 / pow(((t_m * pow(cbrt(l), -2.0)) * cbrt((2.0 * pow(k_m, 2.0)))), 3.0);
	} else {
		tmp = (2.0 * (cos(k_m) * pow(l, 2.0))) / (pow(sin(k_m), 2.0) * (t_m * (k_m * k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 2.4e-164) {
		tmp = (l * (2.0 / (Math.tan(k_m) * (Math.sin(k_m) * Math.pow(t_m, 3.0))))) * (l / (2.0 + Math.pow((k_m / t_m), 2.0)));
	} else if (k_m <= 5.5e-6) {
		tmp = 2.0 / Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * Math.cbrt((2.0 * Math.pow(k_m, 2.0)))), 3.0);
	} else {
		tmp = (2.0 * (Math.cos(k_m) * Math.pow(l, 2.0))) / (Math.pow(Math.sin(k_m), 2.0) * (t_m * (k_m * k_m)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.4e-164)
		tmp = Float64(Float64(l * Float64(2.0 / Float64(tan(k_m) * Float64(sin(k_m) * (t_m ^ 3.0))))) * Float64(l / Float64(2.0 + (Float64(k_m / t_m) ^ 2.0))));
	elseif (k_m <= 5.5e-6)
		tmp = Float64(2.0 / (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * cbrt(Float64(2.0 * (k_m ^ 2.0)))) ^ 3.0));
	else
		tmp = Float64(Float64(2.0 * Float64(cos(k_m) * (l ^ 2.0))) / Float64((sin(k_m) ^ 2.0) * Float64(t_m * Float64(k_m * k_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.4e-164], N[(N[(l * N[(2.0 / N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 5.5e-6], N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.39999999999999983e-164

   1. Initial program 56.0%

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

   3. Simplified 55.5%

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

      1. associate-*r* 61.0%

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

      2. *-un-lft-identity 61.0%

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

      3. times-frac 61.0%

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

      4. associate-/l/ 61.0%

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

   6. Applied egg-rr 61.0%

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

   if 2.39999999999999983e-164 < k < 5.4999999999999999e-6

   1. Initial program 67.2%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in k around 0 85.5%

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

      1. unpow2 85.5%

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

   7. Applied egg-rr 85.5%

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

      1. add-cube-cbrt 85.4%

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

      2. pow3 85.4%

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

   9. Applied egg-rr 96.6%

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

   if 5.4999999999999999e-6 < k

   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 51.7%

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

   5. Taylor expanded in t around 0 71.8%

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

      1. associate-*r/ 71.8%

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

      2. associate-*r* 71.7%

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

   7. Simplified 71.7%

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

      1. unpow2 54.0%

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

   9. Applied egg-rr 71.7%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-164}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 6.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(\sqrt[3]{k\_m} \cdot \sqrt[3]{k\_m \cdot 2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k\_m}^{2} \cdot \left(t\_m \cdot {\sin k\_m}^{2}\right)}{\cos k\_m \cdot {\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 6.6e-6)
    (/
     2.0
     (pow
      (* (* t_m (pow (cbrt l) -2.0)) (* (cbrt k_m) (cbrt (* k_m 2.0))))
      3.0))
    (/
     2.0
     (/
      (* (pow k_m 2.0) (* t_m (pow (sin k_m) 2.0)))
      (* (cos k_m) (pow l 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 6.6e-6) {
		tmp = 2.0 / pow(((t_m * pow(cbrt(l), -2.0)) * (cbrt(k_m) * cbrt((k_m * 2.0)))), 3.0);
	} else {
		tmp = 2.0 / ((pow(k_m, 2.0) * (t_m * pow(sin(k_m), 2.0))) / (cos(k_m) * pow(l, 2.0)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 6.6e-6) {
		tmp = 2.0 / Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * (Math.cbrt(k_m) * Math.cbrt((k_m * 2.0)))), 3.0);
	} else {
		tmp = 2.0 / ((Math.pow(k_m, 2.0) * (t_m * Math.pow(Math.sin(k_m), 2.0))) / (Math.cos(k_m) * Math.pow(l, 2.0)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 6.6e-6)
		tmp = Float64(2.0 / (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * Float64(cbrt(k_m) * cbrt(Float64(k_m * 2.0)))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64((k_m ^ 2.0) * Float64(t_m * (sin(k_m) ^ 2.0))) / Float64(cos(k_m) * (l ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 6.6e-6], N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, 1/3], $MachinePrecision] * N[Power[N[(k$95$m * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.60000000000000034e-6

   1. Initial program 58.0%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in k around 0 60.2%

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

      1. unpow2 60.2%

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

   7. Applied egg-rr 60.2%

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

      1. add-cube-cbrt 60.1%

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

      2. pow3 60.1%

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

   9. Applied egg-rr 68.9%

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

       1. pow2 68.9%

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

       2. associate-*r* 68.9%

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

       3. cbrt-prod 78.7%

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

   11. Applied egg-rr 78.7%

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

   if 6.60000000000000034e-6 < k

   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 51.7%

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

   5. Taylor expanded in t around 0 72.4%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{k \cdot 2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(\sqrt[3]{k\_m} \cdot \sqrt[3]{k\_m \cdot 2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k\_m \cdot {\ell}^{2}\right)}{{\sin k\_m}^{2} \cdot \left(t\_m \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 4e-6)
    (/
     2.0
     (pow
      (* (* t_m (pow (cbrt l) -2.0)) (* (cbrt k_m) (cbrt (* k_m 2.0))))
      3.0))
    (/
     (* 2.0 (* (cos k_m) (pow l 2.0)))
     (* (pow (sin k_m) 2.0) (* t_m (* k_m k_m)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 4e-6) {
		tmp = 2.0 / pow(((t_m * pow(cbrt(l), -2.0)) * (cbrt(k_m) * cbrt((k_m * 2.0)))), 3.0);
	} else {
		tmp = (2.0 * (cos(k_m) * pow(l, 2.0))) / (pow(sin(k_m), 2.0) * (t_m * (k_m * k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 4e-6) {
		tmp = 2.0 / Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * (Math.cbrt(k_m) * Math.cbrt((k_m * 2.0)))), 3.0);
	} else {
		tmp = (2.0 * (Math.cos(k_m) * Math.pow(l, 2.0))) / (Math.pow(Math.sin(k_m), 2.0) * (t_m * (k_m * k_m)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 4e-6)
		tmp = Float64(2.0 / (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * Float64(cbrt(k_m) * cbrt(Float64(k_m * 2.0)))) ^ 3.0));
	else
		tmp = Float64(Float64(2.0 * Float64(cos(k_m) * (l ^ 2.0))) / Float64((sin(k_m) ^ 2.0) * Float64(t_m * Float64(k_m * k_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 4e-6], N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k$95$m, 1/3], $MachinePrecision] * N[Power[N[(k$95$m * 2.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.99999999999999982e-6

   1. Initial program 58.0%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in k around 0 60.2%

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

      1. unpow2 60.2%

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

   7. Applied egg-rr 60.2%

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

      1. add-cube-cbrt 60.1%

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

      2. pow3 60.1%

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

   9. Applied egg-rr 68.9%

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

       1. pow2 68.9%

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

       2. associate-*r* 68.9%

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

       3. cbrt-prod 78.7%

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

   11. Applied egg-rr 78.7%

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

   if 3.99999999999999982e-6 < k

   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 51.7%

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

   5. Taylor expanded in t around 0 71.8%

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

      1. associate-*r/ 71.8%

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

      2. associate-*r* 71.7%

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

   7. Simplified 71.7%

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

      1. unpow2 54.0%

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

   9. Applied egg-rr 71.7%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{k \cdot 2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k\_m}\right)}^{3} \cdot \left(k\_m \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k\_m \cdot {\ell}^{2}\right)}{{\sin k\_m}^{2} \cdot \left(t\_m \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.9e-5)
    (/
     2.0
     (* (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (sin k_m))) 3.0) (* k_m 2.0)))
    (/
     (* 2.0 (* (cos k_m) (pow l 2.0)))
     (* (pow (sin k_m) 2.0) (* t_m (* k_m k_m)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.9e-5) {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(sin(k_m))), 3.0) * (k_m * 2.0));
	} else {
		tmp = (2.0 * (cos(k_m) * pow(l, 2.0))) / (pow(sin(k_m), 2.0) * (t_m * (k_m * k_m)));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 2.9e-5) {
		tmp = 2.0 / (Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * Math.cbrt(Math.sin(k_m))), 3.0) * (k_m * 2.0));
	} else {
		tmp = (2.0 * (Math.cos(k_m) * Math.pow(l, 2.0))) / (Math.pow(Math.sin(k_m), 2.0) * (t_m * (k_m * k_m)));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.9e-5)
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(sin(k_m))) ^ 3.0) * Float64(k_m * 2.0)));
	else
		tmp = Float64(Float64(2.0 * Float64(cos(k_m) * (l ^ 2.0))) / Float64((sin(k_m) ^ 2.0) * Float64(t_m * Float64(k_m * k_m))));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.9e-5], N[(2.0 / N[(N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[k$95$m], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(k$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.9e-5

   1. Initial program 58.0%

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

   3. Simplified 58.0%

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

      1. add-cube-cbrt 58.0%

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

      2. pow3 58.0%

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

      3. associate-/r* 64.3%

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

      4. *-commutative 64.3%

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

      5. cbrt-prod 64.2%

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

      6. associate-/r* 57.9%

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

      7. cbrt-div 59.2%

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

      8. rem-cbrt-cube 66.0%

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

      9. cbrt-prod 78.9%

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

      10. pow2 78.9%

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

   6. Applied egg-rr 78.9%

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

   7. Taylor expanded in k around 0 75.6%

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

   if 2.9e-5 < k

   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 51.7%

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

   5. Taylor expanded in t around 0 71.8%

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

      1. associate-*r/ 71.8%

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

      2. associate-*r* 71.7%

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

   7. Simplified 71.7%

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

      1. unpow2 54.0%

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

   9. Applied egg-rr 71.7%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(k \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.4 \cdot 10^{-65}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k\_m \cdot {\ell}^{2}\right)}{{\sin k\_m}^{2} \cdot \left(t\_m \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{elif}\;t\_m \leq 1.9 \cdot 10^{+96}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{\frac{\ell}{{t\_m}^{3}}}{\sin k\_m \cdot \tan k\_m}\right)}{2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(t\_m \cdot {\ell}^{-0.5}\right)}^{3}}{\sqrt{\ell}} \cdot \left(2 \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.4e-65)
    (/
     (* 2.0 (* (cos k_m) (pow l 2.0)))
     (* (pow (sin k_m) 2.0) (* t_m (* k_m k_m))))
    (if (<= t_m 1.9e+96)
      (/
       (* l (* 2.0 (/ (/ l (pow t_m 3.0)) (* (sin k_m) (tan k_m)))))
       (+ 2.0 (pow (/ k_m t_m) 2.0)))
      (/
       2.0
       (* (/ (pow (* t_m (pow l -0.5)) 3.0) (sqrt l)) (* 2.0 (* k_m k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 2.4e-65) {
		tmp = (2.0 * (cos(k_m) * pow(l, 2.0))) / (pow(sin(k_m), 2.0) * (t_m * (k_m * k_m)));
	} else if (t_m <= 1.9e+96) {
		tmp = (l * (2.0 * ((l / pow(t_m, 3.0)) / (sin(k_m) * tan(k_m))))) / (2.0 + pow((k_m / t_m), 2.0));
	} else {
		tmp = 2.0 / ((pow((t_m * pow(l, -0.5)), 3.0) / sqrt(l)) * (2.0 * (k_m * k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 2.4d-65) then
        tmp = (2.0d0 * (cos(k_m) * (l ** 2.0d0))) / ((sin(k_m) ** 2.0d0) * (t_m * (k_m * k_m)))
    else if (t_m <= 1.9d+96) then
        tmp = (l * (2.0d0 * ((l / (t_m ** 3.0d0)) / (sin(k_m) * tan(k_m))))) / (2.0d0 + ((k_m / t_m) ** 2.0d0))
    else
        tmp = 2.0d0 / ((((t_m * (l ** (-0.5d0))) ** 3.0d0) / sqrt(l)) * (2.0d0 * (k_m * k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (t_m <= 2.4e-65) {
		tmp = (2.0 * (Math.cos(k_m) * Math.pow(l, 2.0))) / (Math.pow(Math.sin(k_m), 2.0) * (t_m * (k_m * k_m)));
	} else if (t_m <= 1.9e+96) {
		tmp = (l * (2.0 * ((l / Math.pow(t_m, 3.0)) / (Math.sin(k_m) * Math.tan(k_m))))) / (2.0 + Math.pow((k_m / t_m), 2.0));
	} else {
		tmp = 2.0 / ((Math.pow((t_m * Math.pow(l, -0.5)), 3.0) / Math.sqrt(l)) * (2.0 * (k_m * k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 2.4e-65:
		tmp = (2.0 * (math.cos(k_m) * math.pow(l, 2.0))) / (math.pow(math.sin(k_m), 2.0) * (t_m * (k_m * k_m)))
	elif t_m <= 1.9e+96:
		tmp = (l * (2.0 * ((l / math.pow(t_m, 3.0)) / (math.sin(k_m) * math.tan(k_m))))) / (2.0 + math.pow((k_m / t_m), 2.0))
	else:
		tmp = 2.0 / ((math.pow((t_m * math.pow(l, -0.5)), 3.0) / math.sqrt(l)) * (2.0 * (k_m * k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 2.4e-65)
		tmp = Float64(Float64(2.0 * Float64(cos(k_m) * (l ^ 2.0))) / Float64((sin(k_m) ^ 2.0) * Float64(t_m * Float64(k_m * k_m))));
	elseif (t_m <= 1.9e+96)
		tmp = Float64(Float64(l * Float64(2.0 * Float64(Float64(l / (t_m ^ 3.0)) / Float64(sin(k_m) * tan(k_m))))) / Float64(2.0 + (Float64(k_m / t_m) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(t_m * (l ^ -0.5)) ^ 3.0) / sqrt(l)) * Float64(2.0 * Float64(k_m * k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 2.4e-65)
		tmp = (2.0 * (cos(k_m) * (l ^ 2.0))) / ((sin(k_m) ^ 2.0) * (t_m * (k_m * k_m)));
	elseif (t_m <= 1.9e+96)
		tmp = (l * (2.0 * ((l / (t_m ^ 3.0)) / (sin(k_m) * tan(k_m))))) / (2.0 + ((k_m / t_m) ^ 2.0));
	else
		tmp = 2.0 / ((((t_m * (l ^ -0.5)) ^ 3.0) / sqrt(l)) * (2.0 * (k_m * k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.4e-65], N[(N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.9e+96], N[(N[(l * N[(2.0 * N[(N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(t$95$m * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.4000000000000002e-65

   1. Initial program 54.1%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in t around 0 63.4%

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

      1. associate-*r/ 63.4%

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

      2. associate-*r* 63.5%

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

   7. Simplified 63.5%

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

      1. unpow2 57.8%

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

   9. Applied egg-rr 63.5%

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

   if 2.4000000000000002e-65 < t < 1.9000000000000001e96

   1. Initial program 69.7%

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

   3. Simplified 69.1%

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

      1. associate-*r* 81.9%

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

      2. *-un-lft-identity 81.9%

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

      3. times-frac 81.8%

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

      4. associate-/l/ 81.8%

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

   6. Applied egg-rr 81.8%

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

      1. /-rgt-identity 81.8%

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

      2. associate-*r/ 81.9%

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

      3. associate-*l/ 81.9%

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

      4. associate-*l* 79.3%

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

   8. Simplified 79.3%

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

      1. associate-/l* 79.3%

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

   10. Applied egg-rr 79.3%

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

       1. associate-/r* 84.3%

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

   12. Simplified 84.3%

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

   if 1.9000000000000001e96 < t

   1. Initial program 53.5%

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

   3. Simplified 52.2%

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

   5. Taylor expanded in k around 0 52.2%

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

      1. unpow2 52.2%

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

   7. Applied egg-rr 52.2%

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

      1. div-inv 52.2%

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

      2. add-sqr-sqrt 24.7%

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

      3. times-frac 21.8%

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

   9. Applied egg-rr 21.8%

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

       1. add-cube-cbrt 21.8%

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

       2. pow3 21.8%

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

   11. Applied egg-rr 27.3%

       \[\leadsto \frac{2}{\color{blue}{{\left(\frac{t}{\sqrt[3]{\sqrt{\ell}}} \cdot {\ell}^{-0.5}\right)}^{3}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*l/ 27.3%

          \[\leadsto \frac{2}{{\color{blue}{\left(\frac{t \cdot {\ell}^{-0.5}}{\sqrt[3]{\sqrt{\ell}}}\right)}}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       2. cube-div 27.3%

          \[\leadsto \frac{2}{\color{blue}{\frac{{\left(t \cdot {\ell}^{-0.5}\right)}^{3}}{{\left(\sqrt[3]{\sqrt{\ell}}\right)}^{3}}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       3. rem-cube-cbrt 27.3%

          \[\leadsto \frac{2}{\frac{{\left(t \cdot {\ell}^{-0.5}\right)}^{3}}{\color{blue}{\sqrt{\ell}}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   13. Simplified 27.3%

       \[\leadsto \frac{2}{\color{blue}{\frac{{\left(t \cdot {\ell}^{-0.5}\right)}^{3}}{\sqrt{\ell}}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-65}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+96}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{\frac{\ell}{{t}^{3}}}{\sin k \cdot \tan k}\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(t \cdot {\ell}^{-0.5}\right)}^{3}}{\sqrt{\ell}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k\_m \cdot {\ell}^{2}\right)}{{\sin k\_m}^{2} \cdot \left(t\_m \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{elif}\;t\_m \leq 5 \cdot 10^{+95}:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k\_m \cdot \left(\tan k\_m \cdot \left(2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)\right)\right) \cdot \frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(t\_m \cdot {\ell}^{-0.5}\right)}^{3}}{\sqrt{\ell}} \cdot \left(2 \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.8e-64)
    (/
     (* 2.0 (* (cos k_m) (pow l 2.0)))
     (* (pow (sin k_m) 2.0) (* t_m (* k_m k_m))))
    (if (<= t_m 5e+95)
      (/
       2.0
       (/
        (*
         (* (sin k_m) (* (tan k_m) (+ 2.0 (pow (/ k_m t_m) 2.0))))
         (/ (pow t_m 3.0) l))
        l))
      (/
       2.0
       (* (/ (pow (* t_m (pow l -0.5)) 3.0) (sqrt l)) (* 2.0 (* k_m k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 4.8e-64) {
		tmp = (2.0 * (cos(k_m) * pow(l, 2.0))) / (pow(sin(k_m), 2.0) * (t_m * (k_m * k_m)));
	} else if (t_m <= 5e+95) {
		tmp = 2.0 / (((sin(k_m) * (tan(k_m) * (2.0 + pow((k_m / t_m), 2.0)))) * (pow(t_m, 3.0) / l)) / l);
	} else {
		tmp = 2.0 / ((pow((t_m * pow(l, -0.5)), 3.0) / sqrt(l)) * (2.0 * (k_m * k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 4.8d-64) then
        tmp = (2.0d0 * (cos(k_m) * (l ** 2.0d0))) / ((sin(k_m) ** 2.0d0) * (t_m * (k_m * k_m)))
    else if (t_m <= 5d+95) then
        tmp = 2.0d0 / (((sin(k_m) * (tan(k_m) * (2.0d0 + ((k_m / t_m) ** 2.0d0)))) * ((t_m ** 3.0d0) / l)) / l)
    else
        tmp = 2.0d0 / ((((t_m * (l ** (-0.5d0))) ** 3.0d0) / sqrt(l)) * (2.0d0 * (k_m * k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (t_m <= 4.8e-64) {
		tmp = (2.0 * (Math.cos(k_m) * Math.pow(l, 2.0))) / (Math.pow(Math.sin(k_m), 2.0) * (t_m * (k_m * k_m)));
	} else if (t_m <= 5e+95) {
		tmp = 2.0 / (((Math.sin(k_m) * (Math.tan(k_m) * (2.0 + Math.pow((k_m / t_m), 2.0)))) * (Math.pow(t_m, 3.0) / l)) / l);
	} else {
		tmp = 2.0 / ((Math.pow((t_m * Math.pow(l, -0.5)), 3.0) / Math.sqrt(l)) * (2.0 * (k_m * k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 4.8e-64:
		tmp = (2.0 * (math.cos(k_m) * math.pow(l, 2.0))) / (math.pow(math.sin(k_m), 2.0) * (t_m * (k_m * k_m)))
	elif t_m <= 5e+95:
		tmp = 2.0 / (((math.sin(k_m) * (math.tan(k_m) * (2.0 + math.pow((k_m / t_m), 2.0)))) * (math.pow(t_m, 3.0) / l)) / l)
	else:
		tmp = 2.0 / ((math.pow((t_m * math.pow(l, -0.5)), 3.0) / math.sqrt(l)) * (2.0 * (k_m * k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 4.8e-64)
		tmp = Float64(Float64(2.0 * Float64(cos(k_m) * (l ^ 2.0))) / Float64((sin(k_m) ^ 2.0) * Float64(t_m * Float64(k_m * k_m))));
	elseif (t_m <= 5e+95)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k_m) * Float64(tan(k_m) * Float64(2.0 + (Float64(k_m / t_m) ^ 2.0)))) * Float64((t_m ^ 3.0) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(t_m * (l ^ -0.5)) ^ 3.0) / sqrt(l)) * Float64(2.0 * Float64(k_m * k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 4.8e-64)
		tmp = (2.0 * (cos(k_m) * (l ^ 2.0))) / ((sin(k_m) ^ 2.0) * (t_m * (k_m * k_m)));
	elseif (t_m <= 5e+95)
		tmp = 2.0 / (((sin(k_m) * (tan(k_m) * (2.0 + ((k_m / t_m) ^ 2.0)))) * ((t_m ^ 3.0) / l)) / l);
	else
		tmp = 2.0 / ((((t_m * (l ^ -0.5)) ^ 3.0) / sqrt(l)) * (2.0 * (k_m * k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 4.8e-64], N[(N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5e+95], N[(2.0 / N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(t$95$m * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.79999999999999997e-64

   1. Initial program 54.1%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in t around 0 63.4%

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

      1. associate-*r/ 63.4%

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

      2. associate-*r* 63.5%

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

   7. Simplified 63.5%

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

      1. unpow2 57.8%

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

   9. Applied egg-rr 63.5%

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

   if 4.79999999999999997e-64 < t < 5.00000000000000025e95

   1. Initial program 69.7%

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

   3. Simplified 69.7%

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

      1. associate-*l* 69.3%

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

      2. associate-/r* 80.5%

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

      3. associate-+r+ 80.5%

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

      4. metadata-eval 80.5%

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

      5. associate-*l* 80.5%

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

      6. associate-*l/ 83.2%

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

      7. associate-*l* 83.2%

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

   6. Applied egg-rr 83.2%

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

   if 5.00000000000000025e95 < t

   1. Initial program 53.5%

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

   3. Simplified 52.2%

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

   5. Taylor expanded in k around 0 52.2%

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

      1. unpow2 52.2%

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

   7. Applied egg-rr 52.2%

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

      1. div-inv 52.2%

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

      2. add-sqr-sqrt 24.7%

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

      3. times-frac 21.8%

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

   9. Applied egg-rr 21.8%

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

       1. add-cube-cbrt 21.8%

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

       2. pow3 21.8%

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

   11. Applied egg-rr 27.3%

       \[\leadsto \frac{2}{\color{blue}{{\left(\frac{t}{\sqrt[3]{\sqrt{\ell}}} \cdot {\ell}^{-0.5}\right)}^{3}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*l/ 27.3%

          \[\leadsto \frac{2}{{\color{blue}{\left(\frac{t \cdot {\ell}^{-0.5}}{\sqrt[3]{\sqrt{\ell}}}\right)}}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       2. cube-div 27.3%

          \[\leadsto \frac{2}{\color{blue}{\frac{{\left(t \cdot {\ell}^{-0.5}\right)}^{3}}{{\left(\sqrt[3]{\sqrt{\ell}}\right)}^{3}}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       3. rem-cube-cbrt 27.3%

          \[\leadsto \frac{2}{\frac{{\left(t \cdot {\ell}^{-0.5}\right)}^{3}}{\color{blue}{\sqrt{\ell}}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   13. Simplified 27.3%

       \[\leadsto \frac{2}{\color{blue}{\frac{{\left(t \cdot {\ell}^{-0.5}\right)}^{3}}{\sqrt{\ell}}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+95}:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \frac{{t}^{3}}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(t \cdot {\ell}^{-0.5}\right)}^{3}}{\sqrt{\ell}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k\_m \cdot {\ell}^{2}\right)}{{\sin k\_m}^{2} \cdot \left(t\_m \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k\_m \cdot \left(\sin k\_m \cdot {t\_m}^{3}\right)}\right) \cdot \frac{\ell}{2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.8e-63)
    (/
     (* 2.0 (* (cos k_m) (pow l 2.0)))
     (* (pow (sin k_m) 2.0) (* t_m (* k_m k_m))))
    (*
     (* l (/ 2.0 (* (tan k_m) (* (sin k_m) (pow t_m 3.0)))))
     (/ l (+ 2.0 (pow (/ k_m t_m) 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 2.8e-63) {
		tmp = (2.0 * (cos(k_m) * pow(l, 2.0))) / (pow(sin(k_m), 2.0) * (t_m * (k_m * k_m)));
	} else {
		tmp = (l * (2.0 / (tan(k_m) * (sin(k_m) * pow(t_m, 3.0))))) * (l / (2.0 + pow((k_m / t_m), 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 2.8d-63) then
        tmp = (2.0d0 * (cos(k_m) * (l ** 2.0d0))) / ((sin(k_m) ** 2.0d0) * (t_m * (k_m * k_m)))
    else
        tmp = (l * (2.0d0 / (tan(k_m) * (sin(k_m) * (t_m ** 3.0d0))))) * (l / (2.0d0 + ((k_m / t_m) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (t_m <= 2.8e-63) {
		tmp = (2.0 * (Math.cos(k_m) * Math.pow(l, 2.0))) / (Math.pow(Math.sin(k_m), 2.0) * (t_m * (k_m * k_m)));
	} else {
		tmp = (l * (2.0 / (Math.tan(k_m) * (Math.sin(k_m) * Math.pow(t_m, 3.0))))) * (l / (2.0 + Math.pow((k_m / t_m), 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 2.8e-63:
		tmp = (2.0 * (math.cos(k_m) * math.pow(l, 2.0))) / (math.pow(math.sin(k_m), 2.0) * (t_m * (k_m * k_m)))
	else:
		tmp = (l * (2.0 / (math.tan(k_m) * (math.sin(k_m) * math.pow(t_m, 3.0))))) * (l / (2.0 + math.pow((k_m / t_m), 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 2.8e-63)
		tmp = Float64(Float64(2.0 * Float64(cos(k_m) * (l ^ 2.0))) / Float64((sin(k_m) ^ 2.0) * Float64(t_m * Float64(k_m * k_m))));
	else
		tmp = Float64(Float64(l * Float64(2.0 / Float64(tan(k_m) * Float64(sin(k_m) * (t_m ^ 3.0))))) * Float64(l / Float64(2.0 + (Float64(k_m / t_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 2.8e-63)
		tmp = (2.0 * (cos(k_m) * (l ^ 2.0))) / ((sin(k_m) ^ 2.0) * (t_m * (k_m * k_m)));
	else
		tmp = (l * (2.0 / (tan(k_m) * (sin(k_m) * (t_m ^ 3.0))))) * (l / (2.0 + ((k_m / t_m) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.8e-63], N[(N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(2.0 / N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.8000000000000002e-63

   1. Initial program 54.1%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in t around 0 63.4%

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

      1. associate-*r/ 63.4%

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

      2. associate-*r* 63.5%

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

   7. Simplified 63.5%

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

      1. unpow2 57.8%

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

   9. Applied egg-rr 63.5%

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

   if 2.8000000000000002e-63 < t

   1. Initial program 61.6%

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

   3. Simplified 61.3%

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

      1. associate-*r* 70.8%

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

      2. *-un-lft-identity 70.8%

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

      3. times-frac 70.8%

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

      4. associate-/l/ 70.8%

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

   6. Applied egg-rr 70.8%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot {t}^{3}\right)}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-65}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k\_m \cdot {\ell}^{2}\right)}{{\sin k\_m}^{2} \cdot \left(t\_m \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{elif}\;t\_m \leq 4.4 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)\right) \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(t\_m \cdot {\ell}^{-0.5}\right)}^{3}}{\sqrt{\ell}} \cdot \left(2 \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4e-65)
    (/
     (* 2.0 (* (cos k_m) (pow l 2.0)))
     (* (pow (sin k_m) 2.0) (* t_m (* k_m k_m))))
    (if (<= t_m 4.4e+149)
      (/
       2.0
       (*
        (* (* (sin k_m) (tan k_m)) (+ 2.0 (pow (/ k_m t_m) 2.0)))
        (* (/ t_m l) (/ (* t_m t_m) l))))
      (/
       2.0
       (* (/ (pow (* t_m (pow l -0.5)) 3.0) (sqrt l)) (* 2.0 (* k_m k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 4e-65) {
		tmp = (2.0 * (cos(k_m) * pow(l, 2.0))) / (pow(sin(k_m), 2.0) * (t_m * (k_m * k_m)));
	} else if (t_m <= 4.4e+149) {
		tmp = 2.0 / (((sin(k_m) * tan(k_m)) * (2.0 + pow((k_m / t_m), 2.0))) * ((t_m / l) * ((t_m * t_m) / l)));
	} else {
		tmp = 2.0 / ((pow((t_m * pow(l, -0.5)), 3.0) / sqrt(l)) * (2.0 * (k_m * k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 4d-65) then
        tmp = (2.0d0 * (cos(k_m) * (l ** 2.0d0))) / ((sin(k_m) ** 2.0d0) * (t_m * (k_m * k_m)))
    else if (t_m <= 4.4d+149) then
        tmp = 2.0d0 / (((sin(k_m) * tan(k_m)) * (2.0d0 + ((k_m / t_m) ** 2.0d0))) * ((t_m / l) * ((t_m * t_m) / l)))
    else
        tmp = 2.0d0 / ((((t_m * (l ** (-0.5d0))) ** 3.0d0) / sqrt(l)) * (2.0d0 * (k_m * k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (t_m <= 4e-65) {
		tmp = (2.0 * (Math.cos(k_m) * Math.pow(l, 2.0))) / (Math.pow(Math.sin(k_m), 2.0) * (t_m * (k_m * k_m)));
	} else if (t_m <= 4.4e+149) {
		tmp = 2.0 / (((Math.sin(k_m) * Math.tan(k_m)) * (2.0 + Math.pow((k_m / t_m), 2.0))) * ((t_m / l) * ((t_m * t_m) / l)));
	} else {
		tmp = 2.0 / ((Math.pow((t_m * Math.pow(l, -0.5)), 3.0) / Math.sqrt(l)) * (2.0 * (k_m * k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 4e-65:
		tmp = (2.0 * (math.cos(k_m) * math.pow(l, 2.0))) / (math.pow(math.sin(k_m), 2.0) * (t_m * (k_m * k_m)))
	elif t_m <= 4.4e+149:
		tmp = 2.0 / (((math.sin(k_m) * math.tan(k_m)) * (2.0 + math.pow((k_m / t_m), 2.0))) * ((t_m / l) * ((t_m * t_m) / l)))
	else:
		tmp = 2.0 / ((math.pow((t_m * math.pow(l, -0.5)), 3.0) / math.sqrt(l)) * (2.0 * (k_m * k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 4e-65)
		tmp = Float64(Float64(2.0 * Float64(cos(k_m) * (l ^ 2.0))) / Float64((sin(k_m) ^ 2.0) * Float64(t_m * Float64(k_m * k_m))));
	elseif (t_m <= 4.4e+149)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k_m) * tan(k_m)) * Float64(2.0 + (Float64(k_m / t_m) ^ 2.0))) * Float64(Float64(t_m / l) * Float64(Float64(t_m * t_m) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(t_m * (l ^ -0.5)) ^ 3.0) / sqrt(l)) * Float64(2.0 * Float64(k_m * k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 4e-65)
		tmp = (2.0 * (cos(k_m) * (l ^ 2.0))) / ((sin(k_m) ^ 2.0) * (t_m * (k_m * k_m)));
	elseif (t_m <= 4.4e+149)
		tmp = 2.0 / (((sin(k_m) * tan(k_m)) * (2.0 + ((k_m / t_m) ^ 2.0))) * ((t_m / l) * ((t_m * t_m) / l)));
	else
		tmp = 2.0 / ((((t_m * (l ^ -0.5)) ^ 3.0) / sqrt(l)) * (2.0 * (k_m * k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 4e-65], N[(N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.4e+149], N[(2.0 / N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(t$95$m * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.99999999999999969e-65

   1. Initial program 54.1%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in t around 0 63.4%

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

      1. associate-*r/ 63.4%

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

      2. associate-*r* 63.5%

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

   7. Simplified 63.5%

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

      1. unpow2 57.8%

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

   9. Applied egg-rr 63.5%

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

   if 3.99999999999999969e-65 < t < 4.4e149

   1. Initial program 62.5%

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

   3. Simplified 71.7%

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

      1. associate-/r* 61.9%

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

      2. unpow3 61.9%

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

      3. times-frac 80.8%

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

      4. pow2 80.8%

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

   6. Applied egg-rr 80.8%

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

      1. unpow2 80.8%

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

   8. Applied egg-rr 80.8%

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

   if 4.4e149 < t

   1. Initial program 60.3%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in k around 0 58.9%

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

      1. unpow2 58.9%

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

   7. Applied egg-rr 58.9%

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

      1. div-inv 58.9%

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

      2. add-sqr-sqrt 26.2%

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

      3. times-frac 22.8%

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

   9. Applied egg-rr 22.8%

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

       1. add-cube-cbrt 22.8%

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

       2. pow3 22.8%

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

   11. Applied egg-rr 26.2%

       \[\leadsto \frac{2}{\color{blue}{{\left(\frac{t}{\sqrt[3]{\sqrt{\ell}}} \cdot {\ell}^{-0.5}\right)}^{3}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*l/ 26.2%

          \[\leadsto \frac{2}{{\color{blue}{\left(\frac{t \cdot {\ell}^{-0.5}}{\sqrt[3]{\sqrt{\ell}}}\right)}}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       2. cube-div 26.2%

          \[\leadsto \frac{2}{\color{blue}{\frac{{\left(t \cdot {\ell}^{-0.5}\right)}^{3}}{{\left(\sqrt[3]{\sqrt{\ell}}\right)}^{3}}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       3. rem-cube-cbrt 26.2%

          \[\leadsto \frac{2}{\frac{{\left(t \cdot {\ell}^{-0.5}\right)}^{3}}{\color{blue}{\sqrt{\ell}}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   13. Simplified 26.2%

       \[\leadsto \frac{2}{\color{blue}{\frac{{\left(t \cdot {\ell}^{-0.5}\right)}^{3}}{\sqrt{\ell}}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-65}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{\sin k}^{2} \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(t \cdot {\ell}^{-0.5}\right)}^{3}}{\sqrt{\ell}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.5 \cdot 10^{-121}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k\_m \cdot {\ell}^{2}\right)}{t\_m \cdot {k\_m}^{4}}\\ \mathbf{elif}\;t\_m \leq 10^{+124}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)\right) \cdot \left(\frac{t\_m}{\ell} \cdot \frac{t\_m \cdot t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(t\_m \cdot {\ell}^{-0.5}\right)}^{3}}{\sqrt{\ell}} \cdot \left(2 \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.5e-121)
    (/ (* 2.0 (* (cos k_m) (pow l 2.0))) (* t_m (pow k_m 4.0)))
    (if (<= t_m 1e+124)
      (/
       2.0
       (*
        (* (* (sin k_m) (tan k_m)) (+ 2.0 (pow (/ k_m t_m) 2.0)))
        (* (/ t_m l) (/ (* t_m t_m) l))))
      (/
       2.0
       (* (/ (pow (* t_m (pow l -0.5)) 3.0) (sqrt l)) (* 2.0 (* k_m k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 5.5e-121) {
		tmp = (2.0 * (cos(k_m) * pow(l, 2.0))) / (t_m * pow(k_m, 4.0));
	} else if (t_m <= 1e+124) {
		tmp = 2.0 / (((sin(k_m) * tan(k_m)) * (2.0 + pow((k_m / t_m), 2.0))) * ((t_m / l) * ((t_m * t_m) / l)));
	} else {
		tmp = 2.0 / ((pow((t_m * pow(l, -0.5)), 3.0) / sqrt(l)) * (2.0 * (k_m * k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 5.5d-121) then
        tmp = (2.0d0 * (cos(k_m) * (l ** 2.0d0))) / (t_m * (k_m ** 4.0d0))
    else if (t_m <= 1d+124) then
        tmp = 2.0d0 / (((sin(k_m) * tan(k_m)) * (2.0d0 + ((k_m / t_m) ** 2.0d0))) * ((t_m / l) * ((t_m * t_m) / l)))
    else
        tmp = 2.0d0 / ((((t_m * (l ** (-0.5d0))) ** 3.0d0) / sqrt(l)) * (2.0d0 * (k_m * k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (t_m <= 5.5e-121) {
		tmp = (2.0 * (Math.cos(k_m) * Math.pow(l, 2.0))) / (t_m * Math.pow(k_m, 4.0));
	} else if (t_m <= 1e+124) {
		tmp = 2.0 / (((Math.sin(k_m) * Math.tan(k_m)) * (2.0 + Math.pow((k_m / t_m), 2.0))) * ((t_m / l) * ((t_m * t_m) / l)));
	} else {
		tmp = 2.0 / ((Math.pow((t_m * Math.pow(l, -0.5)), 3.0) / Math.sqrt(l)) * (2.0 * (k_m * k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 5.5e-121:
		tmp = (2.0 * (math.cos(k_m) * math.pow(l, 2.0))) / (t_m * math.pow(k_m, 4.0))
	elif t_m <= 1e+124:
		tmp = 2.0 / (((math.sin(k_m) * math.tan(k_m)) * (2.0 + math.pow((k_m / t_m), 2.0))) * ((t_m / l) * ((t_m * t_m) / l)))
	else:
		tmp = 2.0 / ((math.pow((t_m * math.pow(l, -0.5)), 3.0) / math.sqrt(l)) * (2.0 * (k_m * k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 5.5e-121)
		tmp = Float64(Float64(2.0 * Float64(cos(k_m) * (l ^ 2.0))) / Float64(t_m * (k_m ^ 4.0)));
	elseif (t_m <= 1e+124)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k_m) * tan(k_m)) * Float64(2.0 + (Float64(k_m / t_m) ^ 2.0))) * Float64(Float64(t_m / l) * Float64(Float64(t_m * t_m) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64((Float64(t_m * (l ^ -0.5)) ^ 3.0) / sqrt(l)) * Float64(2.0 * Float64(k_m * k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 5.5e-121)
		tmp = (2.0 * (cos(k_m) * (l ^ 2.0))) / (t_m * (k_m ^ 4.0));
	elseif (t_m <= 1e+124)
		tmp = 2.0 / (((sin(k_m) * tan(k_m)) * (2.0 + ((k_m / t_m) ^ 2.0))) * ((t_m / l) * ((t_m * t_m) / l)));
	else
		tmp = 2.0 / ((((t_m * (l ^ -0.5)) ^ 3.0) / sqrt(l)) * (2.0 * (k_m * k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 5.5e-121], N[(N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1e+124], N[(2.0 / N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[(t$95$m * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.50000000000000031e-121

   1. Initial program 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in t around 0 62.1%

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

      1. associate-*r/ 62.1%

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

      2. associate-*r* 62.1%

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

   7. Simplified 62.1%

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

   8. Taylor expanded in k around 0 58.3%

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

   if 5.50000000000000031e-121 < t < 9.99999999999999948e123

   1. Initial program 66.0%

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

   3. Simplified 73.0%

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

      1. associate-/r* 65.6%

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

      2. unpow3 65.6%

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

      3. times-frac 80.0%

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

      4. pow2 80.0%

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

   6. Applied egg-rr 80.0%

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

      1. unpow2 80.0%

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

   8. Applied egg-rr 80.0%

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

   if 9.99999999999999948e123 < t

   1. Initial program 58.4%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in k around 0 57.1%

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

      1. unpow2 57.1%

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

   7. Applied egg-rr 57.1%

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

      1. div-inv 57.1%

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

      2. add-sqr-sqrt 25.4%

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

      3. times-frac 22.1%

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

   9. Applied egg-rr 22.1%

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

       1. add-cube-cbrt 22.1%

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

       2. pow3 22.1%

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

   11. Applied egg-rr 25.3%

       \[\leadsto \frac{2}{\color{blue}{{\left(\frac{t}{\sqrt[3]{\sqrt{\ell}}} \cdot {\ell}^{-0.5}\right)}^{3}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*l/ 25.3%

          \[\leadsto \frac{2}{{\color{blue}{\left(\frac{t \cdot {\ell}^{-0.5}}{\sqrt[3]{\sqrt{\ell}}}\right)}}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       2. cube-div 25.4%

          \[\leadsto \frac{2}{\color{blue}{\frac{{\left(t \cdot {\ell}^{-0.5}\right)}^{3}}{{\left(\sqrt[3]{\sqrt{\ell}}\right)}^{3}}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       3. rem-cube-cbrt 25.4%

          \[\leadsto \frac{2}{\frac{{\left(t \cdot {\ell}^{-0.5}\right)}^{3}}{\color{blue}{\sqrt{\ell}}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   13. Simplified 25.4%

       \[\leadsto \frac{2}{\color{blue}{\frac{{\left(t \cdot {\ell}^{-0.5}\right)}^{3}}{\sqrt{\ell}}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-121}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{t \cdot {k}^{4}}\\ \mathbf{elif}\;t \leq 10^{+124}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(t \cdot {\ell}^{-0.5}\right)}^{3}}{\sqrt{\ell}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{\frac{\frac{2}{k\_m}}{k\_m \cdot {t\_m}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 6.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{2}}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(2 \cdot {k\_m}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k\_m \cdot {\ell}^{2}\right)}{t\_m \cdot {k\_m}^{4}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.5e-164)
    (/
     (* (/ (/ 2.0 k_m) (* k_m (pow t_m 3.0))) (* l l))
     (+ 2.0 (pow (/ k_m t_m) 2.0)))
    (if (<= k_m 6.5e+18)
      (/ 2.0 (* (/ (pow t_m 2.0) l) (* (/ t_m l) (* 2.0 (pow k_m 2.0)))))
      (/ (* 2.0 (* (cos k_m) (pow l 2.0))) (* t_m (pow k_m 4.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.5e-164) {
		tmp = (((2.0 / k_m) / (k_m * pow(t_m, 3.0))) * (l * l)) / (2.0 + pow((k_m / t_m), 2.0));
	} else if (k_m <= 6.5e+18) {
		tmp = 2.0 / ((pow(t_m, 2.0) / l) * ((t_m / l) * (2.0 * pow(k_m, 2.0))));
	} else {
		tmp = (2.0 * (cos(k_m) * pow(l, 2.0))) / (t_m * pow(k_m, 4.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.5d-164) then
        tmp = (((2.0d0 / k_m) / (k_m * (t_m ** 3.0d0))) * (l * l)) / (2.0d0 + ((k_m / t_m) ** 2.0d0))
    else if (k_m <= 6.5d+18) then
        tmp = 2.0d0 / (((t_m ** 2.0d0) / l) * ((t_m / l) * (2.0d0 * (k_m ** 2.0d0))))
    else
        tmp = (2.0d0 * (cos(k_m) * (l ** 2.0d0))) / (t_m * (k_m ** 4.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 2.5e-164) {
		tmp = (((2.0 / k_m) / (k_m * Math.pow(t_m, 3.0))) * (l * l)) / (2.0 + Math.pow((k_m / t_m), 2.0));
	} else if (k_m <= 6.5e+18) {
		tmp = 2.0 / ((Math.pow(t_m, 2.0) / l) * ((t_m / l) * (2.0 * Math.pow(k_m, 2.0))));
	} else {
		tmp = (2.0 * (Math.cos(k_m) * Math.pow(l, 2.0))) / (t_m * Math.pow(k_m, 4.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.5e-164:
		tmp = (((2.0 / k_m) / (k_m * math.pow(t_m, 3.0))) * (l * l)) / (2.0 + math.pow((k_m / t_m), 2.0))
	elif k_m <= 6.5e+18:
		tmp = 2.0 / ((math.pow(t_m, 2.0) / l) * ((t_m / l) * (2.0 * math.pow(k_m, 2.0))))
	else:
		tmp = (2.0 * (math.cos(k_m) * math.pow(l, 2.0))) / (t_m * math.pow(k_m, 4.0))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.5e-164)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) / Float64(k_m * (t_m ^ 3.0))) * Float64(l * l)) / Float64(2.0 + (Float64(k_m / t_m) ^ 2.0)));
	elseif (k_m <= 6.5e+18)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 2.0) / l) * Float64(Float64(t_m / l) * Float64(2.0 * (k_m ^ 2.0)))));
	else
		tmp = Float64(Float64(2.0 * Float64(cos(k_m) * (l ^ 2.0))) / Float64(t_m * (k_m ^ 4.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.5e-164)
		tmp = (((2.0 / k_m) / (k_m * (t_m ^ 3.0))) * (l * l)) / (2.0 + ((k_m / t_m) ^ 2.0));
	elseif (k_m <= 6.5e+18)
		tmp = 2.0 / (((t_m ^ 2.0) / l) * ((t_m / l) * (2.0 * (k_m ^ 2.0))));
	else
		tmp = (2.0 * (cos(k_m) * (l ^ 2.0))) / (t_m * (k_m ^ 4.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.5e-164], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] / N[(k$95$m * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 6.5e+18], N[(2.0 / N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.49999999999999981e-164

   1. Initial program 56.0%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in k around 0 55.6%

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

   6. Taylor expanded in k around 0 55.0%

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

   if 2.49999999999999981e-164 < k < 6.5e18

   1. Initial program 58.5%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in k around 0 77.0%

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

      1. unpow2 77.0%

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

   7. Applied egg-rr 77.0%

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

      1. add-cube-cbrt 76.9%

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

      2. pow3 76.9%

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

   9. Applied egg-rr 86.5%

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

   11. Applied egg-rr 82.0%

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

   if 6.5e18 < k

   1. Initial program 55.5%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in t around 0 73.6%

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

      1. associate-*r/ 73.6%

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

      2. associate-*r* 73.6%

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

   7. Simplified 73.6%

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

   8. Taylor expanded in k around 0 70.0%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{k \cdot {t}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{\frac{{t}^{2}}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(2 \cdot {k}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{\frac{{\left(t\_m \cdot {\ell}^{-0.5}\right)}^{3}}{\sqrt{\ell}} \cdot \left(2 \cdot \left(k\_m \cdot k\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k\_m \cdot {\ell}^{2}\right)}{t\_m \cdot {k\_m}^{4}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 3.2e+18)
    (/ 2.0 (* (/ (pow (* t_m (pow l -0.5)) 3.0) (sqrt l)) (* 2.0 (* k_m k_m))))
    (/ (* 2.0 (* (cos k_m) (pow l 2.0))) (* t_m (pow k_m 4.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 3.2e+18) {
		tmp = 2.0 / ((pow((t_m * pow(l, -0.5)), 3.0) / sqrt(l)) * (2.0 * (k_m * k_m)));
	} else {
		tmp = (2.0 * (cos(k_m) * pow(l, 2.0))) / (t_m * pow(k_m, 4.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3.2d+18) then
        tmp = 2.0d0 / ((((t_m * (l ** (-0.5d0))) ** 3.0d0) / sqrt(l)) * (2.0d0 * (k_m * k_m)))
    else
        tmp = (2.0d0 * (cos(k_m) * (l ** 2.0d0))) / (t_m * (k_m ** 4.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 3.2e+18) {
		tmp = 2.0 / ((Math.pow((t_m * Math.pow(l, -0.5)), 3.0) / Math.sqrt(l)) * (2.0 * (k_m * k_m)));
	} else {
		tmp = (2.0 * (Math.cos(k_m) * Math.pow(l, 2.0))) / (t_m * Math.pow(k_m, 4.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 3.2e+18:
		tmp = 2.0 / ((math.pow((t_m * math.pow(l, -0.5)), 3.0) / math.sqrt(l)) * (2.0 * (k_m * k_m)))
	else:
		tmp = (2.0 * (math.cos(k_m) * math.pow(l, 2.0))) / (t_m * math.pow(k_m, 4.0))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 3.2e+18)
		tmp = Float64(2.0 / Float64(Float64((Float64(t_m * (l ^ -0.5)) ^ 3.0) / sqrt(l)) * Float64(2.0 * Float64(k_m * k_m))));
	else
		tmp = Float64(Float64(2.0 * Float64(cos(k_m) * (l ^ 2.0))) / Float64(t_m * (k_m ^ 4.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 3.2e+18)
		tmp = 2.0 / ((((t_m * (l ^ -0.5)) ^ 3.0) / sqrt(l)) * (2.0 * (k_m * k_m)));
	else
		tmp = (2.0 * (cos(k_m) * (l ^ 2.0))) / (t_m * (k_m ^ 4.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 3.2e+18], N[(2.0 / N[(N[(N[Power[N[(t$95$m * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.2e18

   1. Initial program 56.5%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in k around 0 59.2%

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

      1. unpow2 59.2%

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

   7. Applied egg-rr 59.2%

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

      1. div-inv 59.2%

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

      2. add-sqr-sqrt 28.1%

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

      3. times-frac 27.9%

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

   9. Applied egg-rr 27.9%

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

       1. add-cube-cbrt 27.9%

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

       2. pow3 27.9%

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

   11. Applied egg-rr 28.6%

       \[\leadsto \frac{2}{\color{blue}{{\left(\frac{t}{\sqrt[3]{\sqrt{\ell}}} \cdot {\ell}^{-0.5}\right)}^{3}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*l/ 28.6%

          \[\leadsto \frac{2}{{\color{blue}{\left(\frac{t \cdot {\ell}^{-0.5}}{\sqrt[3]{\sqrt{\ell}}}\right)}}^{3} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       2. cube-div 28.6%

          \[\leadsto \frac{2}{\color{blue}{\frac{{\left(t \cdot {\ell}^{-0.5}\right)}^{3}}{{\left(\sqrt[3]{\sqrt{\ell}}\right)}^{3}}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

       3. rem-cube-cbrt 28.6%

          \[\leadsto \frac{2}{\frac{{\left(t \cdot {\ell}^{-0.5}\right)}^{3}}{\color{blue}{\sqrt{\ell}}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   13. Simplified 28.6%

       \[\leadsto \frac{2}{\color{blue}{\frac{{\left(t \cdot {\ell}^{-0.5}\right)}^{3}}{\sqrt{\ell}}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)} \]

   if 3.2e18 < k

   1. Initial program 55.5%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in t around 0 73.6%

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

      1. associate-*r/ 73.6%

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

      2. associate-*r* 73.6%

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

   7. Simplified 73.6%

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

   8. Taylor expanded in k around 0 70.0%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{\frac{{\left(t \cdot {\ell}^{-0.5}\right)}^{3}}{\sqrt{\ell}} \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 62.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k\_m \cdot k\_m\right)\right) \cdot {\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k\_m \cdot {\ell}^{2}\right)}{t\_m \cdot {k\_m}^{4}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 3.1e+18)
    (/ 2.0 (* (* 2.0 (* k_m k_m)) (pow (/ t_m (pow (cbrt l) 2.0)) 3.0)))
    (/ (* 2.0 (* (cos k_m) (pow l 2.0))) (* t_m (pow k_m 4.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 3.1e+18) {
		tmp = 2.0 / ((2.0 * (k_m * k_m)) * pow((t_m / pow(cbrt(l), 2.0)), 3.0));
	} else {
		tmp = (2.0 * (cos(k_m) * pow(l, 2.0))) / (t_m * pow(k_m, 4.0));
	}
	return t_s * tmp;
}

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 3.1e+18) {
		tmp = 2.0 / ((2.0 * (k_m * k_m)) * Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0));
	} else {
		tmp = (2.0 * (Math.cos(k_m) * Math.pow(l, 2.0))) / (t_m * Math.pow(k_m, 4.0));
	}
	return t_s * tmp;
}

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 3.1e+18)
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k_m * k_m)) * (Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0)));
	else
		tmp = Float64(Float64(2.0 * Float64(cos(k_m) * (l ^ 2.0))) / Float64(t_m * (k_m ^ 4.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 3.1e+18], N[(2.0 / N[(N[(2.0 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[Cos[k$95$m], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.1e18

   1. Initial program 56.5%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in k around 0 59.2%

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

      1. unpow2 59.2%

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

   7. Applied egg-rr 59.2%

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

      1. add-cube-cbrt 59.2%

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

      2. pow3 59.2%

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

      3. associate-/r* 54.0%

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

      4. cbrt-div 54.0%

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

      5. rem-cbrt-cube 58.6%

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

      6. cbrt-prod 63.8%

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

      7. pow2 63.8%

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

   9. Applied egg-rr 63.8%

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

   if 3.1e18 < k

   1. Initial program 55.5%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in t around 0 73.6%

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

      1. associate-*r/ 73.6%

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

      2. associate-*r* 73.6%

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

   7. Simplified 73.6%

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

   8. Taylor expanded in k around 0 70.0%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 62.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 2.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{\frac{\frac{2}{k\_m}}{k\_m \cdot {t\_m}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 1.35 \cdot 10^{+88}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{2}}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(2 \cdot {k\_m}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{4}}}{t\_m}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 2.5e-164)
    (/
     (* (/ (/ 2.0 k_m) (* k_m (pow t_m 3.0))) (* l l))
     (+ 2.0 (pow (/ k_m t_m) 2.0)))
    (if (<= k_m 1.35e+88)
      (/ 2.0 (* (/ (pow t_m 2.0) l) (* (/ t_m l) (* 2.0 (pow k_m 2.0)))))
      (* 2.0 (/ (/ (pow l 2.0) (pow k_m 4.0)) t_m))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 2.5e-164) {
		tmp = (((2.0 / k_m) / (k_m * pow(t_m, 3.0))) * (l * l)) / (2.0 + pow((k_m / t_m), 2.0));
	} else if (k_m <= 1.35e+88) {
		tmp = 2.0 / ((pow(t_m, 2.0) / l) * ((t_m / l) * (2.0 * pow(k_m, 2.0))));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 4.0)) / t_m);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 2.5d-164) then
        tmp = (((2.0d0 / k_m) / (k_m * (t_m ** 3.0d0))) * (l * l)) / (2.0d0 + ((k_m / t_m) ** 2.0d0))
    else if (k_m <= 1.35d+88) then
        tmp = 2.0d0 / (((t_m ** 2.0d0) / l) * ((t_m / l) * (2.0d0 * (k_m ** 2.0d0))))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k_m ** 4.0d0)) / t_m)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 2.5e-164) {
		tmp = (((2.0 / k_m) / (k_m * Math.pow(t_m, 3.0))) * (l * l)) / (2.0 + Math.pow((k_m / t_m), 2.0));
	} else if (k_m <= 1.35e+88) {
		tmp = 2.0 / ((Math.pow(t_m, 2.0) / l) * ((t_m / l) * (2.0 * Math.pow(k_m, 2.0))));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 4.0)) / t_m);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 2.5e-164:
		tmp = (((2.0 / k_m) / (k_m * math.pow(t_m, 3.0))) * (l * l)) / (2.0 + math.pow((k_m / t_m), 2.0))
	elif k_m <= 1.35e+88:
		tmp = 2.0 / ((math.pow(t_m, 2.0) / l) * ((t_m / l) * (2.0 * math.pow(k_m, 2.0))))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 4.0)) / t_m)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 2.5e-164)
		tmp = Float64(Float64(Float64(Float64(2.0 / k_m) / Float64(k_m * (t_m ^ 3.0))) * Float64(l * l)) / Float64(2.0 + (Float64(k_m / t_m) ^ 2.0)));
	elseif (k_m <= 1.35e+88)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 2.0) / l) * Float64(Float64(t_m / l) * Float64(2.0 * (k_m ^ 2.0)))));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 4.0)) / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 2.5e-164)
		tmp = (((2.0 / k_m) / (k_m * (t_m ^ 3.0))) * (l * l)) / (2.0 + ((k_m / t_m) ^ 2.0));
	elseif (k_m <= 1.35e+88)
		tmp = 2.0 / (((t_m ^ 2.0) / l) * ((t_m / l) * (2.0 * (k_m ^ 2.0))));
	else
		tmp = 2.0 * (((l ^ 2.0) / (k_m ^ 4.0)) / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 2.5e-164], N[(N[(N[(N[(2.0 / k$95$m), $MachinePrecision] / N[(k$95$m * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.35e+88], N[(2.0 / N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.49999999999999981e-164

   1. Initial program 56.0%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in k around 0 55.6%

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

   6. Taylor expanded in k around 0 55.0%

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

   if 2.49999999999999981e-164 < k < 1.35000000000000008e88

   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 63.7%

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

   5. Taylor expanded in k around 0 71.4%

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

      1. unpow2 71.4%

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

   7. Applied egg-rr 71.4%

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

      1. add-cube-cbrt 71.4%

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

      2. pow3 71.3%

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

   9. Applied egg-rr 78.2%

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

   11. Applied egg-rr 74.9%

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

   if 1.35000000000000008e88 < k

   1. Initial program 57.2%

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

   3. Simplified 57.2%

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

   5. Taylor expanded in t around 0 75.8%

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

      1. associate-*r/ 75.8%

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

      2. associate-*r* 75.8%

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

   7. Simplified 75.8%

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

      1. expm1-log1p-u 75.4%

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

      2. expm1-undefine 75.4%

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

   9. Applied egg-rr 75.4%

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

       1. expm1-define 75.4%

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

   11. Simplified 75.4%

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

   12. Taylor expanded in k around 0 71.9%

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

       1. associate-/r* 71.9%

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

   14. Simplified 71.9%

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

Alternative 20: 60.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-76}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{4}}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{2}}{\ell} \cdot \left(\frac{t\_m}{\ell} \cdot \left(2 \cdot {k\_m}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.6e-76)
    (* 2.0 (/ (/ (pow l 2.0) (pow k_m 4.0)) t_m))
    (/ 2.0 (* (/ (pow t_m 2.0) l) (* (/ t_m l) (* 2.0 (pow k_m 2.0))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 2.6e-76) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 4.0)) / t_m);
	} else {
		tmp = 2.0 / ((pow(t_m, 2.0) / l) * ((t_m / l) * (2.0 * pow(k_m, 2.0))));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 2.6d-76) then
        tmp = 2.0d0 * (((l ** 2.0d0) / (k_m ** 4.0d0)) / t_m)
    else
        tmp = 2.0d0 / (((t_m ** 2.0d0) / l) * ((t_m / l) * (2.0d0 * (k_m ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (t_m <= 2.6e-76) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 4.0)) / t_m);
	} else {
		tmp = 2.0 / ((Math.pow(t_m, 2.0) / l) * ((t_m / l) * (2.0 * Math.pow(k_m, 2.0))));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 2.6e-76:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 4.0)) / t_m)
	else:
		tmp = 2.0 / ((math.pow(t_m, 2.0) / l) * ((t_m / l) * (2.0 * math.pow(k_m, 2.0))))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 2.6e-76)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 4.0)) / t_m));
	else
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 2.0) / l) * Float64(Float64(t_m / l) * Float64(2.0 * (k_m ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 2.6e-76)
		tmp = 2.0 * (((l ^ 2.0) / (k_m ^ 4.0)) / t_m);
	else
		tmp = 2.0 / (((t_m ^ 2.0) / l) * ((t_m / l) * (2.0 * (k_m ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.6e-76], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.6e-76

   1. Initial program 53.9%

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

   3. Simplified 53.9%

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

   5. Taylor expanded in t around 0 62.8%

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

      1. associate-*r/ 62.8%

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

      2. associate-*r* 62.8%

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

   7. Simplified 62.8%

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

      1. expm1-log1p-u 62.5%

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

      2. expm1-undefine 61.5%

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

   9. Applied egg-rr 61.5%

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

       1. expm1-define 62.5%

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

   11. Simplified 62.5%

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

   12. Taylor expanded in k around 0 57.5%

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

       1. associate-/r* 58.7%

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

   14. Simplified 58.7%

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

   if 2.6e-76 < t

   1. Initial program 61.8%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in k around 0 60.3%

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

      1. unpow2 60.3%

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

   7. Applied egg-rr 60.3%

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

      1. add-cube-cbrt 60.2%

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

      2. pow3 60.2%

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

   9. Applied egg-rr 69.4%

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

   11. Applied egg-rr 65.6%

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

Alternative 21: 61.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{-77}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{4}}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k\_m \cdot k\_m\right)\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.8e-77)
    (* 2.0 (/ (/ (pow l 2.0) (pow k_m 4.0)) t_m))
    (/ 2.0 (* (* 2.0 (* k_m k_m)) (pow (/ (pow t_m 1.5) l) 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 5.8e-77) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 4.0)) / t_m);
	} else {
		tmp = 2.0 / ((2.0 * (k_m * k_m)) * pow((pow(t_m, 1.5) / l), 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 5.8d-77) then
        tmp = 2.0d0 * (((l ** 2.0d0) / (k_m ** 4.0d0)) / t_m)
    else
        tmp = 2.0d0 / ((2.0d0 * (k_m * k_m)) * (((t_m ** 1.5d0) / l) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (t_m <= 5.8e-77) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 4.0)) / t_m);
	} else {
		tmp = 2.0 / ((2.0 * (k_m * k_m)) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 5.8e-77:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 4.0)) / t_m)
	else:
		tmp = 2.0 / ((2.0 * (k_m * k_m)) * math.pow((math.pow(t_m, 1.5) / l), 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 5.8e-77)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 4.0)) / t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k_m * k_m)) * (Float64((t_m ^ 1.5) / l) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 5.8e-77)
		tmp = 2.0 * (((l ^ 2.0) / (k_m ^ 4.0)) / t_m);
	else
		tmp = 2.0 / ((2.0 * (k_m * k_m)) * (((t_m ^ 1.5) / l) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 5.8e-77], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.7999999999999997e-77

   1. Initial program 53.9%

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

   3. Simplified 53.9%

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

   5. Taylor expanded in t around 0 62.8%

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

      1. associate-*r/ 62.8%

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

      2. associate-*r* 62.8%

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

   7. Simplified 62.8%

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

      1. expm1-log1p-u 62.5%

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

      2. expm1-undefine 61.5%

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

   9. Applied egg-rr 61.5%

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

       1. expm1-define 62.5%

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

   11. Simplified 62.5%

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

   12. Taylor expanded in k around 0 57.5%

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

       1. associate-/r* 58.7%

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

   14. Simplified 58.7%

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

   if 5.7999999999999997e-77 < t

   1. Initial program 61.8%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in k around 0 60.3%

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

      1. unpow2 60.3%

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

   7. Applied egg-rr 60.3%

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

      1. add-sqr-sqrt 60.3%

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

      2. pow2 60.3%

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

      3. associate-/r* 56.8%

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

      4. sqrt-div 56.8%

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

      5. sqrt-pow1 58.4%

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

      6. metadata-eval 58.4%

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

      7. sqrt-prod 32.9%

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

      8. add-sqr-sqrt 65.4%

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

   9. Applied egg-rr 65.4%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-77}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 59.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-76}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k\_m}^{4}}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k\_m \cdot k\_m\right)\right) \cdot \frac{\frac{t\_m}{\ell} \cdot {t\_m}^{2}}{\ell}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.6e-76)
    (* 2.0 (/ (/ (pow l 2.0) (pow k_m 4.0)) t_m))
    (/ 2.0 (* (* 2.0 (* k_m k_m)) (/ (* (/ t_m l) (pow t_m 2.0)) l))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 2.6e-76) {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k_m, 4.0)) / t_m);
	} else {
		tmp = 2.0 / ((2.0 * (k_m * k_m)) * (((t_m / l) * pow(t_m, 2.0)) / l));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 2.6d-76) then
        tmp = 2.0d0 * (((l ** 2.0d0) / (k_m ** 4.0d0)) / t_m)
    else
        tmp = 2.0d0 / ((2.0d0 * (k_m * k_m)) * (((t_m / l) * (t_m ** 2.0d0)) / l))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (t_m <= 2.6e-76) {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k_m, 4.0)) / t_m);
	} else {
		tmp = 2.0 / ((2.0 * (k_m * k_m)) * (((t_m / l) * Math.pow(t_m, 2.0)) / l));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 2.6e-76:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k_m, 4.0)) / t_m)
	else:
		tmp = 2.0 / ((2.0 * (k_m * k_m)) * (((t_m / l) * math.pow(t_m, 2.0)) / l))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 2.6e-76)
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k_m ^ 4.0)) / t_m));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(k_m * k_m)) * Float64(Float64(Float64(t_m / l) * (t_m ^ 2.0)) / l)));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 2.6e-76)
		tmp = 2.0 * (((l ^ 2.0) / (k_m ^ 4.0)) / t_m);
	else
		tmp = 2.0 / ((2.0 * (k_m * k_m)) * (((t_m / l) * (t_m ^ 2.0)) / l));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.6e-76], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.6e-76

   1. Initial program 53.9%

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

   3. Simplified 53.9%

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

   5. Taylor expanded in t around 0 62.8%

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

      1. associate-*r/ 62.8%

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

      2. associate-*r* 62.8%

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

   7. Simplified 62.8%

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

      1. expm1-log1p-u 62.5%

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

      2. expm1-undefine 61.5%

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

   9. Applied egg-rr 61.5%

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

       1. expm1-define 62.5%

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

   11. Simplified 62.5%

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

   12. Taylor expanded in k around 0 57.5%

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

       1. associate-/r* 58.7%

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

   14. Simplified 58.7%

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

   if 2.6e-76 < t

   1. Initial program 61.8%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in k around 0 60.3%

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

      1. unpow2 60.3%

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

   7. Applied egg-rr 60.3%

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

      1. associate-/r* 59.3%

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

      2. unpow3 59.3%

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

      3. times-frac 71.4%

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

      4. pow2 71.4%

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

   9. Applied egg-rr 64.2%

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

       1. associate-*l/ 64.2%

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

   11. Applied egg-rr 64.2%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-76}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{t}{\ell} \cdot {t}^{2}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 56.9% accurate, 3.6× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ 2.0 (* (* 2.0 (* k_m k_m)) (/ (* (/ t_m l) (pow t_m 2.0)) l)))))

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * (2.0 / ((2.0 * (k_m * k_m)) * (((t_m / l) * Math.pow(t_m, 2.0)) / l)));
}

Python

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

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 / Float64(Float64(2.0 * Float64(k_m * k_m)) * Float64(Float64(Float64(t_m / l) * (t_m ^ 2.0)) / l))))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 / ((2.0 * (k_m * k_m)) * (((t_m / l) * (t_m ^ 2.0)) / l)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(2.0 / N[(N[(2.0 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.2%

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

3. Simplified 58.9%

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

5. Taylor expanded in k around 0 58.4%

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

   1. unpow2 58.4%

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

7. Applied egg-rr 58.4%

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

   1. associate-/r* 53.4%

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

   2. unpow3 53.4%

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

   3. times-frac 62.3%

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

   4. pow2 62.3%

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

9. Applied egg-rr 60.4%

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

    1. associate-*l/ 60.4%

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

11. Applied egg-rr 60.4%

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

12. Final simplification 60.4%

    \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{\frac{t}{\ell} \cdot {t}^{2}}{\ell}} \]
13. Add Preprocessing

Alternative 24: 56.9% accurate, 3.6× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ 2.0 (* (* 2.0 (* k_m k_m)) (/ (* t_m (/ (pow t_m 2.0) l)) l)))))

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * (2.0 / ((2.0 * (k_m * k_m)) * ((t_m * (Math.pow(t_m, 2.0) / l)) / l)));
}

Python

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

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 / Float64(Float64(2.0 * Float64(k_m * k_m)) * Float64(Float64(t_m * Float64((t_m ^ 2.0) / l)) / l))))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 / ((2.0 * (k_m * k_m)) * ((t_m * ((t_m ^ 2.0) / l)) / l)));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(2.0 / N[(N[(2.0 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.2%

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

3. Simplified 58.9%

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

5. Taylor expanded in k around 0 58.4%

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

   1. unpow2 58.4%

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

7. Applied egg-rr 58.4%

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

   1. associate-/r* 53.4%

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

   2. unpow3 53.4%

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

   3. times-frac 62.3%

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

   4. pow2 62.3%

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

9. Applied egg-rr 60.4%

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

    1. associate-*r/ 60.4%

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

11. Applied egg-rr 60.4%

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

12. Final simplification 60.4%

    \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \frac{t \cdot \frac{{t}^{2}}{\ell}}{\ell}} \]
13. Add Preprocessing

Alternative 25: 57.2% accurate, 24.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ 2.0 (* (* 2.0 (* k_m k_m)) (* (/ t_m l) (/ (* t_m t_m) l))))))

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * (2.0 / ((2.0 * (k_m * k_m)) * ((t_m / l) * ((t_m * t_m) / l))));
}

Python

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

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(2.0 / Float64(Float64(2.0 * Float64(k_m * k_m)) * Float64(Float64(t_m / l) * Float64(Float64(t_m * t_m) / l)))))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (2.0 / ((2.0 * (k_m * k_m)) * ((t_m / l) * ((t_m * t_m) / l))));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(2.0 / N[(N[(2.0 * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.2%

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

3. Simplified 58.9%

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

5. Taylor expanded in k around 0 58.4%

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

   1. unpow2 58.4%

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

7. Applied egg-rr 58.4%

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

   1. associate-/r* 53.4%

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

   2. unpow3 53.4%

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

   3. times-frac 62.3%

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

   4. pow2 62.3%

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

9. Applied egg-rr 60.4%

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

    1. unpow2 62.3%

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

11. Applied egg-rr 60.4%

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

12. Final simplification 60.4%

    \[\leadsto \frac{2}{\left(2 \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024157 
(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))))