Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.2% → 90.5%

Time: 17.7s

Alternatives: 20

Speedup: 1.9×

• 28.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.45 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left({\left(\sqrt[3]{k\_m}\right)}^{2} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{elif}\;k\_m \leq 2.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t\_m}\right)\right)\right)}^{2} \cdot \left(\sin k\_m \cdot \tan k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k\_m}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k\_m \cdot \sin k\_m}\right)}^{2}\\ \end{array} \end{array} \]

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 (<= k_m 1.45e-47)
    (/
     2.0
     (pow
      (* (/ t_m (pow (cbrt l) 2.0)) (* (pow (cbrt k_m) 2.0) (cbrt 2.0)))
      3.0))
    (if (<= k_m 2.5e+87)
      (/
       2.0
       (*
        (pow (* (/ (pow t_m 1.5) l) (hypot 1.0 (hypot 1.0 (/ k_m t_m)))) 2.0)
        (* (sin k_m) (tan k_m))))
      (*
       (/ (cos k_m) t_m)
       (pow (* l (/ (sqrt 2.0) (* k_m (sin 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 (k_m <= 1.45e-47) {
		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 <= 2.5e+87) {
		tmp = 2.0 / (pow(((pow(t_m, 1.5) / l) * hypot(1.0, hypot(1.0, (k_m / t_m)))), 2.0) * (sin(k_m) * tan(k_m)));
	} else {
		tmp = (cos(k_m) / t_m) * pow((l * (sqrt(2.0) / (k_m * sin(k_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 (k_m <= 1.45e-47) {
		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 if (k_m <= 2.5e+87) {
		tmp = 2.0 / (Math.pow(((Math.pow(t_m, 1.5) / l) * Math.hypot(1.0, Math.hypot(1.0, (k_m / t_m)))), 2.0) * (Math.sin(k_m) * Math.tan(k_m)));
	} else {
		tmp = (Math.cos(k_m) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k_m * Math.sin(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 (k_m <= 1.45e-47)
		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 <= 2.5e+87)
		tmp = Float64(2.0 / Float64((Float64(Float64((t_m ^ 1.5) / l) * hypot(1.0, hypot(1.0, Float64(k_m / t_m)))) ^ 2.0) * Float64(sin(k_m) * tan(k_m))));
	else
		tmp = Float64(Float64(cos(k_m) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k_m * sin(k_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[k$95$m, 1.45e-47], 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, 2.5e+87], N[(2.0 / N[(N[Power[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], 2.0], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.45e-47

   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 55.2%

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

   5. Taylor expanded in k around 0 55.7%

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

      1. add-cube-cbrt 55.6%

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

      2. pow3 55.6%

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

      3. *-commutative 55.6%

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

      4. cbrt-prod 55.6%

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

      5. associate-/l/ 51.7%

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

      6. cbrt-div 52.1%

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

      7. unpow3 52.1%

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

      8. add-cbrt-cube 59.2%

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

      9. cbrt-unprod 66.0%

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

      10. pow2 66.0%

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

   7. Applied egg-rr 66.0%

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

      1. *-commutative 66.0%

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

   9. Simplified 66.0%

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

       1. *-commutative 66.0%

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

       2. cbrt-prod 66.1%

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

       3. unpow2 66.1%

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

       4. cbrt-prod 76.7%

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

       5. pow2 76.7%

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

   11. Applied egg-rr 76.7%

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

   if 1.45e-47 < k < 2.4999999999999999e87

   1. Initial program 34.7%

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

   3. Simplified 34.7%

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

   6. Applied egg-rr 49.7%

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

      1. associate-*r* 49.6%

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

      2. unpow-prod-down 49.6%

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

      3. pow2 49.6%

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

      4. add-sqr-sqrt 66.2%

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

   8. Applied egg-rr 66.2%

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

   if 2.4999999999999999e87 < k

   1. Initial program 38.1%

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

      1. associate-/r* 38.2%

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

      2. +-commutative 38.2%

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

      3. unpow2 38.2%

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

      4. sqr-neg 38.2%

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

      5. distribute-frac-neg2 38.2%

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

      6. distribute-frac-neg2 38.2%

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

      7. unpow2 38.2%

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

      8. +-commutative 38.2%

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

      9. associate-*l* 38.2%

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

      10. associate-*l/ 38.2%

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

      11. associate-/r/ 38.2%

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

      12. +-commutative 38.2%

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

      13. associate-+r+ 38.2%

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

   3. Simplified 38.2%

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

      1. add-sqr-sqrt 33.9%

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

      2. pow2 33.9%

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

   6. Applied egg-rr 36.3%

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

   7. Taylor expanded in t around 0 57.4%

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

      1. associate-/l* 57.3%

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

   9. Simplified 57.3%

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

       1. *-un-lft-identity 57.3%

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

       2. *-commutative 57.3%

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

       3. unpow-prod-down 53.5%

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

       4. pow2 53.5%

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

       5. add-sqr-sqrt 87.7%

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

       6. associate-*r/ 87.5%

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

   11. Applied egg-rr 87.5%

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

       1. *-lft-identity 87.5%

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

       2. associate-/l* 87.7%

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

   13. Simplified 87.7%

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

Alternative 2: 89.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.02 \cdot 10^{-32}:\\ \;\;\;\;\frac{\cos k\_m}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k\_m \cdot \sin k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k\_m}\right)}^{3} \cdot \left(\tan k\_m \cdot \left(1 + \left(1 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)\right)\right)}\\ \end{array} \end{array} \]

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 1.02e-32)
    (* (/ (cos k_m) t_m) (pow (* l (/ (sqrt 2.0) (* k_m (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 <= 1.02e-32) {
		tmp = (cos(k_m) / t_m) * pow((l * (sqrt(2.0) / (k_m * 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 <= 1.02e-32) {
		tmp = (Math.cos(k_m) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k_m * 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 <= 1.02e-32)
		tmp = Float64(Float64(cos(k_m) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k_m * 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, 1.02e-32], N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 1.02000000000000002e-32

   1. Initial program 50.2%

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

      1. associate-/r* 50.2%

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

      2. +-commutative 50.2%

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

      3. unpow2 50.2%

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

      4. sqr-neg 50.2%

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

      5. distribute-frac-neg2 50.2%

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

      6. distribute-frac-neg2 50.2%

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

      7. unpow2 50.2%

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

      8. +-commutative 50.2%

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

      9. associate-*l* 46.3%

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

      10. associate-*l/ 44.3%

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

      11. associate-/r/ 44.1%

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

      12. +-commutative 44.1%

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

      13. associate-+r+ 44.1%

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

   3. Simplified 44.1%

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

      1. add-sqr-sqrt 26.4%

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

      2. pow2 26.4%

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

   6. Applied egg-rr 29.9%

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

   7. Taylor expanded in t around 0 39.8%

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

      1. associate-/l* 39.8%

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

   9. Simplified 39.8%

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

       1. *-un-lft-identity 39.8%

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

       2. *-commutative 39.8%

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

       3. unpow-prod-down 38.7%

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

       4. pow2 38.7%

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

       5. add-sqr-sqrt 76.1%

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

       6. associate-*r/ 76.1%

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

   11. Applied egg-rr 76.1%

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

       1. *-lft-identity 76.1%

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

       2. associate-/l* 76.1%

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

   13. Simplified 76.1%

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

   if 1.02000000000000002e-32 < t

   1. Initial program 55.0%

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

   3. Simplified 55.0%

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

      1. associate-/r* 61.6%

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

      2. add-cube-cbrt 61.4%

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

      3. *-un-lft-identity 61.4%

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

      4. times-frac 61.4%

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

      5. pow2 61.4%

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

      6. cbrt-div 61.5%

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

      7. rem-cbrt-cube 61.5%

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

      8. cbrt-div 61.4%

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

      9. rem-cbrt-cube 80.8%

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

   6. Applied egg-rr 80.8%

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

      1. add-cube-cbrt 80.7%

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

      2. pow3 80.7%

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

   8. Applied egg-rr 90.5%

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

      1. *-commutative 90.5%

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

   10. Simplified 90.5%

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

Alternative 3: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 8 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left({\left(\sqrt[3]{k\_m}\right)}^{2} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{elif}\;k\_m \leq 2.7 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}}}{\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k\_m}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k\_m \cdot \sin k\_m}\right)}^{2}\\ \end{array} \end{array} \]

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 (<= k_m 8e-25)
    (/
     2.0
     (pow
      (* (/ t_m (pow (cbrt l) 2.0)) (* (pow (cbrt k_m) 2.0) (cbrt 2.0)))
      3.0))
    (if (<= k_m 2.7e+87)
      (/
       (/ 2.0 (pow (/ (pow t_m 1.5) l) 2.0))
       (* (* (sin k_m) (tan k_m)) (+ 2.0 (pow (/ k_m t_m) 2.0))))
      (*
       (/ (cos k_m) t_m)
       (pow (* l (/ (sqrt 2.0) (* k_m (sin 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 (k_m <= 8e-25) {
		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 <= 2.7e+87) {
		tmp = (2.0 / pow((pow(t_m, 1.5) / l), 2.0)) / ((sin(k_m) * tan(k_m)) * (2.0 + pow((k_m / t_m), 2.0)));
	} else {
		tmp = (cos(k_m) / t_m) * pow((l * (sqrt(2.0) / (k_m * sin(k_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 (k_m <= 8e-25) {
		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 if (k_m <= 2.7e+87) {
		tmp = (2.0 / Math.pow((Math.pow(t_m, 1.5) / l), 2.0)) / ((Math.sin(k_m) * Math.tan(k_m)) * (2.0 + Math.pow((k_m / t_m), 2.0)));
	} else {
		tmp = (Math.cos(k_m) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k_m * Math.sin(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 (k_m <= 8e-25)
		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 <= 2.7e+87)
		tmp = Float64(Float64(2.0 / (Float64((t_m ^ 1.5) / l) ^ 2.0)) / Float64(Float64(sin(k_m) * tan(k_m)) * Float64(2.0 + (Float64(k_m / t_m) ^ 2.0))));
	else
		tmp = Float64(Float64(cos(k_m) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k_m * sin(k_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[k$95$m, 8e-25], 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, 2.7e+87], N[(N[(2.0 / N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / 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]), $MachinePrecision], N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 8.00000000000000031e-25

   1. Initial program 55.7%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in k around 0 55.4%

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

      1. add-cube-cbrt 55.3%

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

      2. pow3 55.3%

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

      3. *-commutative 55.3%

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

      4. cbrt-prod 55.3%

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

      5. associate-/l/ 51.5%

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

      6. cbrt-div 51.9%

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

      7. unpow3 51.9%

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

      8. add-cbrt-cube 59.3%

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

      9. cbrt-unprod 66.1%

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

      10. pow2 66.1%

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

   7. Applied egg-rr 66.1%

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

      1. *-commutative 66.1%

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

   9. Simplified 66.1%

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

       1. *-commutative 66.1%

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

       2. cbrt-prod 66.1%

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

       3. unpow2 66.1%

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

       4. cbrt-prod 76.6%

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

       5. pow2 76.6%

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

   11. Applied egg-rr 76.6%

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

   if 8.00000000000000031e-25 < k < 2.70000000000000007e87

   1. Initial program 40.3%

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

   3. Simplified 40.4%

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

   6. Applied egg-rr 46.3%

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

      1. *-un-lft-identity 46.3%

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

      2. unpow-prod-down 39.7%

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

      3. unpow-prod-down 39.7%

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

   8. Applied egg-rr 59.8%

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

      1. *-lft-identity 59.8%

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

      2. associate-/r* 59.8%

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

   10. Simplified 59.8%

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

   if 2.70000000000000007e87 < k

   1. Initial program 38.1%

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

      1. associate-/r* 38.2%

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

      2. +-commutative 38.2%

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

      3. unpow2 38.2%

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

      4. sqr-neg 38.2%

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

      5. distribute-frac-neg2 38.2%

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

      6. distribute-frac-neg2 38.2%

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

      7. unpow2 38.2%

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

      8. +-commutative 38.2%

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

      9. associate-*l* 38.2%

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

      10. associate-*l/ 38.2%

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

      11. associate-/r/ 38.2%

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

      12. +-commutative 38.2%

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

      13. associate-+r+ 38.2%

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

   3. Simplified 38.2%

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

      1. add-sqr-sqrt 33.9%

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

      2. pow2 33.9%

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

   6. Applied egg-rr 36.3%

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

   7. Taylor expanded in t around 0 57.4%

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

      1. associate-/l* 57.3%

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

   9. Simplified 57.3%

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

       1. *-un-lft-identity 57.3%

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

       2. *-commutative 57.3%

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

       3. unpow-prod-down 53.5%

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

       4. pow2 53.5%

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

       5. add-sqr-sqrt 87.7%

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

       6. associate-*r/ 87.5%

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

   11. Applied egg-rr 87.5%

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

       1. *-lft-identity 87.5%

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

       2. associate-/l* 87.7%

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

   13. Simplified 87.7%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{-31}:\\ \;\;\;\;\frac{\cos k\_m}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k\_m \cdot \sin k\_m}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 1.5 \cdot 10^{+205}:\\ \;\;\;\;\frac{2}{\left(\tan k\_m \cdot \left(1 + \left(1 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)\right)\right) \cdot \left(\sin k\_m \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k\_m}\right)}^{3} \cdot \left(k\_m \cdot 2\right)}\\ \end{array} \end{array} \]

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 1.35e-31)
    (* (/ (cos k_m) t_m) (pow (* l (/ (sqrt 2.0) (* k_m (sin k_m)))) 2.0))
    (if (<= t_m 1.5e+205)
      (/
       2.0
       (*
        (* (tan k_m) (+ 1.0 (+ 1.0 (pow (/ k_m t_m) 2.0))))
        (* (sin k_m) (pow (/ (pow t_m 1.5) l) 2.0))))
      (/
       2.0
       (*
        (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (sin k_m))) 3.0)
        (* 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 <= 1.35e-31) {
		tmp = (cos(k_m) / t_m) * pow((l * (sqrt(2.0) / (k_m * sin(k_m)))), 2.0);
	} else if (t_m <= 1.5e+205) {
		tmp = 2.0 / ((tan(k_m) * (1.0 + (1.0 + pow((k_m / t_m), 2.0)))) * (sin(k_m) * pow((pow(t_m, 1.5) / l), 2.0)));
	} else {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(sin(k_m))), 3.0) * (k_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 <= 1.35e-31) {
		tmp = (Math.cos(k_m) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k_m * Math.sin(k_m)))), 2.0);
	} else if (t_m <= 1.5e+205) {
		tmp = 2.0 / ((Math.tan(k_m) * (1.0 + (1.0 + Math.pow((k_m / t_m), 2.0)))) * (Math.sin(k_m) * Math.pow((Math.pow(t_m, 1.5) / l), 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) * (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 <= 1.35e-31)
		tmp = Float64(Float64(cos(k_m) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k_m * sin(k_m)))) ^ 2.0));
	elseif (t_m <= 1.5e+205)
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * Float64(1.0 + Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)))) * Float64(sin(k_m) * (Float64((t_m ^ 1.5) / l) ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(sin(k_m))) ^ 3.0) * Float64(k_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, 1.35e-31], N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.5e+205], N[(2.0 / N[(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] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $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[(k$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.35000000000000007e-31

   1. Initial program 50.0%

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

      1. associate-/r* 50.0%

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

      2. +-commutative 50.0%

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

      3. unpow2 50.0%

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

      4. sqr-neg 50.0%

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

      5. distribute-frac-neg2 50.0%

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

      6. distribute-frac-neg2 50.0%

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

      7. unpow2 50.0%

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

      8. +-commutative 50.0%

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

      9. associate-*l* 46.1%

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

      10. associate-*l/ 44.0%

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

      11. associate-/r/ 43.8%

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

      12. +-commutative 43.8%

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

      13. associate-+r+ 43.8%

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

   3. Simplified 43.8%

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

      1. add-sqr-sqrt 26.2%

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

      2. pow2 26.2%

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

   6. Applied egg-rr 29.7%

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

   7. Taylor expanded in t around 0 39.6%

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

      1. associate-/l* 39.6%

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

   9. Simplified 39.6%

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

       1. *-un-lft-identity 39.6%

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

       2. *-commutative 39.6%

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

       3. unpow-prod-down 38.5%

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

       4. pow2 38.5%

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

       5. add-sqr-sqrt 75.7%

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

       6. associate-*r/ 75.7%

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

   11. Applied egg-rr 75.7%

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

       1. *-lft-identity 75.7%

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

       2. associate-/l* 75.7%

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

   13. Simplified 75.7%

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

   if 1.35000000000000007e-31 < t < 1.5e205

   1. Initial program 64.5%

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

   3. Simplified 64.6%

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

      1. add-sqr-sqrt 64.6%

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

      2. pow2 64.6%

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

      3. sqrt-div 64.6%

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

      4. sqrt-pow1 75.6%

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

      5. metadata-eval 75.6%

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

      6. sqrt-prod 45.7%

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

      7. add-sqr-sqrt 90.5%

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

   6. Applied egg-rr 90.5%

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

   if 1.5e205 < t

   1. Initial program 41.0%

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

   3. Simplified 41.0%

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

      1. associate-/r* 50.8%

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

      2. add-cube-cbrt 50.8%

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

      3. *-un-lft-identity 50.8%

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

      4. times-frac 50.8%

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

      5. pow2 50.8%

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

      6. cbrt-div 50.8%

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

      7. rem-cbrt-cube 50.8%

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

      8. cbrt-div 50.8%

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

      9. rem-cbrt-cube 68.5%

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

   6. Applied egg-rr 68.5%

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

      1. add-cube-cbrt 68.4%

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

      2. pow3 68.4%

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

   8. Applied egg-rr 88.3%

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

      1. *-commutative 88.3%

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

   10. Simplified 88.3%

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

   11. Taylor expanded in k around 0 75.1%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{-31}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+205}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(k \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{t\_m}^{1.5}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(t\_2 \cdot \left(k\_m \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t\_m}\right)\right)\right)\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 8.2 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{2}{{t\_2}^{2}}}{\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k\_m}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k\_m \cdot \sin k\_m}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

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 t_m 1.5) l)))
   (*
    t_s
    (if (<= k_m 3e-5)
      (/ 2.0 (pow (* t_2 (* k_m (hypot 1.0 (hypot 1.0 (/ k_m t_m))))) 2.0))
      (if (<= k_m 8.2e+86)
        (/
         (/ 2.0 (pow t_2 2.0))
         (* (* (sin k_m) (tan k_m)) (+ 2.0 (pow (/ k_m t_m) 2.0))))
        (*
         (/ (cos k_m) t_m)
         (pow (* l (/ (sqrt 2.0) (* k_m (sin 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 t_2 = pow(t_m, 1.5) / l;
	double tmp;
	if (k_m <= 3e-5) {
		tmp = 2.0 / pow((t_2 * (k_m * hypot(1.0, hypot(1.0, (k_m / t_m))))), 2.0);
	} else if (k_m <= 8.2e+86) {
		tmp = (2.0 / pow(t_2, 2.0)) / ((sin(k_m) * tan(k_m)) * (2.0 + pow((k_m / t_m), 2.0)));
	} else {
		tmp = (cos(k_m) / t_m) * pow((l * (sqrt(2.0) / (k_m * sin(k_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 t_2 = Math.pow(t_m, 1.5) / l;
	double tmp;
	if (k_m <= 3e-5) {
		tmp = 2.0 / Math.pow((t_2 * (k_m * Math.hypot(1.0, Math.hypot(1.0, (k_m / t_m))))), 2.0);
	} else if (k_m <= 8.2e+86) {
		tmp = (2.0 / Math.pow(t_2, 2.0)) / ((Math.sin(k_m) * Math.tan(k_m)) * (2.0 + Math.pow((k_m / t_m), 2.0)));
	} else {
		tmp = (Math.cos(k_m) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k_m * Math.sin(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):
	t_2 = math.pow(t_m, 1.5) / l
	tmp = 0
	if k_m <= 3e-5:
		tmp = 2.0 / math.pow((t_2 * (k_m * math.hypot(1.0, math.hypot(1.0, (k_m / t_m))))), 2.0)
	elif k_m <= 8.2e+86:
		tmp = (2.0 / math.pow(t_2, 2.0)) / ((math.sin(k_m) * math.tan(k_m)) * (2.0 + math.pow((k_m / t_m), 2.0)))
	else:
		tmp = (math.cos(k_m) / t_m) * math.pow((l * (math.sqrt(2.0) / (k_m * math.sin(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)
	t_2 = Float64((t_m ^ 1.5) / l)
	tmp = 0.0
	if (k_m <= 3e-5)
		tmp = Float64(2.0 / (Float64(t_2 * Float64(k_m * hypot(1.0, hypot(1.0, Float64(k_m / t_m))))) ^ 2.0));
	elseif (k_m <= 8.2e+86)
		tmp = Float64(Float64(2.0 / (t_2 ^ 2.0)) / Float64(Float64(sin(k_m) * tan(k_m)) * Float64(2.0 + (Float64(k_m / t_m) ^ 2.0))));
	else
		tmp = Float64(Float64(cos(k_m) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k_m * sin(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)
	t_2 = (t_m ^ 1.5) / l;
	tmp = 0.0;
	if (k_m <= 3e-5)
		tmp = 2.0 / ((t_2 * (k_m * hypot(1.0, hypot(1.0, (k_m / t_m))))) ^ 2.0);
	elseif (k_m <= 8.2e+86)
		tmp = (2.0 / (t_2 ^ 2.0)) / ((sin(k_m) * tan(k_m)) * (2.0 + ((k_m / t_m) ^ 2.0)));
	else
		tmp = (cos(k_m) / t_m) * ((l * (sqrt(2.0) / (k_m * sin(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_] := Block[{t$95$2 = N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 3e-5], N[(2.0 / N[Power[N[(t$95$2 * N[(k$95$m * N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 8.2e+86], N[(N[(2.0 / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] / 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]), $MachinePrecision], N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 3.00000000000000008e-5

   1. Initial program 55.6%

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

   3. Simplified 55.6%

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

   6. Applied egg-rr 30.0%

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

   7. Taylor expanded in k around 0 35.8%

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

   if 3.00000000000000008e-5 < k < 8.1999999999999998e86

   1. Initial program 36.2%

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

   3. Simplified 36.4%

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

   6. Applied egg-rr 27.1%

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

      1. *-un-lft-identity 27.1%

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

      2. unpow-prod-down 18.2%

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

      3. unpow-prod-down 18.2%

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

   8. Applied egg-rr 45.3%

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

      1. *-lft-identity 45.3%

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

      2. associate-/r* 45.3%

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

   10. Simplified 45.3%

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

   if 8.1999999999999998e86 < k

   1. Initial program 38.1%

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

      1. associate-/r* 38.2%

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

      2. +-commutative 38.2%

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

      3. unpow2 38.2%

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

      4. sqr-neg 38.2%

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

      5. distribute-frac-neg2 38.2%

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

      6. distribute-frac-neg2 38.2%

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

      7. unpow2 38.2%

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

      8. +-commutative 38.2%

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

      9. associate-*l* 38.2%

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

      10. associate-*l/ 38.2%

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

      11. associate-/r/ 38.2%

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

      12. +-commutative 38.2%

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

      13. associate-+r+ 38.2%

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

   3. Simplified 38.2%

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

      1. add-sqr-sqrt 33.9%

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

      2. pow2 33.9%

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

   6. Applied egg-rr 36.3%

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

   7. Taylor expanded in t around 0 57.4%

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

      1. associate-/l* 57.3%

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

   9. Simplified 57.3%

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

       1. *-un-lft-identity 57.3%

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

       2. *-commutative 57.3%

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

       3. unpow-prod-down 53.5%

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

       4. pow2 53.5%

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

       5. add-sqr-sqrt 87.7%

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

       6. associate-*r/ 87.5%

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

   11. Applied egg-rr 87.5%

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

       1. *-lft-identity 87.5%

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

       2. associate-/l* 87.7%

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

   13. Simplified 87.7%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 8.2 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{{t\_m}^{1.5}}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 8200000:\\ \;\;\;\;\frac{2}{{\left(t\_2 \cdot \left(k\_m \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k\_m}{t\_m}\right)\right)\right)\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 3.1 \cdot 10^{+86}:\\ \;\;\;\;\frac{2}{{t\_2}^{2} \cdot \left(\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(2 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k\_m}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k\_m \cdot \sin k\_m}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

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 t_m 1.5) l)))
   (*
    t_s
    (if (<= k_m 8200000.0)
      (/ 2.0 (pow (* t_2 (* k_m (hypot 1.0 (hypot 1.0 (/ k_m t_m))))) 2.0))
      (if (<= k_m 3.1e+86)
        (/
         2.0
         (*
          (pow t_2 2.0)
          (* (* (sin k_m) (tan k_m)) (+ 2.0 (pow (/ k_m t_m) 2.0)))))
        (*
         (/ (cos k_m) t_m)
         (pow (* l (/ (sqrt 2.0) (* k_m (sin 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 t_2 = pow(t_m, 1.5) / l;
	double tmp;
	if (k_m <= 8200000.0) {
		tmp = 2.0 / pow((t_2 * (k_m * hypot(1.0, hypot(1.0, (k_m / t_m))))), 2.0);
	} else if (k_m <= 3.1e+86) {
		tmp = 2.0 / (pow(t_2, 2.0) * ((sin(k_m) * tan(k_m)) * (2.0 + pow((k_m / t_m), 2.0))));
	} else {
		tmp = (cos(k_m) / t_m) * pow((l * (sqrt(2.0) / (k_m * sin(k_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 t_2 = Math.pow(t_m, 1.5) / l;
	double tmp;
	if (k_m <= 8200000.0) {
		tmp = 2.0 / Math.pow((t_2 * (k_m * Math.hypot(1.0, Math.hypot(1.0, (k_m / t_m))))), 2.0);
	} else if (k_m <= 3.1e+86) {
		tmp = 2.0 / (Math.pow(t_2, 2.0) * ((Math.sin(k_m) * Math.tan(k_m)) * (2.0 + Math.pow((k_m / t_m), 2.0))));
	} else {
		tmp = (Math.cos(k_m) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k_m * Math.sin(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):
	t_2 = math.pow(t_m, 1.5) / l
	tmp = 0
	if k_m <= 8200000.0:
		tmp = 2.0 / math.pow((t_2 * (k_m * math.hypot(1.0, math.hypot(1.0, (k_m / t_m))))), 2.0)
	elif k_m <= 3.1e+86:
		tmp = 2.0 / (math.pow(t_2, 2.0) * ((math.sin(k_m) * math.tan(k_m)) * (2.0 + math.pow((k_m / t_m), 2.0))))
	else:
		tmp = (math.cos(k_m) / t_m) * math.pow((l * (math.sqrt(2.0) / (k_m * math.sin(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)
	t_2 = Float64((t_m ^ 1.5) / l)
	tmp = 0.0
	if (k_m <= 8200000.0)
		tmp = Float64(2.0 / (Float64(t_2 * Float64(k_m * hypot(1.0, hypot(1.0, Float64(k_m / t_m))))) ^ 2.0));
	elseif (k_m <= 3.1e+86)
		tmp = Float64(2.0 / Float64((t_2 ^ 2.0) * Float64(Float64(sin(k_m) * tan(k_m)) * Float64(2.0 + (Float64(k_m / t_m) ^ 2.0)))));
	else
		tmp = Float64(Float64(cos(k_m) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k_m * sin(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)
	t_2 = (t_m ^ 1.5) / l;
	tmp = 0.0;
	if (k_m <= 8200000.0)
		tmp = 2.0 / ((t_2 * (k_m * hypot(1.0, hypot(1.0, (k_m / t_m))))) ^ 2.0);
	elseif (k_m <= 3.1e+86)
		tmp = 2.0 / ((t_2 ^ 2.0) * ((sin(k_m) * tan(k_m)) * (2.0 + ((k_m / t_m) ^ 2.0))));
	else
		tmp = (cos(k_m) / t_m) * ((l * (sqrt(2.0) / (k_m * sin(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_] := Block[{t$95$2 = N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 8200000.0], N[(2.0 / N[Power[N[(t$95$2 * N[(k$95$m * N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 3.1e+86], N[(2.0 / N[(N[Power[t$95$2, 2.0], $MachinePrecision] * 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]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 8.2e6

   1. Initial program 55.8%

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

   3. Simplified 55.8%

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

   6. Applied egg-rr 30.3%

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

   7. Taylor expanded in k around 0 36.1%

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

   if 8.2e6 < k < 3.1000000000000002e86

   1. Initial program 29.8%

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

   3. Simplified 30.0%

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

   6. Applied egg-rr 19.8%

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

      1. unpow-prod-down 10.0%

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

      2. unpow-prod-down 10.0%

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

   8. Applied egg-rr 39.8%

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

   if 3.1000000000000002e86 < k

   1. Initial program 38.1%

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

      1. associate-/r* 38.2%

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

      2. +-commutative 38.2%

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

      3. unpow2 38.2%

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

      4. sqr-neg 38.2%

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

      5. distribute-frac-neg2 38.2%

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

      6. distribute-frac-neg2 38.2%

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

      7. unpow2 38.2%

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

      8. +-commutative 38.2%

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

      9. associate-*l* 38.2%

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

      10. associate-*l/ 38.2%

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

      11. associate-/r/ 38.2%

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

      12. +-commutative 38.2%

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

      13. associate-+r+ 38.2%

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

   3. Simplified 38.2%

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

      1. add-sqr-sqrt 33.9%

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

      2. pow2 33.9%

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

   6. Applied egg-rr 36.3%

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

   7. Taylor expanded in t around 0 57.4%

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

      1. associate-/l* 57.3%

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

   9. Simplified 57.3%

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

       1. *-un-lft-identity 57.3%

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

       2. *-commutative 57.3%

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

       3. unpow-prod-down 53.5%

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

       4. pow2 53.5%

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

       5. add-sqr-sqrt 87.7%

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

       6. associate-*r/ 87.5%

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

   11. Applied egg-rr 87.5%

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

       1. *-lft-identity 87.5%

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

       2. associate-/l* 87.7%

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

   13. Simplified 87.7%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8200000:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(k \cdot \mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 3.1 \cdot 10^{+86}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{\cos k\_m}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k\_m \cdot \sin k\_m}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 5.3 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\left(\tan k\_m \cdot \left(1 + \left(1 + {\left(\frac{k\_m}{t\_m}\right)}^{2}\right)\right)\right) \cdot \left(\sin k\_m \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k\_m}\right)}^{3} \cdot \left(k\_m \cdot 2\right)}\\ \end{array} \end{array} \]

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-32)
    (* (/ (cos k_m) t_m) (pow (* l (/ (sqrt 2.0) (* k_m (sin k_m)))) 2.0))
    (if (<= t_m 5.3e+150)
      (/
       2.0
       (*
        (* (tan k_m) (+ 1.0 (+ 1.0 (pow (/ k_m t_m) 2.0))))
        (* (sin k_m) (* (/ (pow t_m 2.0) l) (/ t_m l)))))
      (/
       2.0
       (*
        (pow (* (/ t_m (pow (cbrt l) 2.0)) (cbrt (sin k_m))) 3.0)
        (* 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 <= 4.8e-32) {
		tmp = (cos(k_m) / t_m) * pow((l * (sqrt(2.0) / (k_m * sin(k_m)))), 2.0);
	} else if (t_m <= 5.3e+150) {
		tmp = 2.0 / ((tan(k_m) * (1.0 + (1.0 + pow((k_m / t_m), 2.0)))) * (sin(k_m) * ((pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		tmp = 2.0 / (pow(((t_m / pow(cbrt(l), 2.0)) * cbrt(sin(k_m))), 3.0) * (k_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 <= 4.8e-32) {
		tmp = (Math.cos(k_m) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k_m * Math.sin(k_m)))), 2.0);
	} else if (t_m <= 5.3e+150) {
		tmp = 2.0 / ((Math.tan(k_m) * (1.0 + (1.0 + Math.pow((k_m / t_m), 2.0)))) * (Math.sin(k_m) * ((Math.pow(t_m, 2.0) / l) * (t_m / l))));
	} else {
		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));
	}
	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-32)
		tmp = Float64(Float64(cos(k_m) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k_m * sin(k_m)))) ^ 2.0));
	elseif (t_m <= 5.3e+150)
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * Float64(1.0 + Float64(1.0 + (Float64(k_m / t_m) ^ 2.0)))) * Float64(sin(k_m) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * cbrt(sin(k_m))) ^ 3.0) * Float64(k_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, 4.8e-32], N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.3e+150], N[(2.0 / N[(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] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(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]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.8000000000000003e-32

   1. Initial program 50.0%

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

      1. associate-/r* 50.0%

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

      2. +-commutative 50.0%

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

      3. unpow2 50.0%

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

      4. sqr-neg 50.0%

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

      5. distribute-frac-neg2 50.0%

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

      6. distribute-frac-neg2 50.0%

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

      7. unpow2 50.0%

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

      8. +-commutative 50.0%

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

      9. associate-*l* 46.1%

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

      10. associate-*l/ 44.0%

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

      11. associate-/r/ 43.8%

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

      12. +-commutative 43.8%

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

      13. associate-+r+ 43.8%

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

   3. Simplified 43.8%

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

      1. add-sqr-sqrt 26.2%

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

      2. pow2 26.2%

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

   6. Applied egg-rr 29.7%

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

   7. Taylor expanded in t around 0 39.6%

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

      1. associate-/l* 39.6%

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

   9. Simplified 39.6%

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

       1. *-un-lft-identity 39.6%

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

       2. *-commutative 39.6%

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

       3. unpow-prod-down 38.5%

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

       4. pow2 38.5%

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

       5. add-sqr-sqrt 75.7%

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

       6. associate-*r/ 75.7%

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

   11. Applied egg-rr 75.7%

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

       1. *-lft-identity 75.7%

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

       2. associate-/l* 75.7%

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

   13. Simplified 75.7%

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

   if 4.8000000000000003e-32 < t < 5.30000000000000013e150

   1. Initial program 65.6%

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

   3. Simplified 65.7%

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

      1. unpow3 65.7%

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

      2. times-frac 87.9%

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

      3. pow2 87.9%

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

   6. Applied egg-rr 87.9%

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

   if 5.30000000000000013e150 < t

   1. Initial program 46.4%

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

   3. Simplified 46.4%

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

      1. associate-/r* 53.8%

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

      2. add-cube-cbrt 53.8%

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

      3. *-un-lft-identity 53.8%

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

      4. times-frac 53.8%

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

      5. pow2 53.8%

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

      6. cbrt-div 53.8%

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

      7. rem-cbrt-cube 53.8%

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

      8. cbrt-div 53.8%

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

      9. rem-cbrt-cube 76.8%

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

   6. Applied egg-rr 76.8%

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

      1. add-cube-cbrt 76.7%

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

      2. pow3 76.7%

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

   8. Applied egg-rr 91.3%

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

      1. *-commutative 91.3%

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

   10. Simplified 91.3%

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

   11. Taylor expanded in k around 0 77.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k}\right)}^{3} \cdot \left(k \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.95 \cdot 10^{-143}:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m \cdot \sqrt{2}}{\ell} \cdot \sqrt{{t\_m}^{3}}\right)}^{2}}\\ \mathbf{elif}\;k\_m \leq 1.3 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot {k\_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k\_m}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k\_m \cdot \sin k\_m}\right)}^{2}\\ \end{array} \end{array} \]

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 (<= k_m 1.95e-143)
    (/ 2.0 (pow (* (/ (* k_m (sqrt 2.0)) l) (sqrt (pow t_m 3.0))) 2.0))
    (if (<= k_m 1.3e-16)
      (/ 2.0 (* (pow (/ t_m (pow (cbrt l) 2.0)) 3.0) (* 2.0 (pow k_m 2.0))))
      (*
       (/ (cos k_m) t_m)
       (pow (* l (/ (sqrt 2.0) (* k_m (sin 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 (k_m <= 1.95e-143) {
		tmp = 2.0 / pow((((k_m * sqrt(2.0)) / l) * sqrt(pow(t_m, 3.0))), 2.0);
	} else if (k_m <= 1.3e-16) {
		tmp = 2.0 / (pow((t_m / pow(cbrt(l), 2.0)), 3.0) * (2.0 * pow(k_m, 2.0)));
	} else {
		tmp = (cos(k_m) / t_m) * pow((l * (sqrt(2.0) / (k_m * sin(k_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 (k_m <= 1.95e-143) {
		tmp = 2.0 / Math.pow((((k_m * Math.sqrt(2.0)) / l) * Math.sqrt(Math.pow(t_m, 3.0))), 2.0);
	} else if (k_m <= 1.3e-16) {
		tmp = 2.0 / (Math.pow((t_m / Math.pow(Math.cbrt(l), 2.0)), 3.0) * (2.0 * Math.pow(k_m, 2.0)));
	} else {
		tmp = (Math.cos(k_m) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k_m * Math.sin(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 (k_m <= 1.95e-143)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k_m * sqrt(2.0)) / l) * sqrt((t_m ^ 3.0))) ^ 2.0));
	elseif (k_m <= 1.3e-16)
		tmp = Float64(2.0 / Float64((Float64(t_m / (cbrt(l) ^ 2.0)) ^ 3.0) * Float64(2.0 * (k_m ^ 2.0))));
	else
		tmp = Float64(Float64(cos(k_m) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k_m * sin(k_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[k$95$m, 1.95e-143], N[(2.0 / N[Power[N[(N[(N[(k$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Power[t$95$m, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k$95$m, 1.3e-16], N[(2.0 / N[(N[Power[N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.95000000000000002e-143

   1. Initial program 56.3%

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

   3. Simplified 56.3%

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

   6. Applied egg-rr 26.4%

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

   7. Taylor expanded in k around 0 28.2%

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

   if 1.95000000000000002e-143 < k < 1.2999999999999999e-16

   1. Initial program 48.6%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in k around 0 67.2%

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

      1. add-cube-cbrt 67.2%

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

      2. pow3 67.2%

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

      3. *-commutative 67.2%

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

      4. cbrt-prod 67.2%

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

      5. associate-/l/ 66.0%

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

      6. cbrt-div 66.0%

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

      7. unpow3 66.0%

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

      8. add-cbrt-cube 70.1%

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

      9. cbrt-unprod 86.3%

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

      10. pow2 86.3%

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

   7. Applied egg-rr 86.3%

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

      1. *-commutative 86.3%

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

      2. cube-prod 78.8%

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

      3. rem-cube-cbrt 78.9%

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

   9. Simplified 78.9%

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

   if 1.2999999999999999e-16 < k

   1. Initial program 39.3%

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

      1. associate-/r* 39.3%

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

      2. +-commutative 39.3%

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

      3. unpow2 39.3%

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

      4. sqr-neg 39.3%

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

      5. distribute-frac-neg2 39.3%

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

      6. distribute-frac-neg2 39.3%

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

      7. unpow2 39.3%

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

      8. +-commutative 39.3%

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

      9. associate-*l* 39.3%

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

      10. associate-*l/ 37.6%

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

      11. associate-/r/ 37.6%

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

      12. +-commutative 37.6%

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

      13. associate-+r+ 37.6%

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

   3. Simplified 37.6%

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

      1. add-sqr-sqrt 31.2%

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

      2. pow2 31.2%

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

   6. Applied egg-rr 34.9%

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

   7. Taylor expanded in t around 0 54.3%

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

      1. associate-/l* 54.2%

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

   9. Simplified 54.2%

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

       1. *-un-lft-identity 54.2%

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

       2. *-commutative 54.2%

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

       3. unpow-prod-down 51.2%

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

       4. pow2 51.2%

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

       5. add-sqr-sqrt 81.4%

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

       6. associate-*r/ 81.4%

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

   11. Applied egg-rr 81.4%

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

       1. *-lft-identity 81.4%

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

       2. associate-/l* 81.4%

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

   13. Simplified 81.4%

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

Alternative 9: 86.6% accurate, 1.0× speedup

Math

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

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 (<= k_m 3.7e-10)
    (/
     2.0
     (pow
      (* (/ (pow t_m 1.5) l) (* k_m (hypot 1.0 (hypot 1.0 (/ k_m t_m)))))
      2.0))
    (* (/ (cos k_m) t_m) (pow (* l (/ (sqrt 2.0) (* k_m (sin 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 (k_m <= 3.7e-10) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (k_m * hypot(1.0, hypot(1.0, (k_m / t_m))))), 2.0);
	} else {
		tmp = (cos(k_m) / t_m) * pow((l * (sqrt(2.0) / (k_m * sin(k_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 (k_m <= 3.7e-10) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (k_m * Math.hypot(1.0, Math.hypot(1.0, (k_m / t_m))))), 2.0);
	} else {
		tmp = (Math.cos(k_m) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k_m * Math.sin(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 k_m <= 3.7e-10:
		tmp = 2.0 / math.pow(((math.pow(t_m, 1.5) / l) * (k_m * math.hypot(1.0, math.hypot(1.0, (k_m / t_m))))), 2.0)
	else:
		tmp = (math.cos(k_m) / t_m) * math.pow((l * (math.sqrt(2.0) / (k_m * math.sin(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 (k_m <= 3.7e-10)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(k_m * hypot(1.0, hypot(1.0, Float64(k_m / t_m))))) ^ 2.0));
	else
		tmp = Float64(Float64(cos(k_m) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k_m * sin(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 (k_m <= 3.7e-10)
		tmp = 2.0 / ((((t_m ^ 1.5) / l) * (k_m * hypot(1.0, hypot(1.0, (k_m / t_m))))) ^ 2.0);
	else
		tmp = (cos(k_m) / t_m) * ((l * (sqrt(2.0) / (k_m * sin(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[k$95$m, 3.7e-10], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k$95$m * N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k$95$m / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.70000000000000015e-10

   1. Initial program 55.6%

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

   3. Simplified 55.6%

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

   6. Applied egg-rr 30.0%

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

   7. Taylor expanded in k around 0 35.8%

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

   if 3.70000000000000015e-10 < k

   1. Initial program 37.8%

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

      1. associate-/r* 37.8%

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

      2. +-commutative 37.8%

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

      3. unpow2 37.8%

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

      4. sqr-neg 37.8%

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

      5. distribute-frac-neg2 37.8%

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

      6. distribute-frac-neg2 37.8%

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

      7. unpow2 37.8%

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

      8. +-commutative 37.8%

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

      9. associate-*l* 37.8%

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

      10. associate-*l/ 37.8%

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

      11. associate-/r/ 37.8%

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

      12. +-commutative 37.8%

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

      13. associate-+r+ 37.8%

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

   3. Simplified 37.8%

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

      1. add-sqr-sqrt 30.9%

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

      2. pow2 30.9%

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

   6. Applied egg-rr 34.8%

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

   7. Taylor expanded in t around 0 53.6%

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

      1. associate-/l* 53.5%

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

   9. Simplified 53.5%

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

       1. *-un-lft-identity 53.5%

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

       2. *-commutative 53.5%

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

       3. unpow-prod-down 50.4%

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

       4. pow2 50.4%

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

       5. add-sqr-sqrt 82.1%

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

       6. associate-*r/ 82.1%

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

   11. Applied egg-rr 82.1%

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

       1. *-lft-identity 82.1%

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

       2. associate-/l* 82.1%

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

   13. Simplified 82.1%

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

4. Final simplification 46.3%

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

Alternative 10: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 6.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m \cdot \sqrt{2}}{\ell} \cdot \sqrt{{t\_m}^{3}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k\_m}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{k\_m \cdot \sin k\_m}\right)}^{2}\\ \end{array} \end{array} \]

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 (<= k_m 6.2e-45)
    (/ 2.0 (pow (* (/ (* k_m (sqrt 2.0)) l) (sqrt (pow t_m 3.0))) 2.0))
    (* (/ (cos k_m) t_m) (pow (* l (/ (sqrt 2.0) (* k_m (sin 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 (k_m <= 6.2e-45) {
		tmp = 2.0 / pow((((k_m * sqrt(2.0)) / l) * sqrt(pow(t_m, 3.0))), 2.0);
	} else {
		tmp = (cos(k_m) / t_m) * pow((l * (sqrt(2.0) / (k_m * sin(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 (k_m <= 6.2d-45) then
        tmp = 2.0d0 / ((((k_m * sqrt(2.0d0)) / l) * sqrt((t_m ** 3.0d0))) ** 2.0d0)
    else
        tmp = (cos(k_m) / t_m) * ((l * (sqrt(2.0d0) / (k_m * sin(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 (k_m <= 6.2e-45) {
		tmp = 2.0 / Math.pow((((k_m * Math.sqrt(2.0)) / l) * Math.sqrt(Math.pow(t_m, 3.0))), 2.0);
	} else {
		tmp = (Math.cos(k_m) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k_m * Math.sin(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 k_m <= 6.2e-45:
		tmp = 2.0 / math.pow((((k_m * math.sqrt(2.0)) / l) * math.sqrt(math.pow(t_m, 3.0))), 2.0)
	else:
		tmp = (math.cos(k_m) / t_m) * math.pow((l * (math.sqrt(2.0) / (k_m * math.sin(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 (k_m <= 6.2e-45)
		tmp = Float64(2.0 / (Float64(Float64(Float64(k_m * sqrt(2.0)) / l) * sqrt((t_m ^ 3.0))) ^ 2.0));
	else
		tmp = Float64(Float64(cos(k_m) / t_m) * (Float64(l * Float64(sqrt(2.0) / Float64(k_m * sin(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 (k_m <= 6.2e-45)
		tmp = 2.0 / ((((k_m * sqrt(2.0)) / l) * sqrt((t_m ^ 3.0))) ^ 2.0);
	else
		tmp = (cos(k_m) / t_m) * ((l * (sqrt(2.0) / (k_m * sin(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[k$95$m, 6.2e-45], N[(2.0 / N[Power[N[(N[(N[(k$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Power[t$95$m, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.2000000000000002e-45

   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 56.5%

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

   6. Applied egg-rr 28.0%

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

   7. Taylor expanded in k around 0 29.6%

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

   if 6.2000000000000002e-45 < k

   1. Initial program 37.2%

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

      1. associate-/r* 37.2%

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

      2. +-commutative 37.2%

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

      3. unpow2 37.2%

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

      4. sqr-neg 37.2%

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

      5. distribute-frac-neg2 37.2%

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

      6. distribute-frac-neg2 37.2%

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

      7. unpow2 37.2%

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

      8. +-commutative 37.2%

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

      9. associate-*l* 37.2%

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

      10. associate-*l/ 35.5%

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

      11. associate-/r/ 35.5%

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

      12. +-commutative 35.5%

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

      13. associate-+r+ 35.5%

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

   3. Simplified 35.5%

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

      1. add-sqr-sqrt 29.5%

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

      2. pow2 29.5%

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

   6. Applied egg-rr 34.6%

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

   7. Taylor expanded in t around 0 54.0%

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

      1. associate-/l* 53.9%

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

   9. Simplified 53.9%

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

       1. *-un-lft-identity 53.9%

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

       2. *-commutative 53.9%

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

       3. unpow-prod-down 51.2%

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

       4. pow2 51.2%

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

       5. add-sqr-sqrt 79.5%

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

       6. associate-*r/ 79.5%

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

   11. Applied egg-rr 79.5%

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

       1. *-lft-identity 79.5%

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

       2. associate-/l* 79.5%

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

   13. Simplified 79.5%

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

Alternative 11: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-62}:\\ \;\;\;\;{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k\_m}^{2}}\right) \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m \cdot \sqrt{2}}{\ell} \cdot \sqrt{{t\_m}^{3}}\right)}^{2}}\\ \end{array} \end{array} \]

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 1.05e-62)
    (pow (* (* l (/ (sqrt 2.0) (pow k_m 2.0))) (sqrt (/ 1.0 t_m))) 2.0)
    (/ 2.0 (pow (* (/ (* k_m (sqrt 2.0)) l) (sqrt (pow t_m 3.0))) 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 <= 1.05e-62) {
		tmp = pow(((l * (sqrt(2.0) / pow(k_m, 2.0))) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 / pow((((k_m * sqrt(2.0)) / l) * sqrt(pow(t_m, 3.0))), 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 <= 1.05d-62) then
        tmp = ((l * (sqrt(2.0d0) / (k_m ** 2.0d0))) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = 2.0d0 / ((((k_m * sqrt(2.0d0)) / l) * sqrt((t_m ** 3.0d0))) ** 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 <= 1.05e-62) {
		tmp = Math.pow(((l * (Math.sqrt(2.0) / Math.pow(k_m, 2.0))) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = 2.0 / Math.pow((((k_m * Math.sqrt(2.0)) / l) * Math.sqrt(Math.pow(t_m, 3.0))), 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 <= 1.05e-62:
		tmp = math.pow(((l * (math.sqrt(2.0) / math.pow(k_m, 2.0))) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = 2.0 / math.pow((((k_m * math.sqrt(2.0)) / l) * math.sqrt(math.pow(t_m, 3.0))), 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 <= 1.05e-62)
		tmp = Float64(Float64(l * Float64(sqrt(2.0) / (k_m ^ 2.0))) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(k_m * sqrt(2.0)) / l) * sqrt((t_m ^ 3.0))) ^ 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 <= 1.05e-62)
		tmp = ((l * (sqrt(2.0) / (k_m ^ 2.0))) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = 2.0 / ((((k_m * sqrt(2.0)) / l) * sqrt((t_m ^ 3.0))) ^ 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, 1.05e-62], N[Power[N[(N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(k$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Power[t$95$m, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.05e-62

   1. Initial program 49.6%

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

      1. associate-/r* 49.6%

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

      2. +-commutative 49.6%

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

      3. unpow2 49.6%

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

      4. sqr-neg 49.6%

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

      5. distribute-frac-neg2 49.6%

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

      6. distribute-frac-neg2 49.6%

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

      7. unpow2 49.6%

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

      8. +-commutative 49.6%

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

      9. associate-*l* 45.6%

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

      10. associate-*l/ 43.4%

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

      11. associate-/r/ 43.3%

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

      12. +-commutative 43.3%

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

      13. associate-+r+ 43.3%

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

   3. Simplified 43.3%

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

      1. add-sqr-sqrt 26.5%

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

      2. pow2 26.5%

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

   6. Applied egg-rr 29.5%

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

   7. Taylor expanded in t around 0 39.9%

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

      1. associate-/l* 39.9%

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

   9. Simplified 39.9%

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

   10. Taylor expanded in k around 0 23.8%

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

       1. associate-/l* 23.8%

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

   12. Simplified 23.8%

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

   if 1.05e-62 < t

   1. Initial program 55.7%

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

   3. Simplified 55.8%

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

   6. Applied egg-rr 48.5%

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

   7. Taylor expanded in k around 0 53.8%

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

Alternative 12: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.65 \cdot 10^{-62}:\\ \;\;\;\;\frac{1}{t\_m} \cdot {\left(\ell \cdot \left(\sqrt{2} \cdot {k\_m}^{-2}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k\_m \cdot \sqrt{2}}{\ell} \cdot \sqrt{{t\_m}^{3}}\right)}^{2}}\\ \end{array} \end{array} \]

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 1.65e-62)
    (* (/ 1.0 t_m) (pow (* l (* (sqrt 2.0) (pow k_m -2.0))) 2.0))
    (/ 2.0 (pow (* (/ (* k_m (sqrt 2.0)) l) (sqrt (pow t_m 3.0))) 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 <= 1.65e-62) {
		tmp = (1.0 / t_m) * pow((l * (sqrt(2.0) * pow(k_m, -2.0))), 2.0);
	} else {
		tmp = 2.0 / pow((((k_m * sqrt(2.0)) / l) * sqrt(pow(t_m, 3.0))), 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 <= 1.65d-62) then
        tmp = (1.0d0 / t_m) * ((l * (sqrt(2.0d0) * (k_m ** (-2.0d0)))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((((k_m * sqrt(2.0d0)) / l) * sqrt((t_m ** 3.0d0))) ** 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 <= 1.65e-62) {
		tmp = (1.0 / t_m) * Math.pow((l * (Math.sqrt(2.0) * Math.pow(k_m, -2.0))), 2.0);
	} else {
		tmp = 2.0 / Math.pow((((k_m * Math.sqrt(2.0)) / l) * Math.sqrt(Math.pow(t_m, 3.0))), 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 <= 1.65e-62:
		tmp = (1.0 / t_m) * math.pow((l * (math.sqrt(2.0) * math.pow(k_m, -2.0))), 2.0)
	else:
		tmp = 2.0 / math.pow((((k_m * math.sqrt(2.0)) / l) * math.sqrt(math.pow(t_m, 3.0))), 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 <= 1.65e-62)
		tmp = Float64(Float64(1.0 / t_m) * (Float64(l * Float64(sqrt(2.0) * (k_m ^ -2.0))) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64(Float64(k_m * sqrt(2.0)) / l) * sqrt((t_m ^ 3.0))) ^ 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 <= 1.65e-62)
		tmp = (1.0 / t_m) * ((l * (sqrt(2.0) * (k_m ^ -2.0))) ^ 2.0);
	else
		tmp = 2.0 / ((((k_m * sqrt(2.0)) / l) * sqrt((t_m ^ 3.0))) ^ 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, 1.65e-62], N[(N[(1.0 / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(N[(k$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Sqrt[N[Power[t$95$m, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.65000000000000002e-62

   1. Initial program 49.6%

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

      1. associate-/r* 49.6%

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

      2. +-commutative 49.6%

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

      3. unpow2 49.6%

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

      4. sqr-neg 49.6%

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

      5. distribute-frac-neg2 49.6%

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

      6. distribute-frac-neg2 49.6%

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

      7. unpow2 49.6%

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

      8. +-commutative 49.6%

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

      9. associate-*l* 45.6%

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

      10. associate-*l/ 43.4%

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

      11. associate-/r/ 43.3%

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

      12. +-commutative 43.3%

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

      13. associate-+r+ 43.3%

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

   3. Simplified 43.3%

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

      1. add-sqr-sqrt 26.5%

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

      2. pow2 26.5%

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

   6. Applied egg-rr 29.5%

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

   7. Taylor expanded in t around 0 39.9%

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

      1. associate-/l* 39.9%

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

   9. Simplified 39.9%

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

   10. Taylor expanded in k around 0 23.8%

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

       1. associate-/l* 23.8%

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

   12. Simplified 23.8%

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

       1. unpow-prod-down 23.8%

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

       2. div-inv 23.8%

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

       3. pow-flip 23.8%

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

       4. metadata-eval 23.8%

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

       5. pow2 23.8%

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

       6. add-sqr-sqrt 60.5%

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

   14. Applied egg-rr 60.5%

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

   if 1.65000000000000002e-62 < t

   1. Initial program 55.7%

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

   3. Simplified 55.8%

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

   6. Applied egg-rr 48.5%

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

   7. Taylor expanded in k around 0 53.8%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{-62}:\\ \;\;\;\;\frac{1}{t} \cdot {\left(\ell \cdot \left(\sqrt{2} \cdot {k}^{-2}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k \cdot \sqrt{2}}{\ell} \cdot \sqrt{{t}^{3}}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.26 \cdot 10^{-61}:\\ \;\;\;\;\frac{1}{t\_m} \cdot {\left(\ell \cdot \left(\sqrt{2} \cdot {k\_m}^{-2}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k\_m \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right) \cdot \left(2 \cdot \tan k\_m\right)}\\ \end{array} \end{array} \]

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 1.26e-61)
    (* (/ 1.0 t_m) (pow (* l (* (sqrt 2.0) (pow k_m -2.0))) 2.0))
    (/ 2.0 (* (* (sin k_m) (/ (pow t_m 3.0) (* l l))) (* 2.0 (tan 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 <= 1.26e-61) {
		tmp = (1.0 / t_m) * pow((l * (sqrt(2.0) * pow(k_m, -2.0))), 2.0);
	} else {
		tmp = 2.0 / ((sin(k_m) * (pow(t_m, 3.0) / (l * l))) * (2.0 * tan(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 <= 1.26d-61) then
        tmp = (1.0d0 / t_m) * ((l * (sqrt(2.0d0) * (k_m ** (-2.0d0)))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((sin(k_m) * ((t_m ** 3.0d0) / (l * l))) * (2.0d0 * tan(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 <= 1.26e-61) {
		tmp = (1.0 / t_m) * Math.pow((l * (Math.sqrt(2.0) * Math.pow(k_m, -2.0))), 2.0);
	} else {
		tmp = 2.0 / ((Math.sin(k_m) * (Math.pow(t_m, 3.0) / (l * l))) * (2.0 * Math.tan(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 <= 1.26e-61:
		tmp = (1.0 / t_m) * math.pow((l * (math.sqrt(2.0) * math.pow(k_m, -2.0))), 2.0)
	else:
		tmp = 2.0 / ((math.sin(k_m) * (math.pow(t_m, 3.0) / (l * l))) * (2.0 * math.tan(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 <= 1.26e-61)
		tmp = Float64(Float64(1.0 / t_m) * (Float64(l * Float64(sqrt(2.0) * (k_m ^ -2.0))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k_m) * Float64((t_m ^ 3.0) / Float64(l * l))) * Float64(2.0 * tan(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 <= 1.26e-61)
		tmp = (1.0 / t_m) * ((l * (sqrt(2.0) * (k_m ^ -2.0))) ^ 2.0);
	else
		tmp = 2.0 / ((sin(k_m) * ((t_m ^ 3.0) / (l * l))) * (2.0 * tan(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, 1.26e-61], N[(N[(1.0 / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.2599999999999999e-61

   1. Initial program 49.4%

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

      1. associate-/r* 49.4%

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

      2. +-commutative 49.4%

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

      3. unpow2 49.4%

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

      4. sqr-neg 49.4%

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

      5. distribute-frac-neg2 49.4%

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

      6. distribute-frac-neg2 49.4%

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

      7. unpow2 49.4%

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

      8. +-commutative 49.4%

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

      9. associate-*l* 45.3%

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

      10. associate-*l/ 43.2%

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

      11. associate-/r/ 43.0%

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

      12. +-commutative 43.0%

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

      13. associate-+r+ 43.0%

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

   3. Simplified 43.0%

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

      1. add-sqr-sqrt 26.3%

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

      2. pow2 26.3%

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

   6. Applied egg-rr 29.4%

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

   7. Taylor expanded in t around 0 39.7%

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

      1. associate-/l* 39.7%

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

   9. Simplified 39.7%

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

   10. Taylor expanded in k around 0 23.7%

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

       1. associate-/l* 23.7%

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

   12. Simplified 23.7%

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

       1. unpow-prod-down 23.7%

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

       2. div-inv 23.7%

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

       3. pow-flip 23.7%

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

       4. metadata-eval 23.7%

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

       5. pow2 23.7%

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

       6. add-sqr-sqrt 60.2%

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

   14. Applied egg-rr 60.2%

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

   if 1.2599999999999999e-61 < t

   1. Initial program 56.4%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in k around 0 49.4%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.26 \cdot 10^{-61}:\\ \;\;\;\;\frac{1}{t} \cdot {\left(\ell \cdot \left(\sqrt{2} \cdot {k}^{-2}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(2 \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-59}:\\ \;\;\;\;\frac{1}{t\_m} \cdot {\left(\ell \cdot \left(\sqrt{2} \cdot {k\_m}^{-2}\right)\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 2.6 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{\left({t\_m}^{2} \cdot \frac{t\_m}{\ell}\right) \cdot \frac{2 \cdot {k\_m}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot \left(\sin k\_m \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

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-59)
    (* (/ 1.0 t_m) (pow (* l (* (sqrt 2.0) (pow k_m -2.0))) 2.0))
    (if (<= t_m 2.6e+148)
      (/ 2.0 (* (* (pow t_m 2.0) (/ t_m l)) (/ (* 2.0 (pow k_m 2.0)) l)))
      (/ 2.0 (* (* k_m 2.0) (* (sin k_m) (/ (pow t_m 3.0) (* 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) {
	double tmp;
	if (t_m <= 4e-59) {
		tmp = (1.0 / t_m) * pow((l * (sqrt(2.0) * pow(k_m, -2.0))), 2.0);
	} else if (t_m <= 2.6e+148) {
		tmp = 2.0 / ((pow(t_m, 2.0) * (t_m / l)) * ((2.0 * pow(k_m, 2.0)) / l));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * (pow(t_m, 3.0) / (l * 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 <= 4d-59) then
        tmp = (1.0d0 / t_m) * ((l * (sqrt(2.0d0) * (k_m ** (-2.0d0)))) ** 2.0d0)
    else if (t_m <= 2.6d+148) then
        tmp = 2.0d0 / (((t_m ** 2.0d0) * (t_m / l)) * ((2.0d0 * (k_m ** 2.0d0)) / l))
    else
        tmp = 2.0d0 / ((k_m * 2.0d0) * (sin(k_m) * ((t_m ** 3.0d0) / (l * 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 <= 4e-59) {
		tmp = (1.0 / t_m) * Math.pow((l * (Math.sqrt(2.0) * Math.pow(k_m, -2.0))), 2.0);
	} else if (t_m <= 2.6e+148) {
		tmp = 2.0 / ((Math.pow(t_m, 2.0) * (t_m / l)) * ((2.0 * Math.pow(k_m, 2.0)) / l));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * (Math.sin(k_m) * (Math.pow(t_m, 3.0) / (l * 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 <= 4e-59:
		tmp = (1.0 / t_m) * math.pow((l * (math.sqrt(2.0) * math.pow(k_m, -2.0))), 2.0)
	elif t_m <= 2.6e+148:
		tmp = 2.0 / ((math.pow(t_m, 2.0) * (t_m / l)) * ((2.0 * math.pow(k_m, 2.0)) / l))
	else:
		tmp = 2.0 / ((k_m * 2.0) * (math.sin(k_m) * (math.pow(t_m, 3.0) / (l * 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 <= 4e-59)
		tmp = Float64(Float64(1.0 / t_m) * (Float64(l * Float64(sqrt(2.0) * (k_m ^ -2.0))) ^ 2.0));
	elseif (t_m <= 2.6e+148)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 2.0) * Float64(t_m / l)) * Float64(Float64(2.0 * (k_m ^ 2.0)) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * Float64(sin(k_m) * Float64((t_m ^ 3.0) / Float64(l * 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 <= 4e-59)
		tmp = (1.0 / t_m) * ((l * (sqrt(2.0) * (k_m ^ -2.0))) ^ 2.0);
	elseif (t_m <= 2.6e+148)
		tmp = 2.0 / (((t_m ^ 2.0) * (t_m / l)) * ((2.0 * (k_m ^ 2.0)) / l));
	else
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * ((t_m ^ 3.0) / (l * 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, 4e-59], N[(N[(1.0 / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.6e+148], N[(2.0 / N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 4.0000000000000001e-59

   1. Initial program 50.2%

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

      1. associate-/r* 50.2%

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

      2. +-commutative 50.2%

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

      3. unpow2 50.2%

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

      4. sqr-neg 50.2%

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

      5. distribute-frac-neg2 50.2%

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

      6. distribute-frac-neg2 50.2%

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

      7. unpow2 50.2%

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

      8. +-commutative 50.2%

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

      9. associate-*l* 46.2%

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

      10. associate-*l/ 44.1%

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

      11. associate-/r/ 44.0%

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

      12. +-commutative 44.0%

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

      13. associate-+r+ 44.0%

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

   3. Simplified 44.0%

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

      1. add-sqr-sqrt 26.4%

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

      2. pow2 26.4%

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

   6. Applied egg-rr 29.5%

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

   7. Taylor expanded in t around 0 39.6%

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

      1. associate-/l* 39.6%

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

   9. Simplified 39.6%

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

   10. Taylor expanded in k around 0 23.9%

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

       1. associate-/l* 23.9%

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

   12. Simplified 23.9%

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

       1. unpow-prod-down 23.9%

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

       2. div-inv 23.9%

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

       3. pow-flip 23.9%

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

       4. metadata-eval 23.9%

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

       5. pow2 23.9%

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

       6. add-sqr-sqrt 59.8%

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

   14. Applied egg-rr 59.8%

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

   if 4.0000000000000001e-59 < t < 2.6e148

   1. Initial program 62.4%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in k around 0 49.0%

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

      1. associate-*l/ 49.3%

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

   7. Applied egg-rr 49.3%

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

      1. associate-/l* 49.3%

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

   9. Simplified 49.3%

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

       1. unpow3 49.3%

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

       2. *-un-lft-identity 49.3%

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

       3. times-frac 54.2%

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

       4. pow2 54.2%

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

   11. Applied egg-rr 54.2%

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

   if 2.6e148 < t

   1. Initial program 46.4%

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

   3. Simplified 46.4%

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

   5. Taylor expanded in k around 0 46.4%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-59}:\\ \;\;\;\;\frac{1}{t} \cdot {\left(\ell \cdot \left(\sqrt{2} \cdot {k}^{-2}\right)\right)}^{2}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{\left({t}^{2} \cdot \frac{t}{\ell}\right) \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 65.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.7 \cdot 10^{-59}:\\ \;\;\;\;\frac{{\left(\ell \cdot \left(\sqrt{2} \cdot {k\_m}^{-2}\right)\right)}^{2}}{t\_m}\\ \mathbf{elif}\;t\_m \leq 4.8 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{\left({t\_m}^{2} \cdot \frac{t\_m}{\ell}\right) \cdot \frac{2 \cdot {k\_m}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot \left(\sin k\_m \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

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 2.7e-59)
    (/ (pow (* l (* (sqrt 2.0) (pow k_m -2.0))) 2.0) t_m)
    (if (<= t_m 4.8e+148)
      (/ 2.0 (* (* (pow t_m 2.0) (/ t_m l)) (/ (* 2.0 (pow k_m 2.0)) l)))
      (/ 2.0 (* (* k_m 2.0) (* (sin k_m) (/ (pow t_m 3.0) (* 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) {
	double tmp;
	if (t_m <= 2.7e-59) {
		tmp = pow((l * (sqrt(2.0) * pow(k_m, -2.0))), 2.0) / t_m;
	} else if (t_m <= 4.8e+148) {
		tmp = 2.0 / ((pow(t_m, 2.0) * (t_m / l)) * ((2.0 * pow(k_m, 2.0)) / l));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * (pow(t_m, 3.0) / (l * 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 <= 2.7d-59) then
        tmp = ((l * (sqrt(2.0d0) * (k_m ** (-2.0d0)))) ** 2.0d0) / t_m
    else if (t_m <= 4.8d+148) then
        tmp = 2.0d0 / (((t_m ** 2.0d0) * (t_m / l)) * ((2.0d0 * (k_m ** 2.0d0)) / l))
    else
        tmp = 2.0d0 / ((k_m * 2.0d0) * (sin(k_m) * ((t_m ** 3.0d0) / (l * 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 <= 2.7e-59) {
		tmp = Math.pow((l * (Math.sqrt(2.0) * Math.pow(k_m, -2.0))), 2.0) / t_m;
	} else if (t_m <= 4.8e+148) {
		tmp = 2.0 / ((Math.pow(t_m, 2.0) * (t_m / l)) * ((2.0 * Math.pow(k_m, 2.0)) / l));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * (Math.sin(k_m) * (Math.pow(t_m, 3.0) / (l * 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 <= 2.7e-59:
		tmp = math.pow((l * (math.sqrt(2.0) * math.pow(k_m, -2.0))), 2.0) / t_m
	elif t_m <= 4.8e+148:
		tmp = 2.0 / ((math.pow(t_m, 2.0) * (t_m / l)) * ((2.0 * math.pow(k_m, 2.0)) / l))
	else:
		tmp = 2.0 / ((k_m * 2.0) * (math.sin(k_m) * (math.pow(t_m, 3.0) / (l * 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 <= 2.7e-59)
		tmp = Float64((Float64(l * Float64(sqrt(2.0) * (k_m ^ -2.0))) ^ 2.0) / t_m);
	elseif (t_m <= 4.8e+148)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 2.0) * Float64(t_m / l)) * Float64(Float64(2.0 * (k_m ^ 2.0)) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * Float64(sin(k_m) * Float64((t_m ^ 3.0) / Float64(l * 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 <= 2.7e-59)
		tmp = ((l * (sqrt(2.0) * (k_m ^ -2.0))) ^ 2.0) / t_m;
	elseif (t_m <= 4.8e+148)
		tmp = 2.0 / (((t_m ^ 2.0) * (t_m / l)) * ((2.0 * (k_m ^ 2.0)) / l));
	else
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * ((t_m ^ 3.0) / (l * 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, 2.7e-59], N[(N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[k$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision], If[LessEqual[t$95$m, 4.8e+148], N[(2.0 / N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.6999999999999999e-59

   1. Initial program 50.2%

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

      1. associate-/r* 50.2%

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

      2. +-commutative 50.2%

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

      3. unpow2 50.2%

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

      4. sqr-neg 50.2%

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

      5. distribute-frac-neg2 50.2%

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

      6. distribute-frac-neg2 50.2%

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

      7. unpow2 50.2%

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

      8. +-commutative 50.2%

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

      9. associate-*l* 46.2%

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

      10. associate-*l/ 44.1%

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

      11. associate-/r/ 44.0%

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

      12. +-commutative 44.0%

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

      13. associate-+r+ 44.0%

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

   3. Simplified 44.0%

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

      1. add-sqr-sqrt 26.4%

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

      2. pow2 26.4%

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

   6. Applied egg-rr 29.5%

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

   7. Taylor expanded in t around 0 39.6%

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

      1. associate-/l* 39.6%

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

   9. Simplified 39.6%

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

   10. Taylor expanded in k around 0 23.9%

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

       1. associate-/l* 23.9%

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

   12. Simplified 23.9%

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

       1. *-un-lft-identity 23.9%

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

       2. *-commutative 23.9%

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

       3. unpow-prod-down 23.9%

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

       4. pow2 23.9%

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

       5. add-sqr-sqrt 59.8%

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

       6. div-inv 59.8%

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

       7. pow-flip 59.8%

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

       8. metadata-eval 59.8%

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

   14. Applied egg-rr 59.8%

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

       1. *-lft-identity 59.8%

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

       2. associate-*l/ 59.8%

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

       3. *-lft-identity 59.8%

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

   16. Simplified 59.8%

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

   if 2.6999999999999999e-59 < t < 4.79999999999999989e148

   1. Initial program 62.4%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in k around 0 49.0%

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

      1. associate-*l/ 49.3%

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

   7. Applied egg-rr 49.3%

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

      1. associate-/l* 49.3%

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

   9. Simplified 49.3%

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

       1. unpow3 49.3%

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

       2. *-un-lft-identity 49.3%

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

       3. times-frac 54.2%

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

       4. pow2 54.2%

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

   11. Applied egg-rr 54.2%

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

   if 4.79999999999999989e148 < t

   1. Initial program 46.4%

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

   3. Simplified 46.4%

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

   5. Taylor expanded in k around 0 46.4%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{-59}:\\ \;\;\;\;\frac{{\left(\ell \cdot \left(\sqrt{2} \cdot {k}^{-2}\right)\right)}^{2}}{t}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{\left({t}^{2} \cdot \frac{t}{\ell}\right) \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.4 \cdot 10^{-138}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t\_m \cdot {k\_m}^{4}}\\ \mathbf{elif}\;t\_m \leq 1.4 \cdot 10^{+145}:\\ \;\;\;\;\frac{2}{\left({t\_m}^{2} \cdot \frac{t\_m}{\ell}\right) \cdot \frac{2 \cdot {k\_m}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot \left(\sin k\_m \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \end{array} \]

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 3.4e-138)
    (* (* l l) (/ 2.0 (* t_m (pow k_m 4.0))))
    (if (<= t_m 1.4e+145)
      (/ 2.0 (* (* (pow t_m 2.0) (/ t_m l)) (/ (* 2.0 (pow k_m 2.0)) l)))
      (/ 2.0 (* (* k_m 2.0) (* (sin k_m) (/ (pow t_m 3.0) (* 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) {
	double tmp;
	if (t_m <= 3.4e-138) {
		tmp = (l * l) * (2.0 / (t_m * pow(k_m, 4.0)));
	} else if (t_m <= 1.4e+145) {
		tmp = 2.0 / ((pow(t_m, 2.0) * (t_m / l)) * ((2.0 * pow(k_m, 2.0)) / l));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * (pow(t_m, 3.0) / (l * 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 <= 3.4d-138) then
        tmp = (l * l) * (2.0d0 / (t_m * (k_m ** 4.0d0)))
    else if (t_m <= 1.4d+145) then
        tmp = 2.0d0 / (((t_m ** 2.0d0) * (t_m / l)) * ((2.0d0 * (k_m ** 2.0d0)) / l))
    else
        tmp = 2.0d0 / ((k_m * 2.0d0) * (sin(k_m) * ((t_m ** 3.0d0) / (l * 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 <= 3.4e-138) {
		tmp = (l * l) * (2.0 / (t_m * Math.pow(k_m, 4.0)));
	} else if (t_m <= 1.4e+145) {
		tmp = 2.0 / ((Math.pow(t_m, 2.0) * (t_m / l)) * ((2.0 * Math.pow(k_m, 2.0)) / l));
	} else {
		tmp = 2.0 / ((k_m * 2.0) * (Math.sin(k_m) * (Math.pow(t_m, 3.0) / (l * 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 <= 3.4e-138:
		tmp = (l * l) * (2.0 / (t_m * math.pow(k_m, 4.0)))
	elif t_m <= 1.4e+145:
		tmp = 2.0 / ((math.pow(t_m, 2.0) * (t_m / l)) * ((2.0 * math.pow(k_m, 2.0)) / l))
	else:
		tmp = 2.0 / ((k_m * 2.0) * (math.sin(k_m) * (math.pow(t_m, 3.0) / (l * 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 <= 3.4e-138)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t_m * (k_m ^ 4.0))));
	elseif (t_m <= 1.4e+145)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 2.0) * Float64(t_m / l)) * Float64(Float64(2.0 * (k_m ^ 2.0)) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * Float64(sin(k_m) * Float64((t_m ^ 3.0) / Float64(l * 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 <= 3.4e-138)
		tmp = (l * l) * (2.0 / (t_m * (k_m ^ 4.0)));
	elseif (t_m <= 1.4e+145)
		tmp = 2.0 / (((t_m ^ 2.0) * (t_m / l)) * ((2.0 * (k_m ^ 2.0)) / l));
	else
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * ((t_m ^ 3.0) / (l * 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, 3.4e-138], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.4e+145], N[(2.0 / N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.4000000000000001e-138

   1. Initial program 47.9%

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

      1. associate-/r* 47.9%

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

      2. +-commutative 47.9%

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

      3. unpow2 47.9%

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

      4. sqr-neg 47.9%

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

      5. distribute-frac-neg2 47.9%

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

      6. distribute-frac-neg2 47.9%

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

      7. unpow2 47.9%

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

      8. +-commutative 47.9%

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

      9. associate-*l* 43.2%

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

      10. associate-*l/ 42.1%

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

      11. associate-/r/ 41.9%

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

      12. +-commutative 41.9%

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

      13. associate-+r+ 41.9%

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

   3. Simplified 41.9%

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

      1. add-sqr-sqrt 22.5%

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

      2. pow2 22.5%

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

   6. Applied egg-rr 26.0%

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

   7. Taylor expanded in t around 0 34.2%

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

      1. associate-/l* 34.1%

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

   9. Simplified 34.1%

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

   10. Taylor expanded in k around 0 52.7%

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

       1. associate-/l* 52.7%

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

       2. unpow2 52.7%

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

       3. rem-square-sqrt 52.7%

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

       4. *-commutative 52.7%

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

   12. Simplified 52.7%

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

       1. unpow2 52.7%

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

   14. Applied egg-rr 52.7%

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

   if 3.4000000000000001e-138 < t < 1.3999999999999999e145

   1. Initial program 62.7%

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

   3. Simplified 67.1%

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

   5. Taylor expanded in k around 0 47.8%

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

      1. associate-*l/ 47.9%

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

   7. Applied egg-rr 47.9%

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

      1. associate-/l* 47.9%

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

   9. Simplified 47.9%

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

       1. unpow3 47.9%

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

       2. *-un-lft-identity 47.9%

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

       3. times-frac 55.3%

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

       4. pow2 55.3%

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

   11. Applied egg-rr 55.3%

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

   if 1.3999999999999999e145 < t

   1. Initial program 46.4%

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

   3. Simplified 46.4%

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

   5. Taylor expanded in k around 0 46.4%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-138}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+145}:\\ \;\;\;\;\frac{2}{\left({t}^{2} \cdot \frac{t}{\ell}\right) \cdot \frac{2 \cdot {k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 58.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.4 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{\left(k\_m \cdot 2\right) \cdot \left(\sin k\_m \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t\_m \cdot {k\_m}^{4}}\\ \end{array} \end{array} \]

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 1.4e-47)
    (/ 2.0 (* (* k_m 2.0) (* (sin k_m) (/ (pow t_m 3.0) (* l l)))))
    (* (* l 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 <= 1.4e-47) {
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * (pow(t_m, 3.0) / (l * l))));
	} else {
		tmp = (l * 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 <= 1.4d-47) then
        tmp = 2.0d0 / ((k_m * 2.0d0) * (sin(k_m) * ((t_m ** 3.0d0) / (l * l))))
    else
        tmp = (l * 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 <= 1.4e-47) {
		tmp = 2.0 / ((k_m * 2.0) * (Math.sin(k_m) * (Math.pow(t_m, 3.0) / (l * l))));
	} else {
		tmp = (l * 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 <= 1.4e-47:
		tmp = 2.0 / ((k_m * 2.0) * (math.sin(k_m) * (math.pow(t_m, 3.0) / (l * l))))
	else:
		tmp = (l * 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 <= 1.4e-47)
		tmp = Float64(2.0 / Float64(Float64(k_m * 2.0) * Float64(sin(k_m) * Float64((t_m ^ 3.0) / Float64(l * l)))));
	else
		tmp = Float64(Float64(l * l) * Float64(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 <= 1.4e-47)
		tmp = 2.0 / ((k_m * 2.0) * (sin(k_m) * ((t_m ^ 3.0) / (l * l))));
	else
		tmp = (l * 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, 1.4e-47], N[(2.0 / N[(N[(k$95$m * 2.0), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.39999999999999996e-47

   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 56.5%

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

   5. Taylor expanded in k around 0 56.8%

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

   if 1.39999999999999996e-47 < k

   1. Initial program 37.2%

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

      1. associate-/r* 37.2%

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

      2. +-commutative 37.2%

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

      3. unpow2 37.2%

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

      4. sqr-neg 37.2%

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

      5. distribute-frac-neg2 37.2%

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

      6. distribute-frac-neg2 37.2%

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

      7. unpow2 37.2%

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

      8. +-commutative 37.2%

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

      9. associate-*l* 37.2%

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

      10. associate-*l/ 35.5%

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

      11. associate-/r/ 35.5%

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

      12. +-commutative 35.5%

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

      13. associate-+r+ 35.5%

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

   3. Simplified 35.5%

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

      1. add-sqr-sqrt 29.5%

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

      2. pow2 29.5%

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

   6. Applied egg-rr 34.6%

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

   7. Taylor expanded in t around 0 54.0%

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

      1. associate-/l* 53.9%

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

   9. Simplified 53.9%

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

   10. Taylor expanded in k around 0 44.5%

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

       1. associate-/l* 44.5%

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

       2. unpow2 44.5%

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

       3. rem-square-sqrt 44.5%

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

       4. *-commutative 44.5%

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

   12. Simplified 44.5%

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

       1. unpow2 44.5%

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

   14. Applied egg-rr 44.5%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{\left(k \cdot 2\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 58.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.2 \cdot 10^{-69}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t\_m \cdot {k\_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k\_m}^{2}}{\ell} \cdot \frac{{t\_m}^{3}}{\ell}}\\ \end{array} \end{array} \]

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 9.2e-69)
    (* (* l l) (/ 2.0 (* t_m (pow k_m 4.0))))
    (/ 2.0 (* (/ (* 2.0 (pow k_m 2.0)) l) (/ (pow t_m 3.0) 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 <= 9.2e-69) {
		tmp = (l * l) * (2.0 / (t_m * pow(k_m, 4.0)));
	} else {
		tmp = 2.0 / (((2.0 * pow(k_m, 2.0)) / l) * (pow(t_m, 3.0) / 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 <= 9.2d-69) then
        tmp = (l * l) * (2.0d0 / (t_m * (k_m ** 4.0d0)))
    else
        tmp = 2.0d0 / (((2.0d0 * (k_m ** 2.0d0)) / l) * ((t_m ** 3.0d0) / 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 <= 9.2e-69) {
		tmp = (l * l) * (2.0 / (t_m * Math.pow(k_m, 4.0)));
	} else {
		tmp = 2.0 / (((2.0 * Math.pow(k_m, 2.0)) / l) * (Math.pow(t_m, 3.0) / 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 <= 9.2e-69:
		tmp = (l * l) * (2.0 / (t_m * math.pow(k_m, 4.0)))
	else:
		tmp = 2.0 / (((2.0 * math.pow(k_m, 2.0)) / l) * (math.pow(t_m, 3.0) / 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 <= 9.2e-69)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t_m * (k_m ^ 4.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(2.0 * (k_m ^ 2.0)) / l) * Float64((t_m ^ 3.0) / 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 <= 9.2e-69)
		tmp = (l * l) * (2.0 / (t_m * (k_m ^ 4.0)));
	else
		tmp = 2.0 / (((2.0 * (k_m ^ 2.0)) / l) * ((t_m ^ 3.0) / 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, 9.2e-69], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.2000000000000003e-69

   1. Initial program 49.6%

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

      1. associate-/r* 49.6%

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

      2. +-commutative 49.6%

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

      3. unpow2 49.6%

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

      4. sqr-neg 49.6%

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

      5. distribute-frac-neg2 49.6%

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

      6. distribute-frac-neg2 49.6%

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

      7. unpow2 49.6%

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

      8. +-commutative 49.6%

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

      9. associate-*l* 45.6%

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

      10. associate-*l/ 43.4%

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

      11. associate-/r/ 43.3%

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

      12. +-commutative 43.3%

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

      13. associate-+r+ 43.3%

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

   3. Simplified 43.3%

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

      1. add-sqr-sqrt 26.5%

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

      2. pow2 26.5%

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

   6. Applied egg-rr 29.5%

      \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

   7. Taylor expanded in t around 0 39.9%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]
   8. Step-by-step derivation

      1. associate-/l* 39.9%

         \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2} \]

   9. Simplified 39.9%

      \[\leadsto {\color{blue}{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]

   10. Taylor expanded in k around 0 55.1%

       \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
   11. Step-by-step derivation

       1. associate-/l* 55.1%

          \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]

       2. unpow2 55.1%

          \[\leadsto {\ell}^{2} \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{{k}^{4} \cdot t} \]

       3. rem-square-sqrt 55.1%

          \[\leadsto {\ell}^{2} \cdot \frac{\color{blue}{2}}{{k}^{4} \cdot t} \]

       4. *-commutative 55.1%

          \[\leadsto {\ell}^{2} \cdot \frac{2}{\color{blue}{t \cdot {k}^{4}}} \]

   12. Simplified 55.1%

       \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{2}{t \cdot {k}^{4}}} \]
   13. Step-by-step derivation

       1. unpow2 55.1%

          \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{2}{t \cdot {k}^{4}} \]

   14. Applied egg-rr 55.1%

       \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{2}{t \cdot {k}^{4}} \]

   if 9.2000000000000003e-69 < t

   1. Initial program 55.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 56.4%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 45.0%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ 45.2%

         \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]

   7. Applied egg-rr 45.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell} \cdot \left(2 \cdot {k}^{2}\right)}{\ell}}} \]
   8. Step-by-step derivation

      1. associate-/l* 44.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]

   9. Simplified 44.9%

      \[\leadsto \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell} \cdot \frac{2 \cdot {k}^{2}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.2 \cdot 10^{-69}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot {k}^{2}}{\ell} \cdot \frac{{t}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 58.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-62}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t\_m \cdot {k\_m}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k\_m}^{2}\right) \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

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 1.6e-62)
    (* (* l l) (/ 2.0 (* t_m (pow k_m 4.0))))
    (/ 2.0 (* (* 2.0 (pow k_m 2.0)) (/ (/ (pow t_m 3.0) 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) {
	double tmp;
	if (t_m <= 1.6e-62) {
		tmp = (l * l) * (2.0 / (t_m * pow(k_m, 4.0)));
	} else {
		tmp = 2.0 / ((2.0 * pow(k_m, 2.0)) * ((pow(t_m, 3.0) / l) / 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 <= 1.6d-62) then
        tmp = (l * l) * (2.0d0 / (t_m * (k_m ** 4.0d0)))
    else
        tmp = 2.0d0 / ((2.0d0 * (k_m ** 2.0d0)) * (((t_m ** 3.0d0) / l) / 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 <= 1.6e-62) {
		tmp = (l * l) * (2.0 / (t_m * Math.pow(k_m, 4.0)));
	} else {
		tmp = 2.0 / ((2.0 * Math.pow(k_m, 2.0)) * ((Math.pow(t_m, 3.0) / l) / 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 <= 1.6e-62:
		tmp = (l * l) * (2.0 / (t_m * math.pow(k_m, 4.0)))
	else:
		tmp = 2.0 / ((2.0 * math.pow(k_m, 2.0)) * ((math.pow(t_m, 3.0) / l) / 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 <= 1.6e-62)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(t_m * (k_m ^ 4.0))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * (k_m ^ 2.0)) * Float64(Float64((t_m ^ 3.0) / l) / 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 <= 1.6e-62)
		tmp = (l * l) * (2.0 / (t_m * (k_m ^ 4.0)));
	else
		tmp = 2.0 / ((2.0 * (k_m ^ 2.0)) * (((t_m ^ 3.0) / l) / 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, 1.6e-62], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.60000000000000011e-62

   1. Initial program 49.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 49.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. +-commutative 49.6%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1} \]

      3. unpow2 49.6%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1} \]

      4. sqr-neg 49.6%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) + 1} \]

      5. distribute-frac-neg2 49.6%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\frac{k}{-t}} \cdot \left(-\frac{k}{t}\right) + 1\right) + 1} \]

      6. distribute-frac-neg2 49.6%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\frac{k}{-t} \cdot \color{blue}{\frac{k}{-t}} + 1\right) + 1} \]

      7. unpow2 49.6%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) + 1} \]

      8. +-commutative 49.6%

         \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} + 1} \]

      9. associate-*l* 45.6%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      10. associate-*l/ 43.4%

          \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      11. associate-/r/ 43.3%

          \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

      12. +-commutative 43.3%

          \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)}} \]

      13. associate-+r+ 43.3%

          \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{-t}\right)}^{2}}} \]

   3. Simplified 43.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 26.5%

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

      2. pow2 26.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}\right)}^{2}} \]

   6. Applied egg-rr 29.5%

      \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

   7. Taylor expanded in t around 0 39.9%

      \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]
   8. Step-by-step derivation

      1. associate-/l* 39.9%

         \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2} \]

   9. Simplified 39.9%

      \[\leadsto {\color{blue}{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]

   10. Taylor expanded in k around 0 55.1%

       \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
   11. Step-by-step derivation

       1. associate-/l* 55.1%

          \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]

       2. unpow2 55.1%

          \[\leadsto {\ell}^{2} \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{{k}^{4} \cdot t} \]

       3. rem-square-sqrt 55.1%

          \[\leadsto {\ell}^{2} \cdot \frac{\color{blue}{2}}{{k}^{4} \cdot t} \]

       4. *-commutative 55.1%

          \[\leadsto {\ell}^{2} \cdot \frac{2}{\color{blue}{t \cdot {k}^{4}}} \]

   12. Simplified 55.1%

       \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{2}{t \cdot {k}^{4}}} \]
   13. Step-by-step derivation

       1. unpow2 55.1%

          \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{2}{t \cdot {k}^{4}} \]

   14. Applied egg-rr 55.1%

       \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{2}{t \cdot {k}^{4}} \]

   if 1.60000000000000011e-62 < t

   1. Initial program 55.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   3. Simplified 56.4%

      \[\leadsto \color{blue}{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around 0 45.0%

      \[\leadsto \frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot \color{blue}{\left(2 \cdot {k}^{2}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-62}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 52.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{2}{t\_m \cdot {k\_m}^{4}}\right) \end{array} \]

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 (* (* l 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) {
	return t_s * ((l * l) * (2.0 / (t_m * pow(k_m, 4.0))));
}

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 * ((l * l) * (2.0d0 / (t_m * (k_m ** 4.0d0))))
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 * ((l * l) * (2.0 / (t_m * Math.pow(k_m, 4.0))));
}

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 * ((l * l) * (2.0 / (t_m * math.pow(k_m, 4.0))))

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(Float64(l * l) * Float64(2.0 / Float64(t_m * (k_m ^ 4.0)))))
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 * ((l * l) * (2.0 / (t_m * (k_m ^ 4.0))));
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[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 51.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   2. +-commutative 51.6%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + 1} \]

   3. unpow2 51.6%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}} + 1\right) + 1} \]

   4. sqr-neg 51.6%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\left(-\frac{k}{t}\right) \cdot \left(-\frac{k}{t}\right)} + 1\right) + 1} \]

   5. distribute-frac-neg2 51.6%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{\frac{k}{-t}} \cdot \left(-\frac{k}{t}\right) + 1\right) + 1} \]

   6. distribute-frac-neg2 51.6%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\frac{k}{-t} \cdot \color{blue}{\frac{k}{-t}} + 1\right) + 1} \]

   7. unpow2 51.6%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(\color{blue}{{\left(\frac{k}{-t}\right)}^{2}} + 1\right) + 1} \]

   8. +-commutative 51.6%

      \[\leadsto \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)} + 1} \]

   9. associate-*l* 47.1%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

   10. associate-*l/ 45.7%

       \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

   11. associate-/r/ 45.6%

       \[\leadsto \frac{\color{blue}{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{-t}\right)}^{2}\right) + 1} \]

   12. +-commutative 45.6%

       \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{1 + \left(1 + {\left(\frac{k}{-t}\right)}^{2}\right)}} \]

   13. associate-+r+ 45.6%

       \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{-t}\right)}^{2}}} \]

3. Simplified 45.6%

   \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 31.7%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \sqrt{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]

   2. pow2 31.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}}\right)}^{2}} \]

6. Applied egg-rr 34.2%

   \[\leadsto \color{blue}{{\left(\frac{\ell \cdot \sqrt{\frac{2}{{t}^{3} \cdot \left(\sin k \cdot \tan k\right)}}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right)}\right)}^{2}} \]

7. Taylor expanded in t around 0 38.0%

   \[\leadsto {\color{blue}{\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k} \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]
8. Step-by-step derivation

   1. associate-/l* 38.0%

      \[\leadsto {\left(\color{blue}{\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)} \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2} \]

9. Simplified 38.0%

   \[\leadsto {\color{blue}{\left(\left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}}^{2} \]

10. Taylor expanded in k around 0 49.7%

    \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]
11. Step-by-step derivation

    1. associate-/l* 49.7%

       \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{{k}^{4} \cdot t}} \]

    2. unpow2 49.7%

       \[\leadsto {\ell}^{2} \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{{k}^{4} \cdot t} \]

    3. rem-square-sqrt 49.7%

       \[\leadsto {\ell}^{2} \cdot \frac{\color{blue}{2}}{{k}^{4} \cdot t} \]

    4. *-commutative 49.7%

       \[\leadsto {\ell}^{2} \cdot \frac{2}{\color{blue}{t \cdot {k}^{4}}} \]

12. Simplified 49.7%

    \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{2}{t \cdot {k}^{4}}} \]
13. Step-by-step derivation

    1. unpow2 49.7%

       \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{2}{t \cdot {k}^{4}} \]

14. Applied egg-rr 49.7%

    \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{2}{t \cdot {k}^{4}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024117 
(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))))