Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.7% → 87.9%

Time: 21.2s

Alternatives: 24

Speedup: 35.0×

• Could not converge on a ground truth

  1. reg0 = (bf #e1.162610331442094115905709805333707687075e-23)
  2. reg1 = (bf "1.671406999580006196020136260926699063261e-171")
  3. reg2 = (bf "5.606228464049008588351230333259030756818e196")

• 40.4% 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: 35.7% 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: 87.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sqrt{l\_m}}\\ t_3 := \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.48:\\ \;\;\;\;{\left(\left(\frac{l\_m}{k\_m} \cdot \frac{\sqrt{2}}{\sin k\_m}\right) \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot \frac{\sqrt{2}}{k\_m}}{{\left(\frac{t\_m}{{\left(t\_2 \cdot t\_2\right)}^{2}} \cdot t\_3\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2} \cdot t\_m}{k\_m}}{t\_m} \cdot {\left(\sqrt[3]{l\_m}\right)}^{2}}{t\_3}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (let* ((t_2 (cbrt (sqrt l_m))) (t_3 (cbrt (* (sin k_m) (tan k_m)))))
   (*
    t_s
    (if (<= k_m 0.48)
      (pow
       (* (* (/ l_m k_m) (/ (sqrt 2.0) (sin k_m))) (sqrt (/ (cos k_m) t_m)))
       2.0)
      (*
       (/
        (* t_m (/ (sqrt 2.0) k_m))
        (pow (* (/ t_m (pow (* t_2 t_2) 2.0)) t_3) 2.0))
       (/ (* (/ (/ (* (sqrt 2.0) t_m) k_m) t_m) (pow (cbrt l_m) 2.0)) t_3))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = cbrt(sqrt(l_m));
	double t_3 = cbrt((sin(k_m) * tan(k_m)));
	double tmp;
	if (k_m <= 0.48) {
		tmp = pow((((l_m / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = ((t_m * (sqrt(2.0) / k_m)) / pow(((t_m / pow((t_2 * t_2), 2.0)) * t_3), 2.0)) * (((((sqrt(2.0) * t_m) / k_m) / t_m) * pow(cbrt(l_m), 2.0)) / t_3);
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double t_2 = Math.cbrt(Math.sqrt(l_m));
	double t_3 = Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double tmp;
	if (k_m <= 0.48) {
		tmp = Math.pow((((l_m / k_m) * (Math.sqrt(2.0) / Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = ((t_m * (Math.sqrt(2.0) / k_m)) / Math.pow(((t_m / Math.pow((t_2 * t_2), 2.0)) * t_3), 2.0)) * (((((Math.sqrt(2.0) * t_m) / k_m) / t_m) * Math.pow(Math.cbrt(l_m), 2.0)) / t_3);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	t_2 = cbrt(sqrt(l_m))
	t_3 = cbrt(Float64(sin(k_m) * tan(k_m)))
	tmp = 0.0
	if (k_m <= 0.48)
		tmp = Float64(Float64(Float64(l_m / k_m) * Float64(sqrt(2.0) / sin(k_m))) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(t_m * Float64(sqrt(2.0) / k_m)) / (Float64(Float64(t_m / (Float64(t_2 * t_2) ^ 2.0)) * t_3) ^ 2.0)) * Float64(Float64(Float64(Float64(Float64(sqrt(2.0) * t_m) / k_m) / t_m) * (cbrt(l_m) ^ 2.0)) / t_3));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := Block[{t$95$2 = N[Power[N[Sqrt[l$95$m], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 0.48], N[Power[N[(N[(N[(l$95$m / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[(t$95$m / N[Power[N[(t$95$2 * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.47999999999999998

   1. Initial program 37.9%

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

   3. Simplified 42.7%

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

      1. add-sqr-sqrt 23.8%

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

      2. pow2 23.8%

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

   6. Applied egg-rr 25.9%

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

      1. associate-*r/ 25.8%

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

      2. *-commutative 25.8%

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

      3. times-frac 26.4%

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

   8. Simplified 26.4%

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

   9. Taylor expanded in l around 0 46.3%

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

       1. times-frac 48.3%

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

   11. Simplified 48.3%

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

   if 0.47999999999999998 < k

   1. Initial program 32.7%

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

   3. Simplified 32.7%

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

   6. Applied egg-rr 75.8%

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

      1. associate-/r/ 75.8%

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

      2. associate-/r* 75.8%

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

      3. associate-/r/ 77.2%

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

   8. Simplified 77.2%

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

      1. pow1/3 38.6%

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

      2. add-sqr-sqrt 38.6%

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

      3. unpow-prod-down 38.6%

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

   10. Applied egg-rr 38.6%

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

       1. unpow1/3 38.8%

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

       2. unpow1/3 39.3%

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

   12. Simplified 39.3%

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

       1. associate-/r/ 39.4%

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

       2. associate-*l/ 39.4%

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

   14. Applied egg-rr 39.4%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.48:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(\frac{t}{{\left(\sqrt[3]{\sqrt{\ell}} \cdot \sqrt[3]{\sqrt{\ell}}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\frac{\sqrt{2} \cdot t}{k}}{t} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sqrt{l\_m}}\\ t_3 := \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t_4 := \frac{\sqrt{2}}{k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.48:\\ \;\;\;\;{\left(\left(\frac{l\_m}{k\_m} \cdot \frac{\sqrt{2}}{\sin k\_m}\right) \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot t\_4}{{\left(\frac{t\_m}{{\left(t\_2 \cdot t\_2\right)}^{2}} \cdot t\_3\right)}^{2}} \cdot \frac{t\_4 \cdot \frac{1}{{\left(\sqrt[3]{l\_m}\right)}^{-2}}}{t\_3}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (let* ((t_2 (cbrt (sqrt l_m)))
        (t_3 (cbrt (* (sin k_m) (tan k_m))))
        (t_4 (/ (sqrt 2.0) k_m)))
   (*
    t_s
    (if (<= k_m 0.48)
      (pow
       (* (* (/ l_m k_m) (/ (sqrt 2.0) (sin k_m))) (sqrt (/ (cos k_m) t_m)))
       2.0)
      (*
       (/ (* t_m t_4) (pow (* (/ t_m (pow (* t_2 t_2) 2.0)) t_3) 2.0))
       (/ (* t_4 (/ 1.0 (pow (cbrt l_m) -2.0))) t_3))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = cbrt(sqrt(l_m));
	double t_3 = cbrt((sin(k_m) * tan(k_m)));
	double t_4 = sqrt(2.0) / k_m;
	double tmp;
	if (k_m <= 0.48) {
		tmp = pow((((l_m / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = ((t_m * t_4) / pow(((t_m / pow((t_2 * t_2), 2.0)) * t_3), 2.0)) * ((t_4 * (1.0 / pow(cbrt(l_m), -2.0))) / t_3);
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double t_2 = Math.cbrt(Math.sqrt(l_m));
	double t_3 = Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double t_4 = Math.sqrt(2.0) / k_m;
	double tmp;
	if (k_m <= 0.48) {
		tmp = Math.pow((((l_m / k_m) * (Math.sqrt(2.0) / Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = ((t_m * t_4) / Math.pow(((t_m / Math.pow((t_2 * t_2), 2.0)) * t_3), 2.0)) * ((t_4 * (1.0 / Math.pow(Math.cbrt(l_m), -2.0))) / t_3);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	t_2 = cbrt(sqrt(l_m))
	t_3 = cbrt(Float64(sin(k_m) * tan(k_m)))
	t_4 = Float64(sqrt(2.0) / k_m)
	tmp = 0.0
	if (k_m <= 0.48)
		tmp = Float64(Float64(Float64(l_m / k_m) * Float64(sqrt(2.0) / sin(k_m))) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(t_m * t_4) / (Float64(Float64(t_m / (Float64(t_2 * t_2) ^ 2.0)) * t_3) ^ 2.0)) * Float64(Float64(t_4 * Float64(1.0 / (cbrt(l_m) ^ -2.0))) / t_3));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := Block[{t$95$2 = N[Power[N[Sqrt[l$95$m], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 0.48], N[Power[N[(N[(N[(l$95$m / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(t$95$m * t$95$4), $MachinePrecision] / N[Power[N[(N[(t$95$m / N[Power[N[(t$95$2 * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$4 * N[(1.0 / N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.47999999999999998

   1. Initial program 37.9%

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

   3. Simplified 42.7%

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

      1. add-sqr-sqrt 23.8%

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

      2. pow2 23.8%

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

   6. Applied egg-rr 25.9%

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

      1. associate-*r/ 25.8%

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

      2. *-commutative 25.8%

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

      3. times-frac 26.4%

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

   8. Simplified 26.4%

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

   9. Taylor expanded in l around 0 46.3%

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

       1. times-frac 48.3%

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

   11. Simplified 48.3%

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

   if 0.47999999999999998 < k

   1. Initial program 32.7%

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

   3. Simplified 32.7%

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

   6. Applied egg-rr 75.8%

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

      1. associate-/r/ 75.8%

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

      2. associate-/r* 75.8%

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

      3. associate-/r/ 77.2%

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

   8. Simplified 77.2%

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

      1. pow1/3 38.6%

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

      2. add-sqr-sqrt 38.6%

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

      3. unpow-prod-down 38.6%

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

   10. Applied egg-rr 38.6%

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

       1. unpow1/3 38.8%

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

       2. unpow1/3 39.3%

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

   12. Simplified 39.3%

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

       1. associate-/l* 39.4%

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

       2. div-inv 39.4%

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

       3. pow-flip 39.4%

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

       4. metadata-eval 39.4%

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

   14. Applied egg-rr 39.4%

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

       1. associate-/r* 39.4%

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

       2. *-inverses 39.4%

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

   16. Simplified 39.4%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.48:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(\frac{t}{{\left(\sqrt[3]{\sqrt{\ell}} \cdot \sqrt[3]{\sqrt{\ell}}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{k} \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.48:\\ \;\;\;\;{\left(\left(\frac{l\_m}{k\_m} \cdot \frac{\sqrt{2}}{\sin k\_m}\right) \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot \frac{\sqrt{2}}{k\_m}}{{\left(t\_2 \cdot \frac{t\_m}{{\left(\sqrt[3]{l\_m}\right)}^{2}}\right)}^{2}} \cdot {\left(\sqrt[3]{\frac{\frac{\sqrt{2} \cdot t\_m}{k\_m}}{t\_2 \cdot \left(t\_m \cdot {\left(\sqrt[3]{l\_m}\right)}^{-2}\right)}}\right)}^{3}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (let* ((t_2 (cbrt (* (sin k_m) (tan k_m)))))
   (*
    t_s
    (if (<= k_m 0.48)
      (pow
       (* (* (/ l_m k_m) (/ (sqrt 2.0) (sin k_m))) (sqrt (/ (cos k_m) t_m)))
       2.0)
      (*
       (/
        (* t_m (/ (sqrt 2.0) k_m))
        (pow (* t_2 (/ t_m (pow (cbrt l_m) 2.0))) 2.0))
       (pow
        (cbrt
         (/ (/ (* (sqrt 2.0) t_m) k_m) (* t_2 (* t_m (pow (cbrt l_m) -2.0)))))
        3.0))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = cbrt((sin(k_m) * tan(k_m)));
	double tmp;
	if (k_m <= 0.48) {
		tmp = pow((((l_m / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = ((t_m * (sqrt(2.0) / k_m)) / pow((t_2 * (t_m / pow(cbrt(l_m), 2.0))), 2.0)) * pow(cbrt((((sqrt(2.0) * t_m) / k_m) / (t_2 * (t_m * pow(cbrt(l_m), -2.0))))), 3.0);
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double t_2 = Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double tmp;
	if (k_m <= 0.48) {
		tmp = Math.pow((((l_m / k_m) * (Math.sqrt(2.0) / Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = ((t_m * (Math.sqrt(2.0) / k_m)) / Math.pow((t_2 * (t_m / Math.pow(Math.cbrt(l_m), 2.0))), 2.0)) * Math.pow(Math.cbrt((((Math.sqrt(2.0) * t_m) / k_m) / (t_2 * (t_m * Math.pow(Math.cbrt(l_m), -2.0))))), 3.0);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	t_2 = cbrt(Float64(sin(k_m) * tan(k_m)))
	tmp = 0.0
	if (k_m <= 0.48)
		tmp = Float64(Float64(Float64(l_m / k_m) * Float64(sqrt(2.0) / sin(k_m))) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(t_m * Float64(sqrt(2.0) / k_m)) / (Float64(t_2 * Float64(t_m / (cbrt(l_m) ^ 2.0))) ^ 2.0)) * (cbrt(Float64(Float64(Float64(sqrt(2.0) * t_m) / k_m) / Float64(t_2 * Float64(t_m * (cbrt(l_m) ^ -2.0))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := Block[{t$95$2 = N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 0.48], N[Power[N[(N[(N[(l$95$m / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[N[(t$95$2 * N[(t$95$m / N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] / N[(t$95$2 * N[(t$95$m * N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.47999999999999998

   1. Initial program 37.9%

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

   3. Simplified 42.7%

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

      1. add-sqr-sqrt 23.8%

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

      2. pow2 23.8%

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

   6. Applied egg-rr 25.9%

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

      1. associate-*r/ 25.8%

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

      2. *-commutative 25.8%

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

      3. times-frac 26.4%

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

   8. Simplified 26.4%

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

   9. Taylor expanded in l around 0 46.3%

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

       1. times-frac 48.3%

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

   11. Simplified 48.3%

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

   if 0.47999999999999998 < k

   1. Initial program 32.7%

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

   3. Simplified 32.7%

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

   6. Applied egg-rr 75.8%

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

      1. associate-/r/ 75.8%

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

      2. associate-/r* 75.8%

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

      3. associate-/r/ 77.2%

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

   8. Simplified 77.2%

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

      1. add-cube-cbrt 77.2%

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

      2. pow3 77.2%

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

      3. associate-/l/ 77.2%

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

      4. associate-*l/ 77.2%

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

      5. *-commutative 77.2%

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

      6. div-inv 77.2%

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

      7. pow-flip 77.3%

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

      8. metadata-eval 77.3%

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

   10. Applied egg-rr 77.3%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.48:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot {\left(\sqrt[3]{\frac{\frac{\sqrt{2} \cdot t}{k}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{l\_m}\right)}^{2}\\ t_3 := \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.48:\\ \;\;\;\;{\left(\left(\frac{l\_m}{k\_m} \cdot \frac{\sqrt{2}}{\sin k\_m}\right) \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot \frac{\sqrt{2}}{k\_m}}{{\left(t\_3 \cdot \frac{t\_m}{t\_2}\right)}^{2}} \cdot \frac{{\left(\sqrt[3]{\frac{\frac{\sqrt{2} \cdot t\_m}{k\_m}}{t\_m} \cdot t\_2}\right)}^{3}}{t\_3}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (let* ((t_2 (pow (cbrt l_m) 2.0)) (t_3 (cbrt (* (sin k_m) (tan k_m)))))
   (*
    t_s
    (if (<= k_m 0.48)
      (pow
       (* (* (/ l_m k_m) (/ (sqrt 2.0) (sin k_m))) (sqrt (/ (cos k_m) t_m)))
       2.0)
      (*
       (/ (* t_m (/ (sqrt 2.0) k_m)) (pow (* t_3 (/ t_m t_2)) 2.0))
       (/ (pow (cbrt (* (/ (/ (* (sqrt 2.0) t_m) k_m) t_m) t_2)) 3.0) t_3))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = pow(cbrt(l_m), 2.0);
	double t_3 = cbrt((sin(k_m) * tan(k_m)));
	double tmp;
	if (k_m <= 0.48) {
		tmp = pow((((l_m / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = ((t_m * (sqrt(2.0) / k_m)) / pow((t_3 * (t_m / t_2)), 2.0)) * (pow(cbrt(((((sqrt(2.0) * t_m) / k_m) / t_m) * t_2)), 3.0) / t_3);
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double t_2 = Math.pow(Math.cbrt(l_m), 2.0);
	double t_3 = Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double tmp;
	if (k_m <= 0.48) {
		tmp = Math.pow((((l_m / k_m) * (Math.sqrt(2.0) / Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = ((t_m * (Math.sqrt(2.0) / k_m)) / Math.pow((t_3 * (t_m / t_2)), 2.0)) * (Math.pow(Math.cbrt(((((Math.sqrt(2.0) * t_m) / k_m) / t_m) * t_2)), 3.0) / t_3);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	t_2 = cbrt(l_m) ^ 2.0
	t_3 = cbrt(Float64(sin(k_m) * tan(k_m)))
	tmp = 0.0
	if (k_m <= 0.48)
		tmp = Float64(Float64(Float64(l_m / k_m) * Float64(sqrt(2.0) / sin(k_m))) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(t_m * Float64(sqrt(2.0) / k_m)) / (Float64(t_3 * Float64(t_m / t_2)) ^ 2.0)) * Float64((cbrt(Float64(Float64(Float64(Float64(sqrt(2.0) * t_m) / k_m) / t_m) * t_2)) ^ 3.0) / t_3));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := Block[{t$95$2 = N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 0.48], N[Power[N[(N[(N[(l$95$m / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[N[(t$95$3 * N[(t$95$m / t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Power[N[(N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.47999999999999998

   1. Initial program 37.9%

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

   3. Simplified 42.7%

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

      1. add-sqr-sqrt 23.8%

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

      2. pow2 23.8%

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

   6. Applied egg-rr 25.9%

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

      1. associate-*r/ 25.8%

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

      2. *-commutative 25.8%

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

      3. times-frac 26.4%

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

   8. Simplified 26.4%

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

   9. Taylor expanded in l around 0 46.3%

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

       1. times-frac 48.3%

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

   11. Simplified 48.3%

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

   if 0.47999999999999998 < k

   1. Initial program 32.7%

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

   3. Simplified 32.7%

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

   6. Applied egg-rr 75.8%

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

      1. associate-/r/ 75.8%

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

      2. associate-/r* 75.8%

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

      3. associate-/r/ 77.2%

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

   8. Simplified 77.2%

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

      1. add-cube-cbrt 77.3%

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

      2. pow3 77.3%

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

      3. associate-/r/ 77.3%

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

      4. associate-*l/ 77.3%

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

   10. Applied egg-rr 77.3%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.48:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{{\left(\sqrt[3]{\frac{\frac{\sqrt{2} \cdot t}{k}}{t} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{\sqrt{2}}{k\_m}\\ t_3 := \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 0.48:\\ \;\;\;\;{\left(\left(\frac{l\_m}{k\_m} \cdot \frac{\sqrt{2}}{\sin k\_m}\right) \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{{\left(t\_3 \cdot \frac{t\_m}{{\left(\sqrt[3]{l\_m}\right)}^{2}}\right)}^{2}} \cdot \frac{\frac{t\_2}{\frac{t\_m}{{\left({\left(\sqrt[3]{\sqrt[3]{l\_m}}\right)}^{3}\right)}^{2}}}}{t\_3}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (let* ((t_2 (* t_m (/ (sqrt 2.0) k_m))) (t_3 (cbrt (* (sin k_m) (tan k_m)))))
   (*
    t_s
    (if (<= k_m 0.48)
      (pow
       (* (* (/ l_m k_m) (/ (sqrt 2.0) (sin k_m))) (sqrt (/ (cos k_m) t_m)))
       2.0)
      (*
       (/ t_2 (pow (* t_3 (/ t_m (pow (cbrt l_m) 2.0))) 2.0))
       (/ (/ t_2 (/ t_m (pow (pow (cbrt (cbrt l_m)) 3.0) 2.0))) t_3))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = t_m * (sqrt(2.0) / k_m);
	double t_3 = cbrt((sin(k_m) * tan(k_m)));
	double tmp;
	if (k_m <= 0.48) {
		tmp = pow((((l_m / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = (t_2 / pow((t_3 * (t_m / pow(cbrt(l_m), 2.0))), 2.0)) * ((t_2 / (t_m / pow(pow(cbrt(cbrt(l_m)), 3.0), 2.0))) / t_3);
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double t_2 = t_m * (Math.sqrt(2.0) / k_m);
	double t_3 = Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double tmp;
	if (k_m <= 0.48) {
		tmp = Math.pow((((l_m / k_m) * (Math.sqrt(2.0) / Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = (t_2 / Math.pow((t_3 * (t_m / Math.pow(Math.cbrt(l_m), 2.0))), 2.0)) * ((t_2 / (t_m / Math.pow(Math.pow(Math.cbrt(Math.cbrt(l_m)), 3.0), 2.0))) / t_3);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	t_2 = Float64(t_m * Float64(sqrt(2.0) / k_m))
	t_3 = cbrt(Float64(sin(k_m) * tan(k_m)))
	tmp = 0.0
	if (k_m <= 0.48)
		tmp = Float64(Float64(Float64(l_m / k_m) * Float64(sqrt(2.0) / sin(k_m))) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(t_2 / (Float64(t_3 * Float64(t_m / (cbrt(l_m) ^ 2.0))) ^ 2.0)) * Float64(Float64(t_2 / Float64(t_m / ((cbrt(cbrt(l_m)) ^ 3.0) ^ 2.0))) / t_3));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := Block[{t$95$2 = N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 0.48], N[Power[N[(N[(N[(l$95$m / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(t$95$2 / N[Power[N[(t$95$3 * N[(t$95$m / N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 / N[(t$95$m / N[Power[N[Power[N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 0.47999999999999998

   1. Initial program 37.9%

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

   3. Simplified 42.7%

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

      1. add-sqr-sqrt 23.8%

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

      2. pow2 23.8%

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

   6. Applied egg-rr 25.9%

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

      1. associate-*r/ 25.8%

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

      2. *-commutative 25.8%

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

      3. times-frac 26.4%

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

   8. Simplified 26.4%

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

   9. Taylor expanded in l around 0 46.3%

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

       1. times-frac 48.3%

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

   11. Simplified 48.3%

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

   if 0.47999999999999998 < k

   1. Initial program 32.7%

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

   3. Simplified 32.7%

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

   6. Applied egg-rr 75.8%

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

      1. associate-/r/ 75.8%

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

      2. associate-/r* 75.8%

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

      3. associate-/r/ 77.2%

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

   8. Simplified 77.2%

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

      1. add-cube-cbrt 77.2%

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

      2. pow3 77.2%

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

   10. Applied egg-rr 77.2%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.48:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{\frac{t \cdot \frac{\sqrt{2}}{k}}{\frac{t}{{\left({\left(\sqrt[3]{\sqrt[3]{\ell}}\right)}^{3}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{{\left(\sqrt[3]{l\_m}\right)}^{2}}\\ t_3 := t\_m \cdot \frac{\sqrt{2}}{k\_m}\\ t_4 := \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 5.4 \cdot 10^{-6}:\\ \;\;\;\;{\left(\left(\frac{l\_m}{k\_m} \cdot \frac{\sqrt{2}}{\sin k\_m}\right) \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{{\left(t\_4 \cdot t\_2\right)}^{2}} \cdot \frac{\frac{t\_3}{t\_2}}{t\_4}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (let* ((t_2 (/ t_m (pow (cbrt l_m) 2.0)))
        (t_3 (* t_m (/ (sqrt 2.0) k_m)))
        (t_4 (cbrt (* (sin k_m) (tan k_m)))))
   (*
    t_s
    (if (<= k_m 5.4e-6)
      (pow
       (* (* (/ l_m k_m) (/ (sqrt 2.0) (sin k_m))) (sqrt (/ (cos k_m) t_m)))
       2.0)
      (* (/ t_3 (pow (* t_4 t_2) 2.0)) (/ (/ t_3 t_2) t_4))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = t_m / pow(cbrt(l_m), 2.0);
	double t_3 = t_m * (sqrt(2.0) / k_m);
	double t_4 = cbrt((sin(k_m) * tan(k_m)));
	double tmp;
	if (k_m <= 5.4e-6) {
		tmp = pow((((l_m / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = (t_3 / pow((t_4 * t_2), 2.0)) * ((t_3 / t_2) / t_4);
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double t_2 = t_m / Math.pow(Math.cbrt(l_m), 2.0);
	double t_3 = t_m * (Math.sqrt(2.0) / k_m);
	double t_4 = Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double tmp;
	if (k_m <= 5.4e-6) {
		tmp = Math.pow((((l_m / k_m) * (Math.sqrt(2.0) / Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = (t_3 / Math.pow((t_4 * t_2), 2.0)) * ((t_3 / t_2) / t_4);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	t_2 = Float64(t_m / (cbrt(l_m) ^ 2.0))
	t_3 = Float64(t_m * Float64(sqrt(2.0) / k_m))
	t_4 = cbrt(Float64(sin(k_m) * tan(k_m)))
	tmp = 0.0
	if (k_m <= 5.4e-6)
		tmp = Float64(Float64(Float64(l_m / k_m) * Float64(sqrt(2.0) / sin(k_m))) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(t_3 / (Float64(t_4 * t_2) ^ 2.0)) * Float64(Float64(t_3 / t_2) / t_4));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := Block[{t$95$2 = N[(t$95$m / N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 5.4e-6], N[Power[N[(N[(N[(l$95$m / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(t$95$3 / N[Power[N[(t$95$4 * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$3 / t$95$2), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.39999999999999997e-6

   1. Initial program 37.9%

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

   3. Simplified 42.7%

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

      1. add-sqr-sqrt 23.8%

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

      2. pow2 23.8%

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

   6. Applied egg-rr 25.9%

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

      1. associate-*r/ 25.8%

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

      2. *-commutative 25.8%

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

      3. times-frac 26.4%

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

   8. Simplified 26.4%

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

   9. Taylor expanded in l around 0 46.3%

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

       1. times-frac 48.3%

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

   11. Simplified 48.3%

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

   if 5.39999999999999997e-6 < k

   1. Initial program 32.7%

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

   3. Simplified 32.7%

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

   6. Applied egg-rr 75.8%

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

      1. associate-/r/ 75.8%

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

      2. associate-/r* 75.8%

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

      3. associate-/r/ 77.2%

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

   8. Simplified 77.2%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.4 \cdot 10^{-6}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{\frac{t \cdot \frac{\sqrt{2}}{k}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t_3 := \frac{\sqrt{2}}{k\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 5.4 \cdot 10^{-6}:\\ \;\;\;\;{\left(\left(\frac{l\_m}{k\_m} \cdot \frac{\sqrt{2}}{\sin k\_m}\right) \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3 \cdot \frac{1}{{\left(\sqrt[3]{l\_m}\right)}^{-2}}}{t\_2} \cdot \frac{t\_m \cdot t\_3}{{\left(t\_2 \cdot \frac{t\_m}{{\left(\sqrt[3]{l\_m}\right)}^{2}}\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (let* ((t_2 (cbrt (* (sin k_m) (tan k_m)))) (t_3 (/ (sqrt 2.0) k_m)))
   (*
    t_s
    (if (<= k_m 5.4e-6)
      (pow
       (* (* (/ l_m k_m) (/ (sqrt 2.0) (sin k_m))) (sqrt (/ (cos k_m) t_m)))
       2.0)
      (*
       (/ (* t_3 (/ 1.0 (pow (cbrt l_m) -2.0))) t_2)
       (/ (* t_m t_3) (pow (* t_2 (/ t_m (pow (cbrt l_m) 2.0))) 2.0)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = cbrt((sin(k_m) * tan(k_m)));
	double t_3 = sqrt(2.0) / k_m;
	double tmp;
	if (k_m <= 5.4e-6) {
		tmp = pow((((l_m / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = ((t_3 * (1.0 / pow(cbrt(l_m), -2.0))) / t_2) * ((t_m * t_3) / pow((t_2 * (t_m / pow(cbrt(l_m), 2.0))), 2.0));
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double t_2 = Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double t_3 = Math.sqrt(2.0) / k_m;
	double tmp;
	if (k_m <= 5.4e-6) {
		tmp = Math.pow((((l_m / k_m) * (Math.sqrt(2.0) / Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = ((t_3 * (1.0 / Math.pow(Math.cbrt(l_m), -2.0))) / t_2) * ((t_m * t_3) / Math.pow((t_2 * (t_m / Math.pow(Math.cbrt(l_m), 2.0))), 2.0));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	t_2 = cbrt(Float64(sin(k_m) * tan(k_m)))
	t_3 = Float64(sqrt(2.0) / k_m)
	tmp = 0.0
	if (k_m <= 5.4e-6)
		tmp = Float64(Float64(Float64(l_m / k_m) * Float64(sqrt(2.0) / sin(k_m))) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(t_3 * Float64(1.0 / (cbrt(l_m) ^ -2.0))) / t_2) * Float64(Float64(t_m * t_3) / (Float64(t_2 * Float64(t_m / (cbrt(l_m) ^ 2.0))) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := Block[{t$95$2 = N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 5.4e-6], N[Power[N[(N[(N[(l$95$m / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(t$95$3 * N[(1.0 / N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[(t$95$m * t$95$3), $MachinePrecision] / N[Power[N[(t$95$2 * N[(t$95$m / N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.39999999999999997e-6

   1. Initial program 37.9%

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

   3. Simplified 42.7%

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

      1. add-sqr-sqrt 23.8%

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

      2. pow2 23.8%

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

   6. Applied egg-rr 25.9%

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

      1. associate-*r/ 25.8%

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

      2. *-commutative 25.8%

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

      3. times-frac 26.4%

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

   8. Simplified 26.4%

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

   9. Taylor expanded in l around 0 46.3%

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

       1. times-frac 48.3%

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

   11. Simplified 48.3%

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

   if 5.39999999999999997e-6 < k

   1. Initial program 32.7%

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

   3. Simplified 32.7%

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

   6. Applied egg-rr 75.8%

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

      1. associate-/r/ 75.8%

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

      2. associate-/r* 75.8%

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

      3. associate-/r/ 77.2%

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

   8. Simplified 77.2%

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

      1. associate-/l* 39.4%

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

      2. div-inv 39.4%

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

      3. pow-flip 39.4%

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

      4. metadata-eval 39.4%

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

   10. Applied egg-rr 77.3%

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

       1. associate-/r* 39.4%

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

       2. *-inverses 39.4%

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

   12. Simplified 77.3%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.4 \cdot 10^{-6}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{k} \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2}}}{\sqrt[3]{\sin k \cdot \tan k}} \cdot \frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k\_m \cdot \tan k\_m}\\ t_3 := {\left(\sqrt[3]{l\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 5.4 \cdot 10^{-6}:\\ \;\;\;\;{\left(\left(\frac{l\_m}{k\_m} \cdot \frac{\sqrt{2}}{\sin k\_m}\right) \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{k\_m} \cdot \frac{\frac{\sqrt{2}}{k\_m} \cdot t\_3}{t\_2 \cdot {\left(t\_2 \cdot \frac{t\_m}{t\_3}\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (let* ((t_2 (cbrt (* (sin k_m) (tan k_m)))) (t_3 (pow (cbrt l_m) 2.0)))
   (*
    t_s
    (if (<= k_m 5.4e-6)
      (pow
       (* (* (/ l_m k_m) (/ (sqrt 2.0) (sin k_m))) (sqrt (/ (cos k_m) t_m)))
       2.0)
      (*
       (/ (* (sqrt 2.0) t_m) k_m)
       (/
        (* (/ (sqrt 2.0) k_m) t_3)
        (* t_2 (pow (* t_2 (/ t_m t_3)) 2.0))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = cbrt((sin(k_m) * tan(k_m)));
	double t_3 = pow(cbrt(l_m), 2.0);
	double tmp;
	if (k_m <= 5.4e-6) {
		tmp = pow((((l_m / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = ((sqrt(2.0) * t_m) / k_m) * (((sqrt(2.0) / k_m) * t_3) / (t_2 * pow((t_2 * (t_m / t_3)), 2.0)));
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double t_2 = Math.cbrt((Math.sin(k_m) * Math.tan(k_m)));
	double t_3 = Math.pow(Math.cbrt(l_m), 2.0);
	double tmp;
	if (k_m <= 5.4e-6) {
		tmp = Math.pow((((l_m / k_m) * (Math.sqrt(2.0) / Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = ((Math.sqrt(2.0) * t_m) / k_m) * (((Math.sqrt(2.0) / k_m) * t_3) / (t_2 * Math.pow((t_2 * (t_m / t_3)), 2.0)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	t_2 = cbrt(Float64(sin(k_m) * tan(k_m)))
	t_3 = cbrt(l_m) ^ 2.0
	tmp = 0.0
	if (k_m <= 5.4e-6)
		tmp = Float64(Float64(Float64(l_m / k_m) * Float64(sqrt(2.0) / sin(k_m))) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) * t_m) / k_m) * Float64(Float64(Float64(sqrt(2.0) / k_m) * t_3) / Float64(t_2 * (Float64(t_2 * Float64(t_m / t_3)) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := Block[{t$95$2 = N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 5.4e-6], N[Power[N[(N[(N[(l$95$m / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / k$95$m), $MachinePrecision] * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision] * t$95$3), $MachinePrecision] / N[(t$95$2 * N[Power[N[(t$95$2 * N[(t$95$m / t$95$3), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.39999999999999997e-6

   1. Initial program 37.9%

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

   3. Simplified 42.7%

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

      1. add-sqr-sqrt 23.8%

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

      2. pow2 23.8%

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

   6. Applied egg-rr 25.9%

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

      1. associate-*r/ 25.8%

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

      2. *-commutative 25.8%

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

      3. times-frac 26.4%

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

   8. Simplified 26.4%

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

   9. Taylor expanded in l around 0 46.3%

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

       1. times-frac 48.3%

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

   11. Simplified 48.3%

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

   if 5.39999999999999997e-6 < k

   1. Initial program 32.7%

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

   3. Simplified 32.7%

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

   6. Applied egg-rr 75.8%

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

      1. associate-/r/ 75.8%

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

      2. associate-/r* 75.8%

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

      3. associate-/r/ 77.2%

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

   8. Simplified 77.2%

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

      1. pow1/3 38.6%

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

      2. add-sqr-sqrt 38.6%

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

      3. unpow-prod-down 38.6%

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

   10. Applied egg-rr 38.6%

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

       1. unpow1/3 38.8%

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

       2. unpow1/3 39.3%

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

   12. Simplified 39.3%

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

       1. associate-/l* 39.4%

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

       2. div-inv 39.4%

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

       3. pow-flip 39.4%

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

       4. metadata-eval 39.4%

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

   14. Applied egg-rr 39.4%

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

       1. associate-/r* 39.4%

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

       2. *-inverses 39.4%

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

   16. Simplified 39.4%

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

       1. frac-times 36.7%

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

       2. pow-flip 36.7%

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

       3. metadata-eval 36.7%

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

   18. Applied egg-rr 70.6%

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

       1. associate-/l* 77.2%

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

       2. associate-*l/ 77.2%

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

       3. *-commutative 77.2%

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

   20. Simplified 77.2%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.4 \cdot 10^{-6}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{k} \cdot \frac{\frac{\sqrt{2}}{k} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k \cdot \tan k} \cdot {\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k\_m}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k\_m \cdot \left(\sin k\_m \cdot \frac{{t\_m}^{3}}{l\_m \cdot l\_m}\right)\right) \cdot \left(\left(1 + t\_2\right) + -1\right)} \leq 0:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{l\_m}\right)}^{2}} \cdot \sqrt[3]{\sin k\_m \cdot \left(\tan k\_m \cdot t\_2\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{l\_m}{k\_m} \cdot \frac{\sqrt{2}}{\sin k\_m}\right) \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (let* ((t_2 (pow (/ k_m t_m) 2.0)))
   (*
    t_s
    (if (<=
         (/
          2.0
          (*
           (* (tan k_m) (* (sin k_m) (/ (pow t_m 3.0) (* l_m l_m))))
           (+ (+ 1.0 t_2) -1.0)))
         0.0)
      (/
       2.0
       (pow
        (* (/ t_m (pow (cbrt l_m) 2.0)) (cbrt (* (sin k_m) (* (tan k_m) t_2))))
        3.0))
      (pow
       (* (* (/ l_m k_m) (/ (sqrt 2.0) (sin k_m))) (sqrt (/ (cos k_m) t_m)))
       2.0)))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = pow((k_m / t_m), 2.0);
	double tmp;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * (pow(t_m, 3.0) / (l_m * l_m)))) * ((1.0 + t_2) + -1.0))) <= 0.0) {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l_m), 2.0)) * cbrt((sin(k_m) * (tan(k_m) * t_2)))), 3.0);
	} else {
		tmp = pow((((l_m / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double t_2 = Math.pow((k_m / t_m), 2.0);
	double tmp;
	if ((2.0 / ((Math.tan(k_m) * (Math.sin(k_m) * (Math.pow(t_m, 3.0) / (l_m * l_m)))) * ((1.0 + t_2) + -1.0))) <= 0.0) {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l_m), 2.0)) * Math.cbrt((Math.sin(k_m) * (Math.tan(k_m) * t_2)))), 3.0);
	} else {
		tmp = Math.pow((((l_m / k_m) * (Math.sqrt(2.0) / Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	}
	return t_s * tmp;
}

Julia

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := Block[{t$95$2 = N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + t$95$2), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Tan[k$95$m], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(l$95$m / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 0.0

   1. Initial program 87.7%

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

   3. Simplified 87.8%

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

      1. add-cube-cbrt 87.7%

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

      2. pow3 87.7%

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

   6. Applied egg-rr 98.1%

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

   if 0.0 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 14.4%

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

   3. Simplified 21.2%

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

      1. add-sqr-sqrt 19.0%

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

      2. pow2 19.0%

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

   6. Applied egg-rr 25.1%

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

      1. associate-*r/ 25.0%

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

      2. *-commutative 25.0%

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

      3. times-frac 25.6%

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

   8. Simplified 25.6%

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

   9. Taylor expanded in l around 0 55.6%

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

       1. times-frac 57.7%

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

   11. Simplified 57.7%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + -1\right)} \leq 0:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k\_m}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k\_m \cdot \left(\sin k\_m \cdot \frac{{t\_m}^{3}}{l\_m \cdot l\_m}\right)\right) \cdot \left(\left(1 + t\_2\right) + -1\right)} \leq 0:\\ \;\;\;\;\frac{2}{\left(\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(\frac{{t\_m}^{2}}{l\_m} \cdot \frac{t\_m}{l\_m}\right)\right) \cdot \left(1 + \left(t\_2 + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{l\_m}{k\_m} \cdot \frac{\sqrt{2}}{\sin k\_m}\right) \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (let* ((t_2 (pow (/ k_m t_m) 2.0)))
   (*
    t_s
    (if (<=
         (/
          2.0
          (*
           (* (tan k_m) (* (sin k_m) (/ (pow t_m 3.0) (* l_m l_m))))
           (+ (+ 1.0 t_2) -1.0)))
         0.0)
      (/
       2.0
       (*
        (* (* (sin k_m) (tan k_m)) (* (/ (pow t_m 2.0) l_m) (/ t_m l_m)))
        (+ 1.0 (+ t_2 -1.0))))
      (pow
       (* (* (/ l_m k_m) (/ (sqrt 2.0) (sin k_m))) (sqrt (/ (cos k_m) t_m)))
       2.0)))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = pow((k_m / t_m), 2.0);
	double tmp;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * (pow(t_m, 3.0) / (l_m * l_m)))) * ((1.0 + t_2) + -1.0))) <= 0.0) {
		tmp = 2.0 / (((sin(k_m) * tan(k_m)) * ((pow(t_m, 2.0) / l_m) * (t_m / l_m))) * (1.0 + (t_2 + -1.0)));
	} else {
		tmp = pow((((l_m / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt((cos(k_m) / t_m))), 2.0);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k_m / t_m) ** 2.0d0
    if ((2.0d0 / ((tan(k_m) * (sin(k_m) * ((t_m ** 3.0d0) / (l_m * l_m)))) * ((1.0d0 + t_2) + (-1.0d0)))) <= 0.0d0) then
        tmp = 2.0d0 / (((sin(k_m) * tan(k_m)) * (((t_m ** 2.0d0) / l_m) * (t_m / l_m))) * (1.0d0 + (t_2 + (-1.0d0))))
    else
        tmp = (((l_m / k_m) * (sqrt(2.0d0) / sin(k_m))) * sqrt((cos(k_m) / t_m))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double t_2 = Math.pow((k_m / t_m), 2.0);
	double tmp;
	if ((2.0 / ((Math.tan(k_m) * (Math.sin(k_m) * (Math.pow(t_m, 3.0) / (l_m * l_m)))) * ((1.0 + t_2) + -1.0))) <= 0.0) {
		tmp = 2.0 / (((Math.sin(k_m) * Math.tan(k_m)) * ((Math.pow(t_m, 2.0) / l_m) * (t_m / l_m))) * (1.0 + (t_2 + -1.0)));
	} else {
		tmp = Math.pow((((l_m / k_m) * (Math.sqrt(2.0) / Math.sin(k_m))) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	t_2 = math.pow((k_m / t_m), 2.0)
	tmp = 0
	if (2.0 / ((math.tan(k_m) * (math.sin(k_m) * (math.pow(t_m, 3.0) / (l_m * l_m)))) * ((1.0 + t_2) + -1.0))) <= 0.0:
		tmp = 2.0 / (((math.sin(k_m) * math.tan(k_m)) * ((math.pow(t_m, 2.0) / l_m) * (t_m / l_m))) * (1.0 + (t_2 + -1.0)))
	else:
		tmp = math.pow((((l_m / k_m) * (math.sqrt(2.0) / math.sin(k_m))) * math.sqrt((math.cos(k_m) / t_m))), 2.0)
	return t_s * tmp

Julia

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

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	t_2 = (k_m / t_m) ^ 2.0;
	tmp = 0.0;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * ((t_m ^ 3.0) / (l_m * l_m)))) * ((1.0 + t_2) + -1.0))) <= 0.0)
		tmp = 2.0 / (((sin(k_m) * tan(k_m)) * (((t_m ^ 2.0) / l_m) * (t_m / l_m))) * (1.0 + (t_2 + -1.0)));
	else
		tmp = (((l_m / k_m) * (sqrt(2.0) / sin(k_m))) * sqrt((cos(k_m) / t_m))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := Block[{t$95$2 = N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + t$95$2), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(2.0 / N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$2 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(l$95$m / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 0.0

   1. Initial program 87.7%

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

   3. Simplified 87.8%

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

      1. unpow3 87.8%

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

      2. times-frac 96.1%

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

      3. pow2 96.1%

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

   6. Applied egg-rr 96.1%

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

   if 0.0 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 14.4%

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

   3. Simplified 21.2%

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

      1. add-sqr-sqrt 19.0%

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

      2. pow2 19.0%

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

   6. Applied egg-rr 25.1%

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

      1. associate-*r/ 25.0%

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

      2. *-commutative 25.0%

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

      3. times-frac 25.6%

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

   8. Simplified 25.6%

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

   9. Taylor expanded in l around 0 55.6%

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

       1. times-frac 57.7%

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

   11. Simplified 57.7%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + -1\right)} \leq 0:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k\_m}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k\_m \cdot \left(\sin k\_m \cdot \frac{{t\_m}^{3}}{l\_m \cdot l\_m}\right)\right) \cdot \left(\left(1 + t\_2\right) + -1\right)} \leq 0:\\ \;\;\;\;\frac{2}{\left(\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(\frac{{t\_m}^{2}}{l\_m} \cdot \frac{t\_m}{l\_m}\right)\right) \cdot \left(1 + \left(t\_2 + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k\_m}{t\_m}} \cdot \left(l\_m \cdot \frac{\sqrt{2}}{k\_m \cdot \sin k\_m}\right)\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (let* ((t_2 (pow (/ k_m t_m) 2.0)))
   (*
    t_s
    (if (<=
         (/
          2.0
          (*
           (* (tan k_m) (* (sin k_m) (/ (pow t_m 3.0) (* l_m l_m))))
           (+ (+ 1.0 t_2) -1.0)))
         0.0)
      (/
       2.0
       (*
        (* (* (sin k_m) (tan k_m)) (* (/ (pow t_m 2.0) l_m) (/ t_m l_m)))
        (+ 1.0 (+ t_2 -1.0))))
      (pow
       (* (sqrt (/ (cos k_m) t_m)) (* l_m (/ (sqrt 2.0) (* k_m (sin k_m)))))
       2.0)))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double t_2 = pow((k_m / t_m), 2.0);
	double tmp;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * (pow(t_m, 3.0) / (l_m * l_m)))) * ((1.0 + t_2) + -1.0))) <= 0.0) {
		tmp = 2.0 / (((sin(k_m) * tan(k_m)) * ((pow(t_m, 2.0) / l_m) * (t_m / l_m))) * (1.0 + (t_2 + -1.0)));
	} else {
		tmp = pow((sqrt((cos(k_m) / t_m)) * (l_m * (sqrt(2.0) / (k_m * sin(k_m))))), 2.0);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (k_m / t_m) ** 2.0d0
    if ((2.0d0 / ((tan(k_m) * (sin(k_m) * ((t_m ** 3.0d0) / (l_m * l_m)))) * ((1.0d0 + t_2) + (-1.0d0)))) <= 0.0d0) then
        tmp = 2.0d0 / (((sin(k_m) * tan(k_m)) * (((t_m ** 2.0d0) / l_m) * (t_m / l_m))) * (1.0d0 + (t_2 + (-1.0d0))))
    else
        tmp = (sqrt((cos(k_m) / t_m)) * (l_m * (sqrt(2.0d0) / (k_m * sin(k_m))))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double t_2 = Math.pow((k_m / t_m), 2.0);
	double tmp;
	if ((2.0 / ((Math.tan(k_m) * (Math.sin(k_m) * (Math.pow(t_m, 3.0) / (l_m * l_m)))) * ((1.0 + t_2) + -1.0))) <= 0.0) {
		tmp = 2.0 / (((Math.sin(k_m) * Math.tan(k_m)) * ((Math.pow(t_m, 2.0) / l_m) * (t_m / l_m))) * (1.0 + (t_2 + -1.0)));
	} else {
		tmp = Math.pow((Math.sqrt((Math.cos(k_m) / t_m)) * (l_m * (Math.sqrt(2.0) / (k_m * Math.sin(k_m))))), 2.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	t_2 = math.pow((k_m / t_m), 2.0)
	tmp = 0
	if (2.0 / ((math.tan(k_m) * (math.sin(k_m) * (math.pow(t_m, 3.0) / (l_m * l_m)))) * ((1.0 + t_2) + -1.0))) <= 0.0:
		tmp = 2.0 / (((math.sin(k_m) * math.tan(k_m)) * ((math.pow(t_m, 2.0) / l_m) * (t_m / l_m))) * (1.0 + (t_2 + -1.0)))
	else:
		tmp = math.pow((math.sqrt((math.cos(k_m) / t_m)) * (l_m * (math.sqrt(2.0) / (k_m * math.sin(k_m))))), 2.0)
	return t_s * tmp

Julia

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

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	t_2 = (k_m / t_m) ^ 2.0;
	tmp = 0.0;
	if ((2.0 / ((tan(k_m) * (sin(k_m) * ((t_m ^ 3.0) / (l_m * l_m)))) * ((1.0 + t_2) + -1.0))) <= 0.0)
		tmp = 2.0 / (((sin(k_m) * tan(k_m)) * (((t_m ^ 2.0) / l_m) * (t_m / l_m))) * (1.0 + (t_2 + -1.0)));
	else
		tmp = (sqrt((cos(k_m) / t_m)) * (l_m * (sqrt(2.0) / (k_m * sin(k_m))))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := Block[{t$95$2 = N[Power[N[(k$95$m / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + t$95$2), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(2.0 / N[(N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$2 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))) < 0.0

   1. Initial program 87.7%

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

   3. Simplified 87.8%

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

      1. unpow3 87.8%

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

      2. times-frac 96.1%

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

      3. pow2 96.1%

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

   6. Applied egg-rr 96.1%

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

   if 0.0 < (/.f64 #s(literal 2 binary64) (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))))

   1. Initial program 14.4%

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

   3. Simplified 21.2%

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

      1. add-sqr-sqrt 19.0%

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

      2. pow2 19.0%

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

   6. Applied egg-rr 25.1%

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

      1. associate-*r/ 25.0%

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

      2. *-commutative 25.0%

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

      3. times-frac 25.6%

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

   8. Simplified 25.6%

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

   9. Taylor expanded in l around 0 55.6%

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

       1. associate-/l* 55.1%

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

   11. Simplified 55.1%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + -1\right)} \leq 0:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\frac{\cos k}{t}} \cdot \left(\ell \cdot \frac{\sqrt{2}}{k \cdot \sin k}\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 9.6 \cdot 10^{-5}:\\ \;\;\;\;{\left(\frac{l\_m \cdot \sqrt{2}}{{k\_m}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \frac{\frac{2}{{k\_m}^{2}}}{t\_m \cdot \frac{{\sin k\_m}^{2}}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 9.6e-5)
    (pow (* (/ (* l_m (sqrt 2.0)) (pow k_m 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
    (*
     (* l_m l_m)
     (/ (/ 2.0 (pow k_m 2.0)) (* t_m (/ (pow (sin k_m) 2.0) (cos k_m))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 9.6e-5) {
		tmp = pow((((l_m * sqrt(2.0)) / pow(k_m, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = (l_m * l_m) * ((2.0 / pow(k_m, 2.0)) / (t_m * (pow(sin(k_m), 2.0) / cos(k_m))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 9.6d-5) then
        tmp = (((l_m * sqrt(2.0d0)) / (k_m ** 2.0d0)) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = (l_m * l_m) * ((2.0d0 / (k_m ** 2.0d0)) / (t_m * ((sin(k_m) ** 2.0d0) / cos(k_m))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double tmp;
	if (k_m <= 9.6e-5) {
		tmp = Math.pow((((l_m * Math.sqrt(2.0)) / Math.pow(k_m, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = (l_m * l_m) * ((2.0 / Math.pow(k_m, 2.0)) / (t_m * (Math.pow(Math.sin(k_m), 2.0) / Math.cos(k_m))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if k_m <= 9.6e-5:
		tmp = math.pow((((l_m * math.sqrt(2.0)) / math.pow(k_m, 2.0)) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = (l_m * l_m) * ((2.0 / math.pow(k_m, 2.0)) / (t_m * (math.pow(math.sin(k_m), 2.0) / math.cos(k_m))))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 9.6e-5)
		tmp = Float64(Float64(Float64(l_m * sqrt(2.0)) / (k_m ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(l_m * l_m) * Float64(Float64(2.0 / (k_m ^ 2.0)) / Float64(t_m * Float64((sin(k_m) ^ 2.0) / cos(k_m)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (k_m <= 9.6e-5)
		tmp = (((l_m * sqrt(2.0)) / (k_m ^ 2.0)) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = (l_m * l_m) * ((2.0 / (k_m ^ 2.0)) / (t_m * ((sin(k_m) ^ 2.0) / cos(k_m))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 9.6e-5], N[Power[N[(N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(N[(2.0 / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.6000000000000002e-5

   1. Initial program 37.9%

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

   3. Simplified 42.7%

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

      1. add-sqr-sqrt 23.8%

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

      2. pow2 23.8%

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

   6. Applied egg-rr 25.9%

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

      1. associate-*r/ 25.8%

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

      2. *-commutative 25.8%

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

      3. times-frac 26.4%

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

   8. Simplified 26.4%

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

   9. Taylor expanded in k around 0 37.0%

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

   if 9.6000000000000002e-5 < k

   1. Initial program 32.7%

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

   3. Simplified 38.5%

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

   5. Taylor expanded in t around 0 63.2%

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

      1. associate-/l* 63.2%

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

   7. Simplified 63.2%

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

      1. *-un-lft-identity 63.2%

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

      2. associate-/r* 63.5%

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

      3. associate-/l* 63.4%

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

   9. Applied egg-rr 63.4%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.6 \cdot 10^{-5}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{{k}^{2}}}{t \cdot \frac{{\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;{\left(\frac{l\_m \cdot \sqrt{2}}{{k\_m}^{2}} \cdot \sqrt{\frac{1}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \frac{2}{\left(k\_m \cdot k\_m\right) \cdot \frac{t\_m \cdot {\sin k\_m}^{2}}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 1.25e-5)
    (pow (* (/ (* l_m (sqrt 2.0)) (pow k_m 2.0)) (sqrt (/ 1.0 t_m))) 2.0)
    (*
     (* l_m l_m)
     (/ 2.0 (* (* k_m k_m) (/ (* t_m (pow (sin k_m) 2.0)) (cos k_m))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 1.25e-5) {
		tmp = pow((((l_m * sqrt(2.0)) / pow(k_m, 2.0)) * sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = (l_m * l_m) * (2.0 / ((k_m * k_m) * ((t_m * pow(sin(k_m), 2.0)) / cos(k_m))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.25d-5) then
        tmp = (((l_m * sqrt(2.0d0)) / (k_m ** 2.0d0)) * sqrt((1.0d0 / t_m))) ** 2.0d0
    else
        tmp = (l_m * l_m) * (2.0d0 / ((k_m * k_m) * ((t_m * (sin(k_m) ** 2.0d0)) / cos(k_m))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double tmp;
	if (k_m <= 1.25e-5) {
		tmp = Math.pow((((l_m * Math.sqrt(2.0)) / Math.pow(k_m, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
	} else {
		tmp = (l_m * l_m) * (2.0 / ((k_m * k_m) * ((t_m * Math.pow(Math.sin(k_m), 2.0)) / Math.cos(k_m))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if k_m <= 1.25e-5:
		tmp = math.pow((((l_m * math.sqrt(2.0)) / math.pow(k_m, 2.0)) * math.sqrt((1.0 / t_m))), 2.0)
	else:
		tmp = (l_m * l_m) * (2.0 / ((k_m * k_m) * ((t_m * math.pow(math.sin(k_m), 2.0)) / math.cos(k_m))))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 1.25e-5)
		tmp = Float64(Float64(Float64(l_m * sqrt(2.0)) / (k_m ^ 2.0)) * sqrt(Float64(1.0 / t_m))) ^ 2.0;
	else
		tmp = Float64(Float64(l_m * l_m) * Float64(2.0 / Float64(Float64(k_m * k_m) * Float64(Float64(t_m * (sin(k_m) ^ 2.0)) / cos(k_m)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (k_m <= 1.25e-5)
		tmp = (((l_m * sqrt(2.0)) / (k_m ^ 2.0)) * sqrt((1.0 / t_m))) ^ 2.0;
	else
		tmp = (l_m * l_m) * (2.0 / ((k_m * k_m) * ((t_m * (sin(k_m) ^ 2.0)) / cos(k_m))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.25e-5], N[Power[N[(N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.25000000000000006e-5

   1. Initial program 37.9%

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

   3. Simplified 42.7%

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

      1. add-sqr-sqrt 23.8%

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

      2. pow2 23.8%

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

   6. Applied egg-rr 25.9%

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

      1. associate-*r/ 25.8%

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

      2. *-commutative 25.8%

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

      3. times-frac 26.4%

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

   8. Simplified 26.4%

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

   9. Taylor expanded in k around 0 37.0%

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

   if 1.25000000000000006e-5 < k

   1. Initial program 32.7%

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

   3. Simplified 38.5%

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

   5. Taylor expanded in t around 0 63.2%

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

      1. associate-/l* 63.2%

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

   7. Simplified 63.2%

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

      1. unpow2 63.2%

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

   9. Applied egg-rr 63.2%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;{\left(\sqrt{\frac{1}{t\_m}} \cdot \left(l\_m \cdot \frac{\sqrt{2}}{{k\_m}^{2}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \frac{2}{\left(k\_m \cdot k\_m\right) \cdot \frac{t\_m \cdot {\sin k\_m}^{2}}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 4.7e-5)
    (pow (* (sqrt (/ 1.0 t_m)) (* l_m (/ (sqrt 2.0) (pow k_m 2.0)))) 2.0)
    (*
     (* l_m l_m)
     (/ 2.0 (* (* k_m k_m) (/ (* t_m (pow (sin k_m) 2.0)) (cos k_m))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 4.7e-5) {
		tmp = pow((sqrt((1.0 / t_m)) * (l_m * (sqrt(2.0) / pow(k_m, 2.0)))), 2.0);
	} else {
		tmp = (l_m * l_m) * (2.0 / ((k_m * k_m) * ((t_m * pow(sin(k_m), 2.0)) / cos(k_m))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 4.7d-5) then
        tmp = (sqrt((1.0d0 / t_m)) * (l_m * (sqrt(2.0d0) / (k_m ** 2.0d0)))) ** 2.0d0
    else
        tmp = (l_m * l_m) * (2.0d0 / ((k_m * k_m) * ((t_m * (sin(k_m) ** 2.0d0)) / cos(k_m))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double tmp;
	if (k_m <= 4.7e-5) {
		tmp = Math.pow((Math.sqrt((1.0 / t_m)) * (l_m * (Math.sqrt(2.0) / Math.pow(k_m, 2.0)))), 2.0);
	} else {
		tmp = (l_m * l_m) * (2.0 / ((k_m * k_m) * ((t_m * Math.pow(Math.sin(k_m), 2.0)) / Math.cos(k_m))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if k_m <= 4.7e-5:
		tmp = math.pow((math.sqrt((1.0 / t_m)) * (l_m * (math.sqrt(2.0) / math.pow(k_m, 2.0)))), 2.0)
	else:
		tmp = (l_m * l_m) * (2.0 / ((k_m * k_m) * ((t_m * math.pow(math.sin(k_m), 2.0)) / math.cos(k_m))))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 4.7e-5)
		tmp = Float64(sqrt(Float64(1.0 / t_m)) * Float64(l_m * Float64(sqrt(2.0) / (k_m ^ 2.0)))) ^ 2.0;
	else
		tmp = Float64(Float64(l_m * l_m) * Float64(2.0 / Float64(Float64(k_m * k_m) * Float64(Float64(t_m * (sin(k_m) ^ 2.0)) / cos(k_m)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (k_m <= 4.7e-5)
		tmp = (sqrt((1.0 / t_m)) * (l_m * (sqrt(2.0) / (k_m ^ 2.0)))) ^ 2.0;
	else
		tmp = (l_m * l_m) * (2.0 / ((k_m * k_m) * ((t_m * (sin(k_m) ^ 2.0)) / cos(k_m))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 4.7e-5], N[Power[N[(N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 4.69999999999999972e-5

   1. Initial program 37.9%

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

   3. Simplified 42.7%

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

      1. add-sqr-sqrt 23.8%

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

      2. pow2 23.8%

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

   6. Applied egg-rr 25.9%

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

      1. associate-*r/ 25.8%

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

      2. *-commutative 25.8%

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

      3. times-frac 26.4%

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

   8. Simplified 26.4%

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

   9. Taylor expanded in k around 0 37.0%

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

       1. associate-/l* 36.5%

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

   11. Simplified 36.5%

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

   if 4.69999999999999972e-5 < k

   1. Initial program 32.7%

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

   3. Simplified 38.5%

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

   5. Taylor expanded in t around 0 63.2%

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

      1. associate-/l* 63.2%

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

   7. Simplified 63.2%

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

      1. unpow2 63.2%

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

   9. Applied egg-rr 63.2%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;{\left(\sqrt{\frac{1}{t}} \cdot \left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{{l\_m}^{2}}{t\_m} \cdot \frac{2}{{k\_m}^{4}}\\ \mathbf{elif}\;t\_m \leq 2.4 \cdot 10^{+86}:\\ \;\;\;\;{\left(\frac{l\_m}{\frac{k\_m}{t\_m}} \cdot \frac{\sqrt{\frac{2}{{t\_m}^{3}}}}{k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \frac{2}{{\left({k\_m}^{2} \cdot \sqrt{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3.5e-98)
    (* (/ (pow l_m 2.0) t_m) (/ 2.0 (pow k_m 4.0)))
    (if (<= t_m 2.4e+86)
      (pow (* (/ l_m (/ k_m t_m)) (/ (sqrt (/ 2.0 (pow t_m 3.0))) k_m)) 2.0)
      (* (* l_m l_m) (/ 2.0 (pow (* (pow k_m 2.0) (sqrt t_m)) 2.0)))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (t_m <= 3.5e-98) {
		tmp = (pow(l_m, 2.0) / t_m) * (2.0 / pow(k_m, 4.0));
	} else if (t_m <= 2.4e+86) {
		tmp = pow(((l_m / (k_m / t_m)) * (sqrt((2.0 / pow(t_m, 3.0))) / k_m)), 2.0);
	} else {
		tmp = (l_m * l_m) * (2.0 / pow((pow(k_m, 2.0) * sqrt(t_m)), 2.0));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 3.5d-98) then
        tmp = ((l_m ** 2.0d0) / t_m) * (2.0d0 / (k_m ** 4.0d0))
    else if (t_m <= 2.4d+86) then
        tmp = ((l_m / (k_m / t_m)) * (sqrt((2.0d0 / (t_m ** 3.0d0))) / k_m)) ** 2.0d0
    else
        tmp = (l_m * l_m) * (2.0d0 / (((k_m ** 2.0d0) * sqrt(t_m)) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double tmp;
	if (t_m <= 3.5e-98) {
		tmp = (Math.pow(l_m, 2.0) / t_m) * (2.0 / Math.pow(k_m, 4.0));
	} else if (t_m <= 2.4e+86) {
		tmp = Math.pow(((l_m / (k_m / t_m)) * (Math.sqrt((2.0 / Math.pow(t_m, 3.0))) / k_m)), 2.0);
	} else {
		tmp = (l_m * l_m) * (2.0 / Math.pow((Math.pow(k_m, 2.0) * Math.sqrt(t_m)), 2.0));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if t_m <= 3.5e-98:
		tmp = (math.pow(l_m, 2.0) / t_m) * (2.0 / math.pow(k_m, 4.0))
	elif t_m <= 2.4e+86:
		tmp = math.pow(((l_m / (k_m / t_m)) * (math.sqrt((2.0 / math.pow(t_m, 3.0))) / k_m)), 2.0)
	else:
		tmp = (l_m * l_m) * (2.0 / math.pow((math.pow(k_m, 2.0) * math.sqrt(t_m)), 2.0))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (t_m <= 3.5e-98)
		tmp = Float64(Float64((l_m ^ 2.0) / t_m) * Float64(2.0 / (k_m ^ 4.0)));
	elseif (t_m <= 2.4e+86)
		tmp = Float64(Float64(l_m / Float64(k_m / t_m)) * Float64(sqrt(Float64(2.0 / (t_m ^ 3.0))) / k_m)) ^ 2.0;
	else
		tmp = Float64(Float64(l_m * l_m) * Float64(2.0 / (Float64((k_m ^ 2.0) * sqrt(t_m)) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (t_m <= 3.5e-98)
		tmp = ((l_m ^ 2.0) / t_m) * (2.0 / (k_m ^ 4.0));
	elseif (t_m <= 2.4e+86)
		tmp = ((l_m / (k_m / t_m)) * (sqrt((2.0 / (t_m ^ 3.0))) / k_m)) ^ 2.0;
	else
		tmp = (l_m * l_m) * (2.0 / (((k_m ^ 2.0) * sqrt(t_m)) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 3.5e-98], N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(2.0 / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.4e+86], N[Power[N[(N[(l$95$m / N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(2.0 / N[Power[N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 3.5000000000000002e-98

   1. Initial program 35.9%

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

   3. Simplified 41.0%

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

   5. Taylor expanded in k around 0 60.4%

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

      1. associate-*r/ 60.4%

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

      2. *-commutative 60.4%

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

      3. *-commutative 60.4%

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

      4. times-frac 59.0%

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

   7. Simplified 59.0%

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

   if 3.5000000000000002e-98 < t < 2.4e86

   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 68.6%

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

      1. add-sqr-sqrt 61.7%

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

      2. pow2 61.7%

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

   6. Applied egg-rr 68.3%

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

      1. associate-*r/ 67.9%

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

      2. *-commutative 67.9%

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

      3. times-frac 70.6%

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

   8. Simplified 70.6%

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

   9. Taylor expanded in k around 0 79.8%

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

   if 2.4e86 < t

   1. Initial program 7.5%

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

   3. Simplified 15.0%

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

      1. add-sqr-sqrt 7.5%

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

      2. pow2 7.5%

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

      3. *-commutative 7.5%

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

      4. sqrt-prod 7.5%

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

      5. associate-*r* 7.5%

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

      6. sqrt-prod 10.0%

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

      7. sqrt-pow1 15.2%

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

      8. metadata-eval 15.2%

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

      9. pow1 15.2%

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

      10. sqrt-pow1 32.3%

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

      11. metadata-eval 32.3%

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

   6. Applied egg-rr 32.3%

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

      1. associate-*l* 32.4%

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

   8. Simplified 32.4%

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

   9. Taylor expanded in k around 0 64.5%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{{\ell}^{2}}{t} \cdot \frac{2}{{k}^{4}}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+86}:\\ \;\;\;\;{\left(\frac{\ell}{\frac{k}{t}} \cdot \frac{\sqrt{\frac{2}{{t}^{3}}}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left({k}^{2} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 74.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.55 \cdot 10^{-176}:\\ \;\;\;\;{\left(\frac{l\_m}{\frac{k\_m}{t\_m}} \cdot \frac{\sqrt{\frac{2}{{t\_m}^{3}}}}{k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \frac{2}{\left(k\_m \cdot k\_m\right) \cdot \frac{t\_m \cdot {\sin k\_m}^{2}}{\cos k\_m}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 1.55e-176)
    (pow (* (/ l_m (/ k_m t_m)) (/ (sqrt (/ 2.0 (pow t_m 3.0))) k_m)) 2.0)
    (*
     (* l_m l_m)
     (/ 2.0 (* (* k_m k_m) (/ (* t_m (pow (sin k_m) 2.0)) (cos k_m))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (l_m <= 1.55e-176) {
		tmp = pow(((l_m / (k_m / t_m)) * (sqrt((2.0 / pow(t_m, 3.0))) / k_m)), 2.0);
	} else {
		tmp = (l_m * l_m) * (2.0 / ((k_m * k_m) * ((t_m * pow(sin(k_m), 2.0)) / cos(k_m))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l_m <= 1.55d-176) then
        tmp = ((l_m / (k_m / t_m)) * (sqrt((2.0d0 / (t_m ** 3.0d0))) / k_m)) ** 2.0d0
    else
        tmp = (l_m * l_m) * (2.0d0 / ((k_m * k_m) * ((t_m * (sin(k_m) ** 2.0d0)) / cos(k_m))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double tmp;
	if (l_m <= 1.55e-176) {
		tmp = Math.pow(((l_m / (k_m / t_m)) * (Math.sqrt((2.0 / Math.pow(t_m, 3.0))) / k_m)), 2.0);
	} else {
		tmp = (l_m * l_m) * (2.0 / ((k_m * k_m) * ((t_m * Math.pow(Math.sin(k_m), 2.0)) / Math.cos(k_m))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if l_m <= 1.55e-176:
		tmp = math.pow(((l_m / (k_m / t_m)) * (math.sqrt((2.0 / math.pow(t_m, 3.0))) / k_m)), 2.0)
	else:
		tmp = (l_m * l_m) * (2.0 / ((k_m * k_m) * ((t_m * math.pow(math.sin(k_m), 2.0)) / math.cos(k_m))))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (l_m <= 1.55e-176)
		tmp = Float64(Float64(l_m / Float64(k_m / t_m)) * Float64(sqrt(Float64(2.0 / (t_m ^ 3.0))) / k_m)) ^ 2.0;
	else
		tmp = Float64(Float64(l_m * l_m) * Float64(2.0 / Float64(Float64(k_m * k_m) * Float64(Float64(t_m * (sin(k_m) ^ 2.0)) / cos(k_m)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (l_m <= 1.55e-176)
		tmp = ((l_m / (k_m / t_m)) * (sqrt((2.0 / (t_m ^ 3.0))) / k_m)) ^ 2.0;
	else
		tmp = (l_m * l_m) * (2.0 / ((k_m * k_m) * ((t_m * (sin(k_m) ^ 2.0)) / cos(k_m))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 1.55e-176], N[Power[N[(N[(l$95$m / N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(2.0 / N[(N[(k$95$m * k$95$m), $MachinePrecision] * N[(N[(t$95$m * N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.54999999999999996e-176

   1. Initial program 36.0%

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

   3. Simplified 42.4%

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

      1. add-sqr-sqrt 28.1%

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

      2. pow2 28.1%

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

   6. Applied egg-rr 27.4%

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

      1. associate-*r/ 27.3%

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

      2. *-commutative 27.3%

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

      3. times-frac 28.0%

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

   8. Simplified 28.0%

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

   9. Taylor expanded in k around 0 32.9%

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

   if 1.54999999999999996e-176 < l

   1. Initial program 37.1%

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

   3. Simplified 40.4%

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

   5. Taylor expanded in t around 0 70.6%

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

      1. associate-/l* 70.6%

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

   7. Simplified 70.6%

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

      1. unpow2 70.6%

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

   9. Applied egg-rr 70.6%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.55 \cdot 10^{-176}:\\ \;\;\;\;{\left(\frac{\ell}{\frac{k}{t}} \cdot \frac{\sqrt{\frac{2}{{t}^{3}}}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(k \cdot k\right) \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 74.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.56 \cdot 10^{-176}:\\ \;\;\;\;{\left(\frac{l\_m}{\frac{k\_m}{t\_m}} \cdot \frac{\sqrt{\frac{2}{{t\_m}^{3}}}}{k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \frac{2}{{k\_m}^{2} \cdot \left(t\_m \cdot \left(\sin k\_m \cdot \tan k\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 1.56e-176)
    (pow (* (/ l_m (/ k_m t_m)) (/ (sqrt (/ 2.0 (pow t_m 3.0))) k_m)) 2.0)
    (*
     (* l_m l_m)
     (/ 2.0 (* (pow k_m 2.0) (* t_m (* (sin k_m) (tan k_m)))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (l_m <= 1.56e-176) {
		tmp = pow(((l_m / (k_m / t_m)) * (sqrt((2.0 / pow(t_m, 3.0))) / k_m)), 2.0);
	} else {
		tmp = (l_m * l_m) * (2.0 / (pow(k_m, 2.0) * (t_m * (sin(k_m) * tan(k_m)))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l_m <= 1.56d-176) then
        tmp = ((l_m / (k_m / t_m)) * (sqrt((2.0d0 / (t_m ** 3.0d0))) / k_m)) ** 2.0d0
    else
        tmp = (l_m * l_m) * (2.0d0 / ((k_m ** 2.0d0) * (t_m * (sin(k_m) * tan(k_m)))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double tmp;
	if (l_m <= 1.56e-176) {
		tmp = Math.pow(((l_m / (k_m / t_m)) * (Math.sqrt((2.0 / Math.pow(t_m, 3.0))) / k_m)), 2.0);
	} else {
		tmp = (l_m * l_m) * (2.0 / (Math.pow(k_m, 2.0) * (t_m * (Math.sin(k_m) * Math.tan(k_m)))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if l_m <= 1.56e-176:
		tmp = math.pow(((l_m / (k_m / t_m)) * (math.sqrt((2.0 / math.pow(t_m, 3.0))) / k_m)), 2.0)
	else:
		tmp = (l_m * l_m) * (2.0 / (math.pow(k_m, 2.0) * (t_m * (math.sin(k_m) * math.tan(k_m)))))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (l_m <= 1.56e-176)
		tmp = Float64(Float64(l_m / Float64(k_m / t_m)) * Float64(sqrt(Float64(2.0 / (t_m ^ 3.0))) / k_m)) ^ 2.0;
	else
		tmp = Float64(Float64(l_m * l_m) * Float64(2.0 / Float64((k_m ^ 2.0) * Float64(t_m * Float64(sin(k_m) * tan(k_m))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (l_m <= 1.56e-176)
		tmp = ((l_m / (k_m / t_m)) * (sqrt((2.0 / (t_m ^ 3.0))) / k_m)) ^ 2.0;
	else
		tmp = (l_m * l_m) * (2.0 / ((k_m ^ 2.0) * (t_m * (sin(k_m) * tan(k_m)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 1.56e-176], N[Power[N[(N[(l$95$m / N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.56000000000000006e-176

   1. Initial program 36.0%

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

   3. Simplified 42.4%

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

      1. add-sqr-sqrt 28.1%

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

      2. pow2 28.1%

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

   6. Applied egg-rr 27.4%

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

      1. associate-*r/ 27.3%

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

      2. *-commutative 27.3%

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

      3. times-frac 28.0%

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

   8. Simplified 28.0%

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

   9. Taylor expanded in k around 0 32.9%

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

   if 1.56000000000000006e-176 < l

   1. Initial program 37.1%

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

   3. Simplified 40.4%

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

      1. add-log-exp 24.7%

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

      2. exp-prod 27.1%

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

   6. Applied egg-rr 27.1%

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

   7. Taylor expanded in k around inf 70.6%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.56 \cdot 10^{-176}:\\ \;\;\;\;{\left(\frac{\ell}{\frac{k}{t}} \cdot \frac{\sqrt{\frac{2}{{t}^{3}}}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 74.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.2 \cdot 10^{-178}:\\ \;\;\;\;{\left(\frac{l\_m}{\frac{k\_m}{t\_m}} \cdot \frac{\sqrt{\frac{2}{{t\_m}^{3}}}}{k\_m}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \frac{2}{\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(t\_m \cdot {k\_m}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 1.2e-178)
    (pow (* (/ l_m (/ k_m t_m)) (/ (sqrt (/ 2.0 (pow t_m 3.0))) k_m)) 2.0)
    (*
     (* l_m l_m)
     (/ 2.0 (* (* (sin k_m) (tan k_m)) (* t_m (pow k_m 2.0))))))))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (l_m <= 1.2e-178) {
		tmp = pow(((l_m / (k_m / t_m)) * (sqrt((2.0 / pow(t_m, 3.0))) / k_m)), 2.0);
	} else {
		tmp = (l_m * l_m) * (2.0 / ((sin(k_m) * tan(k_m)) * (t_m * pow(k_m, 2.0))));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l_m <= 1.2d-178) then
        tmp = ((l_m / (k_m / t_m)) * (sqrt((2.0d0 / (t_m ** 3.0d0))) / k_m)) ** 2.0d0
    else
        tmp = (l_m * l_m) * (2.0d0 / ((sin(k_m) * tan(k_m)) * (t_m * (k_m ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double tmp;
	if (l_m <= 1.2e-178) {
		tmp = Math.pow(((l_m / (k_m / t_m)) * (Math.sqrt((2.0 / Math.pow(t_m, 3.0))) / k_m)), 2.0);
	} else {
		tmp = (l_m * l_m) * (2.0 / ((Math.sin(k_m) * Math.tan(k_m)) * (t_m * Math.pow(k_m, 2.0))));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if l_m <= 1.2e-178:
		tmp = math.pow(((l_m / (k_m / t_m)) * (math.sqrt((2.0 / math.pow(t_m, 3.0))) / k_m)), 2.0)
	else:
		tmp = (l_m * l_m) * (2.0 / ((math.sin(k_m) * math.tan(k_m)) * (t_m * math.pow(k_m, 2.0))))
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (l_m <= 1.2e-178)
		tmp = Float64(Float64(l_m / Float64(k_m / t_m)) * Float64(sqrt(Float64(2.0 / (t_m ^ 3.0))) / k_m)) ^ 2.0;
	else
		tmp = Float64(Float64(l_m * l_m) * Float64(2.0 / Float64(Float64(sin(k_m) * tan(k_m)) * Float64(t_m * (k_m ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (l_m <= 1.2e-178)
		tmp = ((l_m / (k_m / t_m)) * (sqrt((2.0 / (t_m ^ 3.0))) / k_m)) ^ 2.0;
	else
		tmp = (l_m * l_m) * (2.0 / ((sin(k_m) * tan(k_m)) * (t_m * (k_m ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 1.2e-178], N[Power[N[(N[(l$95$m / N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(2.0 / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.20000000000000002e-178

   1. Initial program 36.0%

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

   3. Simplified 42.4%

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

      1. add-sqr-sqrt 28.1%

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

      2. pow2 28.1%

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

   6. Applied egg-rr 27.4%

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

      1. associate-*r/ 27.3%

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

      2. *-commutative 27.3%

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

      3. times-frac 28.0%

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

   8. Simplified 28.0%

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

   9. Taylor expanded in k around 0 32.9%

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

   if 1.20000000000000002e-178 < l

   1. Initial program 37.1%

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

   3. Simplified 40.4%

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

      1. add-log-exp 24.7%

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

      2. exp-prod 27.1%

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

   6. Applied egg-rr 27.1%

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

   7. Taylor expanded in k around inf 70.6%

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

      1. associate-*r* 70.6%

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

      2. *-commutative 70.6%

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

   9. Simplified 70.6%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.2 \cdot 10^{-178}:\\ \;\;\;\;{\left(\frac{\ell}{\frac{k}{t}} \cdot \frac{\sqrt{\frac{2}{{t}^{3}}}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot {k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 63.1% accurate, 1.4× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (* t_s (* (* l_m l_m) (/ 2.0 (pow (* (pow k_m 2.0) (sqrt t_m)) 2.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.5%

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

3. Simplified 41.6%

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

   1. add-sqr-sqrt 19.2%

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

   2. pow2 19.2%

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

   3. *-commutative 19.2%

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

   4. sqrt-prod 14.2%

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

   5. associate-*r* 14.2%

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

   6. sqrt-prod 14.6%

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

   7. sqrt-pow1 16.7%

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

   8. metadata-eval 16.7%

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

   9. pow1 16.7%

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

   10. sqrt-pow1 21.7%

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

   11. metadata-eval 21.7%

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

6. Applied egg-rr 21.7%

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

   1. associate-*l* 21.7%

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

8. Simplified 21.7%

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

9. Taylor expanded in k around 0 28.7%

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

10. Final simplification 28.7%

    \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{{\left({k}^{2} \cdot \sqrt{t}\right)}^{2}} \]
11. Add Preprocessing

Alternative 20: 61.5% accurate, 2.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (* t_s (* (* l_m l_m) (/ 2.0 (* t_m (* (pow k_m 2.0) (pow k_m 2.0)))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.5%

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

3. Simplified 41.6%

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

5. Taylor expanded in k around 0 60.5%

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

   1. metadata-eval 60.5%

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

   2. pow-prod-up 60.6%

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

7. Applied egg-rr 60.6%

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

8. Final simplification 60.6%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot \left({k}^{2} \cdot {k}^{2}\right)} \]
9. Add Preprocessing

Alternative 21: 61.5% accurate, 2.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (* t_s (* (pow l_m 2.0) (/ (/ 2.0 (pow k_m 4.0)) t_m))))

C

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

Fortran

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

Java

l_m = Math.abs(l);
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_m, double k_m) {
	return t_s * (Math.pow(l_m, 2.0) * ((2.0 / Math.pow(k_m, 4.0)) / t_m));
}

Python

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

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	return Float64(t_s * Float64((l_m ^ 2.0) * Float64(Float64(2.0 / (k_m ^ 4.0)) / t_m)))
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k_m)
	tmp = t_s * ((l_m ^ 2.0) * ((2.0 / (k_m ^ 4.0)) / t_m));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(2.0 / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.5%

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

3. Simplified 36.5%

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

6. Applied egg-rr 77.2%

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

   1. associate-/r/ 77.2%

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

   2. associate-/r* 77.2%

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

   3. associate-/r/ 77.6%

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

8. Simplified 77.6%

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

   1. pow1/3 38.2%

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

   2. add-sqr-sqrt 38.2%

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

   3. unpow-prod-down 38.2%

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

10. Applied egg-rr 38.2%

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

    1. unpow1/3 38.4%

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

    2. unpow1/3 39.0%

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

12. Simplified 39.0%

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

13. Taylor expanded in k around 0 60.5%

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

    1. associate-/l* 60.5%

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

    2. unpow2 60.5%

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

    3. rem-square-sqrt 60.5%

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

    4. associate-/r* 60.5%

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

15. Simplified 60.5%

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

Alternative 22: 61.5% accurate, 3.8× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (* t_s (* (* l_m l_m) (/ 2.0 (* t_m (pow k_m 4.0))))))

C

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

Fortran

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

Java

l_m = Math.abs(l);
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_m, double k_m) {
	return t_s * ((l_m * l_m) * (2.0 / (t_m * Math.pow(k_m, 4.0))));
}

Python

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

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	return Float64(t_s * Float64(Float64(l_m * l_m) * Float64(2.0 / Float64(t_m * (k_m ^ 4.0)))))
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k_m)
	tmp = t_s * ((l_m * l_m) * (2.0 / (t_m * (k_m ^ 4.0))));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.5%

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

3. Simplified 41.6%

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

5. Taylor expanded in k around 0 60.5%

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

6. Final simplification 60.5%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{t \cdot {k}^{4}} \]
7. Add Preprocessing

Alternative 23: 55.3% accurate, 35.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 1.65 \cdot 10^{-36}:\\ \;\;\;\;\left(l\_m \cdot l\_m\right) \cdot \frac{4}{0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
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_m k_m)
 :precision binary64
 (* t_s (if (<= k_m 1.65e-36) (* (* l_m l_m) (/ 4.0 0.0)) 0.0)))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	double tmp;
	if (k_m <= 1.65e-36) {
		tmp = (l_m * l_m) * (4.0 / 0.0);
	} else {
		tmp = 0.0;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 1.65d-36) then
        tmp = (l_m * l_m) * (4.0d0 / 0.0d0)
    else
        tmp = 0.0d0
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	double tmp;
	if (k_m <= 1.65e-36) {
		tmp = (l_m * l_m) * (4.0 / 0.0);
	} else {
		tmp = 0.0;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	tmp = 0
	if k_m <= 1.65e-36:
		tmp = (l_m * l_m) * (4.0 / 0.0)
	else:
		tmp = 0.0
	return t_s * tmp

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	tmp = 0.0
	if (k_m <= 1.65e-36)
		tmp = Float64(Float64(l_m * l_m) * Float64(4.0 / 0.0));
	else
		tmp = 0.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k_m)
	tmp = 0.0;
	if (k_m <= 1.65e-36)
		tmp = (l_m * l_m) * (4.0 / 0.0);
	else
		tmp = 0.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 1.65e-36], N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(4.0 / 0.0), $MachinePrecision]), $MachinePrecision], 0.0]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.64999999999999995e-36

   1. Initial program 39.0%

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

   3. Simplified 43.5%

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

      1. add-log-exp 22.6%

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

      2. exp-prod 25.2%

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

   6. Applied egg-rr 25.2%

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

   7. Taylor expanded in t around 0 19.4%

      \[\leadsto \frac{2}{\log \color{blue}{1}} \cdot \left(\ell \cdot \ell\right) \]
   8. Step-by-step derivation

      1. div-inv 19.4%

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

      2. metadata-eval 19.4%

         \[\leadsto \left(2 \cdot \frac{1}{\color{blue}{0}}\right) \cdot \left(\ell \cdot \ell\right) \]

      3. metadata-eval 19.4%

         \[\leadsto \left(2 \cdot \frac{1}{\color{blue}{\frac{0}{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      4. metadata-eval 19.4%

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

      5. clear-num 19.4%

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

      6. metadata-eval 19.4%

         \[\leadsto \left(2 \cdot \frac{2}{\color{blue}{0}}\right) \cdot \left(\ell \cdot \ell\right) \]

   9. Applied egg-rr 19.4%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{2}{0}\right)} \cdot \left(\ell \cdot \ell\right) \]
   10. Step-by-step derivation

       1. associate-*r/ 19.4%

          \[\leadsto \color{blue}{\frac{2 \cdot 2}{0}} \cdot \left(\ell \cdot \ell\right) \]

       2. metadata-eval 19.4%

          \[\leadsto \frac{\color{blue}{4}}{0} \cdot \left(\ell \cdot \ell\right) \]

   11. Simplified 19.4%

       \[\leadsto \color{blue}{\frac{4}{0}} \cdot \left(\ell \cdot \ell\right) \]

   if 1.64999999999999995e-36 < k

   1. Initial program 30.8%

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

   3. Simplified 37.2%

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

      1. add-log-exp 33.2%

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

      2. exp-prod 25.1%

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

   6. Applied egg-rr 25.1%

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

   7. Taylor expanded in t around 0 8.7%

      \[\leadsto \frac{2}{\log \color{blue}{1}} \cdot \left(\ell \cdot \ell\right) \]
   8. Step-by-step derivation

      1. add-cube-cbrt 8.7%

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

      2. pow2 8.7%

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

      3. clear-num 8.7%

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

      4. metadata-eval 8.7%

         \[\leadsto \left({\left(\sqrt[3]{\frac{1}{\frac{\color{blue}{0}}{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      5. metadata-eval 8.7%

         \[\leadsto \left({\left(\sqrt[3]{\frac{1}{\color{blue}{0}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      6. metadata-eval 8.7%

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

      7. cbrt-div 8.7%

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

      8. metadata-eval 8.7%

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

      9. metadata-eval 8.7%

         \[\leadsto \left({\left(\frac{1}{\sqrt[3]{\color{blue}{0}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      10. clear-num 8.7%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{1}{\frac{\log 1}{2}}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      11. metadata-eval 8.7%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\frac{\color{blue}{0}}{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      12. metadata-eval 8.7%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\color{blue}{0}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      13. metadata-eval 8.7%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\color{blue}{\log 1}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      14. cbrt-div 8.7%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\log 1}}}\right) \cdot \left(\ell \cdot \ell\right) \]

      15. metadata-eval 8.7%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{\color{blue}{1}}{\sqrt[3]{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

      16. metadata-eval 8.7%

          \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{1}{\sqrt[3]{\color{blue}{0}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   9. Applied egg-rr 8.7%

      \[\leadsto \color{blue}{\left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{1}{\sqrt[3]{0}}\right)} \cdot \left(\ell \cdot \ell\right) \]
   10. Step-by-step derivation

       1. pow-plus 8.7%

          \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt[3]{0}}\right)}^{\left(2 + 1\right)}} \cdot \left(\ell \cdot \ell\right) \]

       2. metadata-eval 8.7%

          \[\leadsto {\left(\frac{1}{\sqrt[3]{0}}\right)}^{\color{blue}{3}} \cdot \left(\ell \cdot \ell\right) \]

       3. cube-div 8.7%

          \[\leadsto \color{blue}{\frac{{1}^{3}}{{\left(\sqrt[3]{0}\right)}^{3}}} \cdot \left(\ell \cdot \ell\right) \]

       4. metadata-eval 8.7%

          \[\leadsto \frac{\color{blue}{1}}{{\left(\sqrt[3]{0}\right)}^{3}} \cdot \left(\ell \cdot \ell\right) \]

       5. rem-cube-cbrt 8.7%

          \[\leadsto \frac{1}{\color{blue}{0}} \cdot \left(\ell \cdot \ell\right) \]

       6. unpow-1 8.7%

          \[\leadsto \color{blue}{{0}^{-1}} \cdot \left(\ell \cdot \ell\right) \]

       7. pow-base-0 42.9%

          \[\leadsto \color{blue}{0} \cdot \left(\ell \cdot \ell\right) \]

   11. Simplified 42.9%

       \[\leadsto \color{blue}{0} \cdot \left(\ell \cdot \ell\right) \]

   12. Taylor expanded in l around 0 45.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.65 \cdot 10^{-36}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{4}{0}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 27.5% accurate, 421.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot 0 \end{array} \]

FPCore

l_m = (fabs.f64 l)
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_m k_m) :precision binary64 (* t_s 0.0))

C

l_m = fabs(l);
k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k_m) {
	return t_s * 0.0;
}

Fortran

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k_m
    code = t_s * 0.0d0
end function

Java

l_m = Math.abs(l);
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_m, double k_m) {
	return t_s * 0.0;
}

Python

l_m = math.fabs(l)
k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k_m):
	return t_s * 0.0

Julia

l_m = abs(l)
k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k_m)
	return Float64(t_s * 0.0)
end

MATLAB

l_m = abs(l);
k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k_m)
	tmp = t_s * 0.0;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
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$95$m_, k$95$m_] := N[(t$95$s * 0.0), $MachinePrecision]

Derivation

1. Initial program 36.5%

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

3. Simplified 41.6%

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

   1. add-log-exp 25.9%

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

   2. exp-prod 25.2%

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

6. Applied egg-rr 25.2%

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

7. Taylor expanded in t around 0 16.1%

   \[\leadsto \frac{2}{\log \color{blue}{1}} \cdot \left(\ell \cdot \ell\right) \]
8. Step-by-step derivation

   1. add-cube-cbrt 16.1%

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

   2. pow2 16.1%

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

   3. clear-num 16.1%

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

   4. metadata-eval 16.1%

      \[\leadsto \left({\left(\sqrt[3]{\frac{1}{\frac{\color{blue}{0}}{2}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

   5. metadata-eval 16.1%

      \[\leadsto \left({\left(\sqrt[3]{\frac{1}{\color{blue}{0}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

   6. metadata-eval 16.1%

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

   7. cbrt-div 16.1%

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

   8. metadata-eval 16.1%

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

   9. metadata-eval 16.1%

      \[\leadsto \left({\left(\frac{1}{\sqrt[3]{\color{blue}{0}}}\right)}^{2} \cdot \sqrt[3]{\frac{2}{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

   10. clear-num 16.1%

       \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{1}{\frac{\log 1}{2}}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   11. metadata-eval 16.1%

       \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\frac{\color{blue}{0}}{2}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   12. metadata-eval 16.1%

       \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\color{blue}{0}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   13. metadata-eval 16.1%

       \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{\color{blue}{\log 1}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   14. cbrt-div 16.1%

       \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\log 1}}}\right) \cdot \left(\ell \cdot \ell\right) \]

   15. metadata-eval 16.1%

       \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{\color{blue}{1}}{\sqrt[3]{\log 1}}\right) \cdot \left(\ell \cdot \ell\right) \]

   16. metadata-eval 16.1%

       \[\leadsto \left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{1}{\sqrt[3]{\color{blue}{0}}}\right) \cdot \left(\ell \cdot \ell\right) \]

9. Applied egg-rr 16.1%

   \[\leadsto \color{blue}{\left({\left(\frac{1}{\sqrt[3]{0}}\right)}^{2} \cdot \frac{1}{\sqrt[3]{0}}\right)} \cdot \left(\ell \cdot \ell\right) \]
10. Step-by-step derivation

    1. pow-plus 16.1%

       \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt[3]{0}}\right)}^{\left(2 + 1\right)}} \cdot \left(\ell \cdot \ell\right) \]

    2. metadata-eval 16.1%

       \[\leadsto {\left(\frac{1}{\sqrt[3]{0}}\right)}^{\color{blue}{3}} \cdot \left(\ell \cdot \ell\right) \]

    3. cube-div 16.1%

       \[\leadsto \color{blue}{\frac{{1}^{3}}{{\left(\sqrt[3]{0}\right)}^{3}}} \cdot \left(\ell \cdot \ell\right) \]

    4. metadata-eval 16.1%

       \[\leadsto \frac{\color{blue}{1}}{{\left(\sqrt[3]{0}\right)}^{3}} \cdot \left(\ell \cdot \ell\right) \]

    5. rem-cube-cbrt 16.1%

       \[\leadsto \frac{1}{\color{blue}{0}} \cdot \left(\ell \cdot \ell\right) \]

    6. unpow-1 16.1%

       \[\leadsto \color{blue}{{0}^{-1}} \cdot \left(\ell \cdot \ell\right) \]

    7. pow-base-0 26.6%

       \[\leadsto \color{blue}{0} \cdot \left(\ell \cdot \ell\right) \]

11. Simplified 26.6%

    \[\leadsto \color{blue}{0} \cdot \left(\ell \cdot \ell\right) \]

12. Taylor expanded in l around 0 27.9%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(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))))