Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 88.6%

Time: 18.3s

Alternatives: 17

Speedup: 3.8×

• 39.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{l\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 8.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{k}{l\_m} \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{elif}\;l\_m \leq 2.3 \cdot 10^{+165}:\\ \;\;\;\;\frac{2}{t\_m \cdot {k}^{2}} \cdot \frac{\cos k \cdot {l\_m}^{2}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\frac{k}{t\_m}}}{{\left(\frac{t\_m}{t\_2} \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\sin k}\right)\right)}^{2}} \cdot \frac{t\_m \cdot \frac{\sqrt{2}}{k}}{\frac{t\_m \cdot \sqrt[3]{\sin k \cdot \tan k}}{t\_2}}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (pow (cbrt l_m) 2.0)))
   (*
    t_s
    (if (<= l_m 8.2e-148)
      (/ 2.0 (pow (* k (* (/ k l_m) (sqrt t_m))) 2.0))
      (if (<= l_m 2.3e+165)
        (*
         (/ 2.0 (* t_m (pow k 2.0)))
         (/ (* (cos k) (pow l_m 2.0)) (pow (sin k) 2.0)))
        (*
         (/
          (/ (sqrt 2.0) (/ k t_m))
          (pow (* (/ t_m t_2) (* (cbrt (tan k)) (cbrt (sin k)))) 2.0))
         (/
          (* t_m (/ (sqrt 2.0) k))
          (/ (* t_m (cbrt (* (sin k) (tan k)))) t_2))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = pow(cbrt(l_m), 2.0);
	double tmp;
	if (l_m <= 8.2e-148) {
		tmp = 2.0 / pow((k * ((k / l_m) * sqrt(t_m))), 2.0);
	} else if (l_m <= 2.3e+165) {
		tmp = (2.0 / (t_m * pow(k, 2.0))) * ((cos(k) * pow(l_m, 2.0)) / pow(sin(k), 2.0));
	} else {
		tmp = ((sqrt(2.0) / (k / t_m)) / pow(((t_m / t_2) * (cbrt(tan(k)) * cbrt(sin(k)))), 2.0)) * ((t_m * (sqrt(2.0) / k)) / ((t_m * cbrt((sin(k) * tan(k)))) / t_2));
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
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) {
	double t_2 = Math.pow(Math.cbrt(l_m), 2.0);
	double tmp;
	if (l_m <= 8.2e-148) {
		tmp = 2.0 / Math.pow((k * ((k / l_m) * Math.sqrt(t_m))), 2.0);
	} else if (l_m <= 2.3e+165) {
		tmp = (2.0 / (t_m * Math.pow(k, 2.0))) * ((Math.cos(k) * Math.pow(l_m, 2.0)) / Math.pow(Math.sin(k), 2.0));
	} else {
		tmp = ((Math.sqrt(2.0) / (k / t_m)) / Math.pow(((t_m / t_2) * (Math.cbrt(Math.tan(k)) * Math.cbrt(Math.sin(k)))), 2.0)) * ((t_m * (Math.sqrt(2.0) / k)) / ((t_m * Math.cbrt((Math.sin(k) * Math.tan(k)))) / t_2));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = cbrt(l_m) ^ 2.0
	tmp = 0.0
	if (l_m <= 8.2e-148)
		tmp = Float64(2.0 / (Float64(k * Float64(Float64(k / l_m) * sqrt(t_m))) ^ 2.0));
	elseif (l_m <= 2.3e+165)
		tmp = Float64(Float64(2.0 / Float64(t_m * (k ^ 2.0))) * Float64(Float64(cos(k) * (l_m ^ 2.0)) / (sin(k) ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / Float64(k / t_m)) / (Float64(Float64(t_m / t_2) * Float64(cbrt(tan(k)) * cbrt(sin(k)))) ^ 2.0)) * Float64(Float64(t_m * Float64(sqrt(2.0) / k)) / Float64(Float64(t_m * cbrt(Float64(sin(k) * tan(k)))) / t_2)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := Block[{t$95$2 = N[Power[N[Power[l$95$m, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 8.2e-148], N[(2.0 / N[Power[N[(k * N[(N[(k / l$95$m), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 2.3e+165], N[(N[(2.0 / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[(t$95$m / t$95$2), $MachinePrecision] * N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if l < 8.2000000000000005e-148

   1. Initial program 30.1%

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

      1. *-commutative 30.1%

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

      2. associate-/r* 30.2%

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

   3. Simplified 37.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 27.0%

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

      2. pow2 27.0%

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

      3. sqrt-prod 17.9%

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

      4. sqrt-div 17.9%

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

      5. sqrt-pow1 20.6%

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

      6. metadata-eval 20.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 6.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 28.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 28.0%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

   7. Taylor expanded in k around 0 31.4%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 31.4%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2}}} \]

      2. associate-/l/ 30.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 30.9%

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

      4. pow-prod-down 32.7%

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

      5. *-commutative 32.7%

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

   9. Applied egg-rr 32.7%

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

       1. *-lft-identity 32.7%

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

       2. associate-*l* 32.7%

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

       3. times-frac 29.5%

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

       4. associate-/l* 29.5%

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

   11. Simplified 29.5%

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

   12. Taylor expanded in t around 0 42.2%

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

   if 8.2000000000000005e-148 < l < 2.30000000000000016e165

   1. Initial program 46.1%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in t around 0 91.1%

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

      1. associate-*r/ 91.1%

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

      2. associate-*r* 91.1%

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

      3. times-frac 93.8%

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

      4. *-commutative 93.8%

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

   7. Simplified 93.8%

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

   if 2.30000000000000016e165 < l

   1. Initial program 48.5%

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

      1. *-commutative 48.5%

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

      2. associate-/r* 48.5%

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

   3. Simplified 48.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 48.5%

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

      2. add-cube-cbrt 48.5%

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

      3. times-frac 48.5%

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

   6. Applied egg-rr 82.8%

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

      1. associate-*l/ 82.8%

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

   8. Applied egg-rr 82.8%

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

      1. pow1/3 65.7%

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

      2. *-commutative 65.7%

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

      3. unpow-prod-down 41.6%

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

      4. pow1/3 45.1%

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

      5. pow1/3 83.0%

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

   10. Applied egg-rr 83.0%

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

       1. associate-/r/ 83.0%

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

   12. Applied egg-rr 83.0%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+165}:\\ \;\;\;\;\frac{2}{t \cdot {k}^{2}} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\sin k}\right)\right)}^{2}} \cdot \frac{t \cdot \frac{\sqrt{2}}{k}}{\frac{t \cdot \sqrt[3]{\sin k \cdot \tan k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{{\left(\sqrt[3]{l\_m}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3.5 \cdot 10^{-148}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{k}{l\_m} \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{elif}\;l\_m \leq 3.5 \cdot 10^{+165}:\\ \;\;\;\;\frac{2}{t\_m \cdot {k}^{2}} \cdot \frac{\cos k \cdot {l\_m}^{2}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\frac{k}{t\_m}}}{{t\_2}^{2}} \cdot \frac{\frac{t\_m \cdot \sqrt{2}}{k}}{t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (* (/ t_m (pow (cbrt l_m) 2.0)) (cbrt (* (sin k) (tan k))))))
   (*
    t_s
    (if (<= l_m 3.5e-148)
      (/ 2.0 (pow (* k (* (/ k l_m) (sqrt t_m))) 2.0))
      (if (<= l_m 3.5e+165)
        (*
         (/ 2.0 (* t_m (pow k 2.0)))
         (/ (* (cos k) (pow l_m 2.0)) (pow (sin k) 2.0)))
        (*
         (/ (/ (sqrt 2.0) (/ k t_m)) (pow t_2 2.0))
         (/ (/ (* t_m (sqrt 2.0)) k) t_2)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = (t_m / pow(cbrt(l_m), 2.0)) * cbrt((sin(k) * tan(k)));
	double tmp;
	if (l_m <= 3.5e-148) {
		tmp = 2.0 / pow((k * ((k / l_m) * sqrt(t_m))), 2.0);
	} else if (l_m <= 3.5e+165) {
		tmp = (2.0 / (t_m * pow(k, 2.0))) * ((cos(k) * pow(l_m, 2.0)) / pow(sin(k), 2.0));
	} else {
		tmp = ((sqrt(2.0) / (k / t_m)) / pow(t_2, 2.0)) * (((t_m * sqrt(2.0)) / k) / t_2);
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
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) {
	double t_2 = (t_m / Math.pow(Math.cbrt(l_m), 2.0)) * Math.cbrt((Math.sin(k) * Math.tan(k)));
	double tmp;
	if (l_m <= 3.5e-148) {
		tmp = 2.0 / Math.pow((k * ((k / l_m) * Math.sqrt(t_m))), 2.0);
	} else if (l_m <= 3.5e+165) {
		tmp = (2.0 / (t_m * Math.pow(k, 2.0))) * ((Math.cos(k) * Math.pow(l_m, 2.0)) / Math.pow(Math.sin(k), 2.0));
	} else {
		tmp = ((Math.sqrt(2.0) / (k / t_m)) / Math.pow(t_2, 2.0)) * (((t_m * Math.sqrt(2.0)) / k) / t_2);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(Float64(t_m / (cbrt(l_m) ^ 2.0)) * cbrt(Float64(sin(k) * tan(k))))
	tmp = 0.0
	if (l_m <= 3.5e-148)
		tmp = Float64(2.0 / (Float64(k * Float64(Float64(k / l_m) * sqrt(t_m))) ^ 2.0));
	elseif (l_m <= 3.5e+165)
		tmp = Float64(Float64(2.0 / Float64(t_m * (k ^ 2.0))) * Float64(Float64(cos(k) * (l_m ^ 2.0)) / (sin(k) ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / Float64(k / t_m)) / (t_2 ^ 2.0)) * Float64(Float64(Float64(t_m * sqrt(2.0)) / k) / t_2));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := Block[{t$95$2 = 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], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 3.5e-148], N[(2.0 / N[Power[N[(k * N[(N[(k / l$95$m), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 3.5e+165], N[(N[(2.0 / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if l < 3.5e-148

   1. Initial program 30.1%

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

      1. *-commutative 30.1%

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

      2. associate-/r* 30.2%

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

   3. Simplified 37.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 27.0%

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

      2. pow2 27.0%

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

      3. sqrt-prod 17.9%

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

      4. sqrt-div 17.9%

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

      5. sqrt-pow1 20.6%

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

      6. metadata-eval 20.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 6.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 28.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 28.0%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

   7. Taylor expanded in k around 0 31.4%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 31.4%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2}}} \]

      2. associate-/l/ 30.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 30.9%

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

      4. pow-prod-down 32.7%

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

      5. *-commutative 32.7%

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

   9. Applied egg-rr 32.7%

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

       1. *-lft-identity 32.7%

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

       2. associate-*l* 32.7%

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

       3. times-frac 29.5%

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

       4. associate-/l* 29.5%

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

   11. Simplified 29.5%

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

   12. Taylor expanded in t around 0 42.2%

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

   if 3.5e-148 < l < 3.49999999999999996e165

   1. Initial program 46.1%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in t around 0 91.1%

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

      1. associate-*r/ 91.1%

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

      2. associate-*r* 91.1%

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

      3. times-frac 93.8%

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

      4. *-commutative 93.8%

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

   7. Simplified 93.8%

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

   if 3.49999999999999996e165 < l

   1. Initial program 48.5%

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

      1. *-commutative 48.5%

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

      2. associate-/r* 48.5%

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

   3. Simplified 48.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 48.5%

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

      2. add-cube-cbrt 48.5%

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

      3. times-frac 48.5%

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

   6. Applied egg-rr 82.8%

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

   7. Taylor expanded in k around 0 82.9%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.5 \cdot 10^{-148}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+165}:\\ \;\;\;\;\frac{2}{t \cdot {k}^{2}} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\frac{t \cdot \sqrt{2}}{k}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{{\left(\sqrt[3]{l\_m}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 2.05 \cdot 10^{-147}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{k}{l\_m} \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{elif}\;l\_m \leq 2.6 \cdot 10^{+165}:\\ \;\;\;\;\frac{2}{t\_m \cdot {k}^{2}} \cdot \frac{\cos k \cdot {l\_m}^{2}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot \frac{\sqrt{2}}{k}}{{t\_2}^{2}} \cdot \frac{\sqrt{2}}{\frac{k}{t\_m} \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (let* ((t_2 (* (/ t_m (pow (cbrt l_m) 2.0)) (cbrt (* (sin k) (tan k))))))
   (*
    t_s
    (if (<= l_m 2.05e-147)
      (/ 2.0 (pow (* k (* (/ k l_m) (sqrt t_m))) 2.0))
      (if (<= l_m 2.6e+165)
        (*
         (/ 2.0 (* t_m (pow k 2.0)))
         (/ (* (cos k) (pow l_m 2.0)) (pow (sin k) 2.0)))
        (*
         (/ (* t_m (/ (sqrt 2.0) k)) (pow t_2 2.0))
         (/ (sqrt 2.0) (* (/ k t_m) t_2))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double t_2 = (t_m / pow(cbrt(l_m), 2.0)) * cbrt((sin(k) * tan(k)));
	double tmp;
	if (l_m <= 2.05e-147) {
		tmp = 2.0 / pow((k * ((k / l_m) * sqrt(t_m))), 2.0);
	} else if (l_m <= 2.6e+165) {
		tmp = (2.0 / (t_m * pow(k, 2.0))) * ((cos(k) * pow(l_m, 2.0)) / pow(sin(k), 2.0));
	} else {
		tmp = ((t_m * (sqrt(2.0) / k)) / pow(t_2, 2.0)) * (sqrt(2.0) / ((k / t_m) * t_2));
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
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) {
	double t_2 = (t_m / Math.pow(Math.cbrt(l_m), 2.0)) * Math.cbrt((Math.sin(k) * Math.tan(k)));
	double tmp;
	if (l_m <= 2.05e-147) {
		tmp = 2.0 / Math.pow((k * ((k / l_m) * Math.sqrt(t_m))), 2.0);
	} else if (l_m <= 2.6e+165) {
		tmp = (2.0 / (t_m * Math.pow(k, 2.0))) * ((Math.cos(k) * Math.pow(l_m, 2.0)) / Math.pow(Math.sin(k), 2.0));
	} else {
		tmp = ((t_m * (Math.sqrt(2.0) / k)) / Math.pow(t_2, 2.0)) * (Math.sqrt(2.0) / ((k / t_m) * t_2));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	t_2 = Float64(Float64(t_m / (cbrt(l_m) ^ 2.0)) * cbrt(Float64(sin(k) * tan(k))))
	tmp = 0.0
	if (l_m <= 2.05e-147)
		tmp = Float64(2.0 / (Float64(k * Float64(Float64(k / l_m) * sqrt(t_m))) ^ 2.0));
	elseif (l_m <= 2.6e+165)
		tmp = Float64(Float64(2.0 / Float64(t_m * (k ^ 2.0))) * Float64(Float64(cos(k) * (l_m ^ 2.0)) / (sin(k) ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(t_m * Float64(sqrt(2.0) / k)) / (t_2 ^ 2.0)) * Float64(sqrt(2.0) / Float64(Float64(k / t_m) * t_2)));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := Block[{t$95$2 = 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], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[l$95$m, 2.05e-147], N[(2.0 / N[Power[N[(k * N[(N[(k / l$95$m), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[l$95$m, 2.6e+165], N[(N[(2.0 / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if l < 2.05e-147

   1. Initial program 30.1%

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

      1. *-commutative 30.1%

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

      2. associate-/r* 30.2%

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

   3. Simplified 37.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 27.0%

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

      2. pow2 27.0%

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

      3. sqrt-prod 17.9%

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

      4. sqrt-div 17.9%

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

      5. sqrt-pow1 20.6%

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

      6. metadata-eval 20.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 6.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 28.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 28.0%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

   7. Taylor expanded in k around 0 31.4%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 31.4%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2}}} \]

      2. associate-/l/ 30.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 30.9%

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

      4. pow-prod-down 32.7%

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

      5. *-commutative 32.7%

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

   9. Applied egg-rr 32.7%

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

       1. *-lft-identity 32.7%

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

       2. associate-*l* 32.7%

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

       3. times-frac 29.5%

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

       4. associate-/l* 29.5%

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

   11. Simplified 29.5%

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

   12. Taylor expanded in t around 0 42.2%

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

   if 2.05e-147 < l < 2.6000000000000001e165

   1. Initial program 46.1%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in t around 0 91.1%

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

      1. associate-*r/ 91.1%

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

      2. associate-*r* 91.1%

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

      3. times-frac 93.8%

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

      4. *-commutative 93.8%

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

   7. Simplified 93.8%

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

   if 2.6000000000000001e165 < l

   1. Initial program 48.5%

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

      1. *-commutative 48.5%

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

      2. associate-/r* 48.5%

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

   3. Simplified 48.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 48.5%

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

      2. add-cube-cbrt 48.5%

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

      3. times-frac 48.5%

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

   6. Applied egg-rr 82.8%

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

      1. associate-/r/ 82.8%

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

      2. associate-/l/ 82.9%

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

   8. Simplified 82.9%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.05 \cdot 10^{-147}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+165}:\\ \;\;\;\;\frac{2}{t \cdot {k}^{2}} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}^{2}} \cdot \frac{\sqrt{2}}{\frac{k}{t} \cdot \left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 28500000:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{\sin k}{l\_m}\right) \cdot \sqrt{\frac{t\_m}{\cos k}}\right)}^{2}}\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{+210}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \tan k}}{{\left(k \cdot \frac{{t\_m}^{1.5}}{l\_m \cdot t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{l\_m}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t\_m}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 28500000.0)
    (/ 2.0 (pow (* (* k (/ (sin k) l_m)) (sqrt (/ t_m (cos k)))) 2.0))
    (if (<= k 5.8e+210)
      (/
       (/ 2.0 (* (sin k) (tan k)))
       (pow (* k (/ (pow t_m 1.5) (* l_m t_m))) 2.0))
      (/
       2.0
       (pow
        (*
         (/ t_m (pow (cbrt l_m) 2.0))
         (cbrt (* (sin k) (* (tan k) (pow (/ k t_m) 2.0)))))
        3.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 28500000.0) {
		tmp = 2.0 / pow(((k * (sin(k) / l_m)) * sqrt((t_m / cos(k)))), 2.0);
	} else if (k <= 5.8e+210) {
		tmp = (2.0 / (sin(k) * tan(k))) / pow((k * (pow(t_m, 1.5) / (l_m * t_m))), 2.0);
	} else {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l_m), 2.0)) * cbrt((sin(k) * (tan(k) * pow((k / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (k <= 28500000.0) {
		tmp = 2.0 / Math.pow(((k * (Math.sin(k) / l_m)) * Math.sqrt((t_m / Math.cos(k)))), 2.0);
	} else if (k <= 5.8e+210) {
		tmp = (2.0 / (Math.sin(k) * Math.tan(k))) / Math.pow((k * (Math.pow(t_m, 1.5) / (l_m * t_m))), 2.0);
	} else {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l_m), 2.0)) * Math.cbrt((Math.sin(k) * (Math.tan(k) * Math.pow((k / t_m), 2.0))))), 3.0);
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 28500000.0)
		tmp = Float64(2.0 / (Float64(Float64(k * Float64(sin(k) / l_m)) * sqrt(Float64(t_m / cos(k)))) ^ 2.0));
	elseif (k <= 5.8e+210)
		tmp = Float64(Float64(2.0 / Float64(sin(k) * tan(k))) / (Float64(k * Float64((t_m ^ 1.5) / Float64(l_m * t_m))) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l_m) ^ 2.0)) * cbrt(Float64(sin(k) * Float64(tan(k) * (Float64(k / t_m) ^ 2.0))))) ^ 3.0));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[k, 28500000.0], N[(2.0 / N[Power[N[(N[(k * N[(N[Sin[k], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.8e+210], N[(N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / N[(l$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 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], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.85e7

   1. Initial program 39.3%

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

      1. *-commutative 39.3%

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

      2. associate-/r* 39.3%

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

   3. Simplified 45.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.2%

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

      2. pow2 28.2%

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

      3. sqrt-prod 21.8%

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

      4. sqrt-div 22.3%

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

      5. sqrt-pow1 25.5%

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

      6. metadata-eval 25.5%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 17.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 31.7%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 31.7%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 31.7%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. +-rgt-identity 31.7%

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

      3. associate-/l/ 31.3%

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

      4. pow-prod-down 33.3%

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

      5. *-commutative 33.3%

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

   8. Applied egg-rr 33.3%

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

      1. *-lft-identity 33.3%

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

      2. associate-*l* 33.3%

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

   10. Simplified 33.3%

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

   11. Taylor expanded in k around inf 53.9%

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

       1. associate-/l* 56.3%

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

   13. Simplified 56.3%

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

   if 2.85e7 < k < 5.79999999999999984e210

   1. Initial program 18.8%

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

      1. *-commutative 18.8%

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

      2. associate-/r* 18.8%

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

   3. Simplified 29.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 20.9%

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

      2. pow2 20.9%

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

      3. sqrt-prod 7.7%

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

      4. sqrt-div 7.7%

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

      5. sqrt-pow1 10.4%

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

      6. metadata-eval 10.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 5.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 10.5%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 10.5%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 10.5%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. +-rgt-identity 10.5%

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

      3. associate-/l/ 10.5%

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

      4. pow-prod-down 13.0%

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

      5. *-commutative 13.0%

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

   8. Applied egg-rr 13.0%

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

      1. *-lft-identity 13.0%

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

      2. associate-*l* 12.9%

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

   10. Simplified 12.9%

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

       1. *-un-lft-identity 12.9%

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

       2. unpow-prod-down 13.0%

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

       3. pow2 13.0%

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

       4. add-sqr-sqrt 23.2%

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

       5. frac-times 30.9%

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

   12. Applied egg-rr 30.9%

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

       1. *-lft-identity 30.9%

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

       2. associate-/r* 30.8%

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

       3. *-commutative 30.8%

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

       4. associate-/l* 30.8%

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

   14. Simplified 30.8%

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

   if 5.79999999999999984e210 < k

   1. Initial program 42.9%

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

   3. Simplified 42.9%

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

      1. add-cube-cbrt 42.9%

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

      2. pow3 42.9%

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

   6. Applied egg-rr 88.6%

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

Alternative 5: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.75 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{\sin k}{l\_m}\right) \cdot \sqrt{\frac{t\_m}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\cos k}{{k}^{2}}}{t\_m \cdot {\sin k}^{2}}\right) \cdot \left(l\_m \cdot l\_m\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.75e-22)
    (/ 2.0 (pow (* (* k (/ (sin k) l_m)) (sqrt (/ t_m (cos k)))) 2.0))
    (*
     (* 2.0 (/ (/ (cos k) (pow k 2.0)) (* t_m (pow (sin k) 2.0))))
     (* l_m l_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2.75e-22) {
		tmp = 2.0 / pow(((k * (sin(k) / l_m)) * sqrt((t_m / cos(k)))), 2.0);
	} else {
		tmp = (2.0 * ((cos(k) / pow(k, 2.0)) / (t_m * pow(sin(k), 2.0)))) * (l_m * l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.75d-22) then
        tmp = 2.0d0 / (((k * (sin(k) / l_m)) * sqrt((t_m / cos(k)))) ** 2.0d0)
    else
        tmp = (2.0d0 * ((cos(k) / (k ** 2.0d0)) / (t_m * (sin(k) ** 2.0d0)))) * (l_m * l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (k <= 2.75e-22) {
		tmp = 2.0 / Math.pow(((k * (Math.sin(k) / l_m)) * Math.sqrt((t_m / Math.cos(k)))), 2.0);
	} else {
		tmp = (2.0 * ((Math.cos(k) / Math.pow(k, 2.0)) / (t_m * Math.pow(Math.sin(k), 2.0)))) * (l_m * l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 2.75e-22:
		tmp = 2.0 / math.pow(((k * (math.sin(k) / l_m)) * math.sqrt((t_m / math.cos(k)))), 2.0)
	else:
		tmp = (2.0 * ((math.cos(k) / math.pow(k, 2.0)) / (t_m * math.pow(math.sin(k), 2.0)))) * (l_m * l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 2.75e-22)
		tmp = Float64(2.0 / (Float64(Float64(k * Float64(sin(k) / l_m)) * sqrt(Float64(t_m / cos(k)))) ^ 2.0));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k) / (k ^ 2.0)) / Float64(t_m * (sin(k) ^ 2.0)))) * Float64(l_m * l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 2.75e-22)
		tmp = 2.0 / (((k * (sin(k) / l_m)) * sqrt((t_m / cos(k)))) ^ 2.0);
	else
		tmp = (2.0 * ((cos(k) / (k ^ 2.0)) / (t_m * (sin(k) ^ 2.0)))) * (l_m * l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[k, 2.75e-22], N[(2.0 / N[Power[N[(N[(k * N[(N[Sin[k], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.7500000000000001e-22

   1. Initial program 40.2%

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

      1. *-commutative 40.2%

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

      2. associate-/r* 40.2%

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

   3. Simplified 46.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.3%

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

      2. pow2 28.3%

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

      3. sqrt-prod 21.7%

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

      4. sqrt-div 22.2%

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

      5. sqrt-pow1 25.6%

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

      6. metadata-eval 25.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 17.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 31.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 31.8%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 31.8%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. +-rgt-identity 31.8%

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

      3. associate-/l/ 31.4%

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

      4. pow-prod-down 33.5%

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

      5. *-commutative 33.5%

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

   8. Applied egg-rr 33.5%

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

      1. *-lft-identity 33.5%

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

      2. associate-*l* 33.5%

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

   10. Simplified 33.5%

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

   11. Taylor expanded in k around inf 54.0%

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

       1. associate-/l* 56.5%

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

   13. Simplified 56.5%

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

   if 2.7500000000000001e-22 < k

   1. Initial program 27.2%

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

   3. Simplified 41.2%

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

   5. Taylor expanded in t around 0 78.9%

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

      1. associate-/r* 79.0%

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

   7. Simplified 79.0%

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

Alternative 6: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{k}{l\_m} \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t\_m}{\cos k}} \cdot \frac{k \cdot \sin k}{l\_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 5e-30)
    (/ 2.0 (pow (* k (* (/ k l_m) (sqrt t_m))) 2.0))
    (/ 2.0 (pow (* (sqrt (/ t_m (cos k))) (/ (* k (sin k)) l_m)) 2.0)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((l_m * l_m) <= 5e-30) {
		tmp = 2.0 / pow((k * ((k / l_m) * sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 / pow((sqrt((t_m / cos(k))) * ((k * sin(k)) / l_m)), 2.0);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l_m * l_m) <= 5d-30) then
        tmp = 2.0d0 / ((k * ((k / l_m) * sqrt(t_m))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((sqrt((t_m / cos(k))) * ((k * sin(k)) / l_m)) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if ((l_m * l_m) <= 5e-30) {
		tmp = 2.0 / Math.pow((k * ((k / l_m) * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 / Math.pow((Math.sqrt((t_m / Math.cos(k))) * ((k * Math.sin(k)) / l_m)), 2.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if (l_m * l_m) <= 5e-30:
		tmp = 2.0 / math.pow((k * ((k / l_m) * math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 / math.pow((math.sqrt((t_m / math.cos(k))) * ((k * math.sin(k)) / l_m)), 2.0)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 5e-30)
		tmp = Float64(2.0 / (Float64(k * Float64(Float64(k / l_m) * sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(sqrt(Float64(t_m / cos(k))) * Float64(Float64(k * sin(k)) / l_m)) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if ((l_m * l_m) <= 5e-30)
		tmp = 2.0 / ((k * ((k / l_m) * sqrt(t_m))) ^ 2.0);
	else
		tmp = 2.0 / ((sqrt((t_m / cos(k))) * ((k * sin(k)) / l_m)) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 5e-30], N[(2.0 / N[Power[N[(k * N[(N[(k / l$95$m), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.99999999999999972e-30

   1. Initial program 29.3%

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

      1. *-commutative 29.3%

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

      2. associate-/r* 29.3%

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

   3. Simplified 40.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 27.6%

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

      2. pow2 27.6%

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

      3. sqrt-prod 16.8%

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

      4. sqrt-div 17.6%

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

      5. sqrt-pow1 21.9%

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

      6. metadata-eval 21.9%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 15.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 30.9%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 30.9%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

   7. Taylor expanded in k around 0 39.2%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 39.2%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2}}} \]

      2. associate-/l/ 38.6%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 38.6%

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

      4. pow-prod-down 40.9%

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

      5. *-commutative 40.9%

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

   9. Applied egg-rr 40.9%

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

       1. *-lft-identity 40.9%

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

       2. associate-*l* 40.9%

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

       3. times-frac 35.1%

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

       4. associate-/l* 35.1%

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

   11. Simplified 35.1%

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

   12. Taylor expanded in t around 0 51.7%

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

   if 4.99999999999999972e-30 < (*.f64 l l)

   1. Initial program 43.0%

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

      1. *-commutative 43.0%

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

      2. associate-/r* 43.0%

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

   3. Simplified 47.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 27.5%

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

      2. pow2 27.5%

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

      3. sqrt-prod 21.3%

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

      4. sqrt-div 21.3%

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

      5. sqrt-pow1 23.6%

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

      6. metadata-eval 23.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 13.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 24.9%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 24.9%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 24.9%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. +-rgt-identity 24.9%

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

      3. associate-/l/ 24.9%

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

      4. pow-prod-down 26.6%

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

      5. *-commutative 26.6%

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

   8. Applied egg-rr 26.6%

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

      1. *-lft-identity 26.6%

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

      2. associate-*l* 26.6%

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

   10. Simplified 26.6%

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

   11. Taylor expanded in k around inf 51.8%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\frac{t}{\cos k}} \cdot \frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{k}{l\_m} \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{\sin k}{l\_m}\right) \cdot \sqrt{\frac{t\_m}{\cos k}}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 5e-30)
    (/ 2.0 (pow (* k (* (/ k l_m) (sqrt t_m))) 2.0))
    (/ 2.0 (pow (* (* k (/ (sin k) l_m)) (sqrt (/ t_m (cos k)))) 2.0)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if ((l_m * l_m) <= 5e-30) {
		tmp = 2.0 / pow((k * ((k / l_m) * sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 / pow(((k * (sin(k) / l_m)) * sqrt((t_m / cos(k)))), 2.0);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l_m * l_m) <= 5d-30) then
        tmp = 2.0d0 / ((k * ((k / l_m) * sqrt(t_m))) ** 2.0d0)
    else
        tmp = 2.0d0 / (((k * (sin(k) / l_m)) * sqrt((t_m / cos(k)))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if ((l_m * l_m) <= 5e-30) {
		tmp = 2.0 / Math.pow((k * ((k / l_m) * Math.sqrt(t_m))), 2.0);
	} else {
		tmp = 2.0 / Math.pow(((k * (Math.sin(k) / l_m)) * Math.sqrt((t_m / Math.cos(k)))), 2.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if (l_m * l_m) <= 5e-30:
		tmp = 2.0 / math.pow((k * ((k / l_m) * math.sqrt(t_m))), 2.0)
	else:
		tmp = 2.0 / math.pow(((k * (math.sin(k) / l_m)) * math.sqrt((t_m / math.cos(k)))), 2.0)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 5e-30)
		tmp = Float64(2.0 / (Float64(k * Float64(Float64(k / l_m) * sqrt(t_m))) ^ 2.0));
	else
		tmp = Float64(2.0 / (Float64(Float64(k * Float64(sin(k) / l_m)) * sqrt(Float64(t_m / cos(k)))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if ((l_m * l_m) <= 5e-30)
		tmp = 2.0 / ((k * ((k / l_m) * sqrt(t_m))) ^ 2.0);
	else
		tmp = 2.0 / (((k * (sin(k) / l_m)) * sqrt((t_m / cos(k)))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 5e-30], N[(2.0 / N[Power[N[(k * N[(N[(k / l$95$m), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(k * N[(N[Sin[k], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.99999999999999972e-30

   1. Initial program 29.3%

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

      1. *-commutative 29.3%

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

      2. associate-/r* 29.3%

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

   3. Simplified 40.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 27.6%

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

      2. pow2 27.6%

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

      3. sqrt-prod 16.8%

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

      4. sqrt-div 17.6%

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

      5. sqrt-pow1 21.9%

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

      6. metadata-eval 21.9%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 15.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 30.9%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 30.9%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

   7. Taylor expanded in k around 0 39.2%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 39.2%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2}}} \]

      2. associate-/l/ 38.6%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 38.6%

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

      4. pow-prod-down 40.9%

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

      5. *-commutative 40.9%

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

   9. Applied egg-rr 40.9%

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

       1. *-lft-identity 40.9%

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

       2. associate-*l* 40.9%

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

       3. times-frac 35.1%

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

       4. associate-/l* 35.1%

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

   11. Simplified 35.1%

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

   12. Taylor expanded in t around 0 51.7%

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

   if 4.99999999999999972e-30 < (*.f64 l l)

   1. Initial program 43.0%

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

      1. *-commutative 43.0%

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

      2. associate-/r* 43.0%

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

   3. Simplified 47.6%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 27.5%

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

      2. pow2 27.5%

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

      3. sqrt-prod 21.3%

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

      4. sqrt-div 21.3%

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

      5. sqrt-pow1 23.6%

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

      6. metadata-eval 23.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 13.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 24.9%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 24.9%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 24.9%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. +-rgt-identity 24.9%

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

      3. associate-/l/ 24.9%

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

      4. pow-prod-down 26.6%

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

      5. *-commutative 26.6%

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

   8. Applied egg-rr 26.6%

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

      1. *-lft-identity 26.6%

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

      2. associate-*l* 26.6%

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

   10. Simplified 26.6%

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

   11. Taylor expanded in k around inf 51.8%

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

       1. associate-/l* 51.8%

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

   13. Simplified 51.8%

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

Alternative 8: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.75 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{\sin k}{l\_m}\right) \cdot \sqrt{\frac{t\_m}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{l\_m}^{2} \cdot \left(\frac{2}{{\left(k \cdot \sin k\right)}^{2}} \cdot \frac{\cos k}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.75e-22)
    (/ 2.0 (pow (* (* k (/ (sin k) l_m)) (sqrt (/ t_m (cos k)))) 2.0))
    (* (pow l_m 2.0) (* (/ 2.0 (pow (* k (sin k)) 2.0)) (/ (cos k) t_m))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2.75e-22) {
		tmp = 2.0 / pow(((k * (sin(k) / l_m)) * sqrt((t_m / cos(k)))), 2.0);
	} else {
		tmp = pow(l_m, 2.0) * ((2.0 / pow((k * sin(k)), 2.0)) * (cos(k) / t_m));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.75d-22) then
        tmp = 2.0d0 / (((k * (sin(k) / l_m)) * sqrt((t_m / cos(k)))) ** 2.0d0)
    else
        tmp = (l_m ** 2.0d0) * ((2.0d0 / ((k * sin(k)) ** 2.0d0)) * (cos(k) / t_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (k <= 2.75e-22) {
		tmp = 2.0 / Math.pow(((k * (Math.sin(k) / l_m)) * Math.sqrt((t_m / Math.cos(k)))), 2.0);
	} else {
		tmp = Math.pow(l_m, 2.0) * ((2.0 / Math.pow((k * Math.sin(k)), 2.0)) * (Math.cos(k) / t_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 2.75e-22:
		tmp = 2.0 / math.pow(((k * (math.sin(k) / l_m)) * math.sqrt((t_m / math.cos(k)))), 2.0)
	else:
		tmp = math.pow(l_m, 2.0) * ((2.0 / math.pow((k * math.sin(k)), 2.0)) * (math.cos(k) / t_m))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 2.75e-22)
		tmp = Float64(2.0 / (Float64(Float64(k * Float64(sin(k) / l_m)) * sqrt(Float64(t_m / cos(k)))) ^ 2.0));
	else
		tmp = Float64((l_m ^ 2.0) * Float64(Float64(2.0 / (Float64(k * sin(k)) ^ 2.0)) * Float64(cos(k) / t_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 2.75e-22)
		tmp = 2.0 / (((k * (sin(k) / l_m)) * sqrt((t_m / cos(k)))) ^ 2.0);
	else
		tmp = (l_m ^ 2.0) * ((2.0 / ((k * sin(k)) ^ 2.0)) * (cos(k) / t_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[k, 2.75e-22], N[(2.0 / N[Power[N[(N[(k * N[(N[Sin[k], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(2.0 / N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.7500000000000001e-22

   1. Initial program 40.2%

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

      1. *-commutative 40.2%

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

      2. associate-/r* 40.2%

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

   3. Simplified 46.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.3%

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

      2. pow2 28.3%

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

      3. sqrt-prod 21.7%

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

      4. sqrt-div 22.2%

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

      5. sqrt-pow1 25.6%

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

      6. metadata-eval 25.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 17.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 31.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 31.8%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 31.8%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. +-rgt-identity 31.8%

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

      3. associate-/l/ 31.4%

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

      4. pow-prod-down 33.5%

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

      5. *-commutative 33.5%

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

   8. Applied egg-rr 33.5%

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

      1. *-lft-identity 33.5%

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

      2. associate-*l* 33.5%

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

   10. Simplified 33.5%

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

   11. Taylor expanded in k around inf 54.0%

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

       1. associate-/l* 56.5%

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

   13. Simplified 56.5%

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

   if 2.7500000000000001e-22 < k

   1. Initial program 27.2%

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

      1. *-commutative 27.2%

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

      2. associate-/r* 27.3%

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

   3. Simplified 38.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 38.7%

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

      2. add-cube-cbrt 38.6%

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

      3. times-frac 38.7%

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

   6. Applied egg-rr 71.7%

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

   7. Taylor expanded in l around -inf 78.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   8. Step-by-step derivation

      1. associate-/l* 78.8%

         \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. *-commutative 78.8%

         \[\leadsto {\ell}^{2} \cdot \frac{\color{blue}{\left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right) \cdot {\left(\sqrt[3]{-1}\right)}^{6}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      3. *-commutative 78.8%

         \[\leadsto {\ell}^{2} \cdot \frac{\left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right) \cdot {\left(\sqrt[3]{-1}\right)}^{6}}{{k}^{2} \cdot \color{blue}{\left({\sin k}^{2} \cdot t\right)}} \]

   9. Simplified 78.8%

      \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{\left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right) \cdot {\left(\sqrt[3]{-1}\right)}^{6}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   10. Step-by-step derivation

       1. associate-/l* 78.8%

          \[\leadsto {\ell}^{2} \cdot \color{blue}{\left(\left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right) \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{6}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}\right)} \]

       2. sqrt-pow2 78.9%

          \[\leadsto {\ell}^{2} \cdot \left(\left(\cos k \cdot \color{blue}{{2}^{\left(\frac{2}{2}\right)}}\right) \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{6}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}\right) \]

       3. metadata-eval 78.9%

          \[\leadsto {\ell}^{2} \cdot \left(\left(\cos k \cdot {2}^{\color{blue}{1}}\right) \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{6}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}\right) \]

       4. metadata-eval 78.9%

          \[\leadsto {\ell}^{2} \cdot \left(\left(\cos k \cdot \color{blue}{2}\right) \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{6}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}\right) \]

       5. *-commutative 78.9%

          \[\leadsto {\ell}^{2} \cdot \left(\color{blue}{\left(2 \cdot \cos k\right)} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{6}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}\right) \]

       6. pow1/3 0.0%

          \[\leadsto {\ell}^{2} \cdot \left(\left(2 \cdot \cos k\right) \cdot \frac{{\color{blue}{\left({-1}^{0.3333333333333333}\right)}}^{6}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}\right) \]

       7. pow-pow 78.9%

          \[\leadsto {\ell}^{2} \cdot \left(\left(2 \cdot \cos k\right) \cdot \frac{\color{blue}{{-1}^{\left(0.3333333333333333 \cdot 6\right)}}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}\right) \]

       8. metadata-eval 78.9%

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

       9. metadata-eval 78.9%

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

       10. associate-*r* 78.9%

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

       11. pow-prod-down 78.9%

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

   11. Applied egg-rr 78.9%

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

       1. associate-*r/ 78.9%

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

       2. *-rgt-identity 78.9%

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

       3. times-frac 78.9%

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

   13. Simplified 78.9%

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

Alternative 9: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.75 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{{\left(\left(k \cdot \frac{\sin k}{l\_m}\right) \cdot \sqrt{\frac{t\_m}{\cos k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{l\_m}^{2}}{{\left(k \cdot \sin k\right)}^{2}} \cdot \frac{2 \cdot \cos k}{t\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.75e-22)
    (/ 2.0 (pow (* (* k (/ (sin k) l_m)) (sqrt (/ t_m (cos k)))) 2.0))
    (* (/ (pow l_m 2.0) (pow (* k (sin k)) 2.0)) (/ (* 2.0 (cos k)) t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 2.75e-22) {
		tmp = 2.0 / pow(((k * (sin(k) / l_m)) * sqrt((t_m / cos(k)))), 2.0);
	} else {
		tmp = (pow(l_m, 2.0) / pow((k * sin(k)), 2.0)) * ((2.0 * cos(k)) / t_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.75d-22) then
        tmp = 2.0d0 / (((k * (sin(k) / l_m)) * sqrt((t_m / cos(k)))) ** 2.0d0)
    else
        tmp = ((l_m ** 2.0d0) / ((k * sin(k)) ** 2.0d0)) * ((2.0d0 * cos(k)) / t_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (k <= 2.75e-22) {
		tmp = 2.0 / Math.pow(((k * (Math.sin(k) / l_m)) * Math.sqrt((t_m / Math.cos(k)))), 2.0);
	} else {
		tmp = (Math.pow(l_m, 2.0) / Math.pow((k * Math.sin(k)), 2.0)) * ((2.0 * Math.cos(k)) / t_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 2.75e-22:
		tmp = 2.0 / math.pow(((k * (math.sin(k) / l_m)) * math.sqrt((t_m / math.cos(k)))), 2.0)
	else:
		tmp = (math.pow(l_m, 2.0) / math.pow((k * math.sin(k)), 2.0)) * ((2.0 * math.cos(k)) / t_m)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 2.75e-22)
		tmp = Float64(2.0 / (Float64(Float64(k * Float64(sin(k) / l_m)) * sqrt(Float64(t_m / cos(k)))) ^ 2.0));
	else
		tmp = Float64(Float64((l_m ^ 2.0) / (Float64(k * sin(k)) ^ 2.0)) * Float64(Float64(2.0 * cos(k)) / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 2.75e-22)
		tmp = 2.0 / (((k * (sin(k) / l_m)) * sqrt((t_m / cos(k)))) ^ 2.0);
	else
		tmp = ((l_m ^ 2.0) / ((k * sin(k)) ^ 2.0)) * ((2.0 * cos(k)) / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[k, 2.75e-22], N[(2.0 / N[Power[N[(N[(k * N[(N[Sin[k], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$m / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l$95$m, 2.0], $MachinePrecision] / N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.7500000000000001e-22

   1. Initial program 40.2%

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

      1. *-commutative 40.2%

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

      2. associate-/r* 40.2%

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

   3. Simplified 46.2%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.3%

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

      2. pow2 28.3%

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

      3. sqrt-prod 21.7%

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

      4. sqrt-div 22.2%

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

      5. sqrt-pow1 25.6%

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

      6. metadata-eval 25.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 17.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 31.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 31.8%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 31.8%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. +-rgt-identity 31.8%

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

      3. associate-/l/ 31.4%

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

      4. pow-prod-down 33.5%

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

      5. *-commutative 33.5%

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

   8. Applied egg-rr 33.5%

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

      1. *-lft-identity 33.5%

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

      2. associate-*l* 33.5%

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

   10. Simplified 33.5%

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

   11. Taylor expanded in k around inf 54.0%

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

       1. associate-/l* 56.5%

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

   13. Simplified 56.5%

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

   if 2.7500000000000001e-22 < k

   1. Initial program 27.2%

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

      1. *-commutative 27.2%

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

      2. associate-/r* 27.3%

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

   3. Simplified 38.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 38.7%

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

      2. add-cube-cbrt 38.6%

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

      3. times-frac 38.7%

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

   6. Applied egg-rr 71.7%

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

   7. Taylor expanded in l around -inf 78.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   8. Step-by-step derivation

      1. associate-/l* 78.8%

         \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{6} \cdot \left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

      2. *-commutative 78.8%

         \[\leadsto {\ell}^{2} \cdot \frac{\color{blue}{\left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right) \cdot {\left(\sqrt[3]{-1}\right)}^{6}}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      3. *-commutative 78.8%

         \[\leadsto {\ell}^{2} \cdot \frac{\left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right) \cdot {\left(\sqrt[3]{-1}\right)}^{6}}{{k}^{2} \cdot \color{blue}{\left({\sin k}^{2} \cdot t\right)}} \]

   9. Simplified 78.8%

      \[\leadsto \color{blue}{{\ell}^{2} \cdot \frac{\left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right) \cdot {\left(\sqrt[3]{-1}\right)}^{6}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
   10. Step-by-step derivation

       1. pow2 78.8%

          \[\leadsto \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{\left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right) \cdot {\left(\sqrt[3]{-1}\right)}^{6}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

       2. associate-*r/ 78.8%

          \[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot \left(\left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right) \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]

       3. pow2 78.8%

          \[\leadsto \frac{\color{blue}{{\ell}^{2}} \cdot \left(\left(\cos k \cdot {\left(\sqrt{2}\right)}^{2}\right) \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

       4. sqrt-pow2 78.9%

          \[\leadsto \frac{{\ell}^{2} \cdot \left(\left(\cos k \cdot \color{blue}{{2}^{\left(\frac{2}{2}\right)}}\right) \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

       5. metadata-eval 78.9%

          \[\leadsto \frac{{\ell}^{2} \cdot \left(\left(\cos k \cdot {2}^{\color{blue}{1}}\right) \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

       6. metadata-eval 78.9%

          \[\leadsto \frac{{\ell}^{2} \cdot \left(\left(\cos k \cdot \color{blue}{2}\right) \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

       7. associate-*l* 78.9%

          \[\leadsto \frac{{\ell}^{2} \cdot \color{blue}{\left(\cos k \cdot \left(2 \cdot {\left(\sqrt[3]{-1}\right)}^{6}\right)\right)}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

       8. pow1/3 0.0%

          \[\leadsto \frac{{\ell}^{2} \cdot \left(\cos k \cdot \left(2 \cdot {\color{blue}{\left({-1}^{0.3333333333333333}\right)}}^{6}\right)\right)}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

       9. pow-pow 78.9%

          \[\leadsto \frac{{\ell}^{2} \cdot \left(\cos k \cdot \left(2 \cdot \color{blue}{{-1}^{\left(0.3333333333333333 \cdot 6\right)}}\right)\right)}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

       10. metadata-eval 78.9%

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

       11. metadata-eval 78.9%

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

       12. metadata-eval 78.9%

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

       13. *-commutative 78.9%

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

       14. associate-*r* 78.9%

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

       15. pow-prod-down 78.9%

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

   11. Applied egg-rr 78.9%

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

       1. times-frac 78.9%

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

   13. Simplified 78.9%

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

Alternative 10: 76.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{k}{l\_m} \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{+135}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t\_m}^{2}}{l\_m} \cdot \frac{t\_m}{l\_m}\right)\right) \cdot \left(\left(k \cdot \frac{k}{t\_m}\right) \cdot \frac{1}{t\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(k \cdot \frac{\sqrt{t\_m}}{l\_m}\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.1e+18)
    (/ 2.0 (pow (* k (* (/ k l_m) (sqrt t_m))) 2.0))
    (if (<= k 2.5e+135)
      (/
       2.0
       (*
        (* (* (sin k) (tan k)) (* (/ (pow t_m 2.0) l_m) (/ t_m l_m)))
        (* (* k (/ k t_m)) (/ 1.0 t_m))))
      (/ 2.0 (pow (* k (* k (/ (sqrt t_m) l_m))) 2.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.1e+18) {
		tmp = 2.0 / pow((k * ((k / l_m) * sqrt(t_m))), 2.0);
	} else if (k <= 2.5e+135) {
		tmp = 2.0 / (((sin(k) * tan(k)) * ((pow(t_m, 2.0) / l_m) * (t_m / l_m))) * ((k * (k / t_m)) * (1.0 / t_m)));
	} else {
		tmp = 2.0 / pow((k * (k * (sqrt(t_m) / l_m))), 2.0);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.1d+18) then
        tmp = 2.0d0 / ((k * ((k / l_m) * sqrt(t_m))) ** 2.0d0)
    else if (k <= 2.5d+135) then
        tmp = 2.0d0 / (((sin(k) * tan(k)) * (((t_m ** 2.0d0) / l_m) * (t_m / l_m))) * ((k * (k / t_m)) * (1.0d0 / t_m)))
    else
        tmp = 2.0d0 / ((k * (k * (sqrt(t_m) / l_m))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (k <= 1.1e+18) {
		tmp = 2.0 / Math.pow((k * ((k / l_m) * Math.sqrt(t_m))), 2.0);
	} else if (k <= 2.5e+135) {
		tmp = 2.0 / (((Math.sin(k) * Math.tan(k)) * ((Math.pow(t_m, 2.0) / l_m) * (t_m / l_m))) * ((k * (k / t_m)) * (1.0 / t_m)));
	} else {
		tmp = 2.0 / Math.pow((k * (k * (Math.sqrt(t_m) / l_m))), 2.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 1.1e+18:
		tmp = 2.0 / math.pow((k * ((k / l_m) * math.sqrt(t_m))), 2.0)
	elif k <= 2.5e+135:
		tmp = 2.0 / (((math.sin(k) * math.tan(k)) * ((math.pow(t_m, 2.0) / l_m) * (t_m / l_m))) * ((k * (k / t_m)) * (1.0 / t_m)))
	else:
		tmp = 2.0 / math.pow((k * (k * (math.sqrt(t_m) / l_m))), 2.0)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 1.1e+18)
		tmp = Float64(2.0 / (Float64(k * Float64(Float64(k / l_m) * sqrt(t_m))) ^ 2.0));
	elseif (k <= 2.5e+135)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * tan(k)) * Float64(Float64((t_m ^ 2.0) / l_m) * Float64(t_m / l_m))) * Float64(Float64(k * Float64(k / t_m)) * Float64(1.0 / t_m))));
	else
		tmp = Float64(2.0 / (Float64(k * Float64(k * Float64(sqrt(t_m) / l_m))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 1.1e+18)
		tmp = 2.0 / ((k * ((k / l_m) * sqrt(t_m))) ^ 2.0);
	elseif (k <= 2.5e+135)
		tmp = 2.0 / (((sin(k) * tan(k)) * (((t_m ^ 2.0) / l_m) * (t_m / l_m))) * ((k * (k / t_m)) * (1.0 / t_m)));
	else
		tmp = 2.0 / ((k * (k * (sqrt(t_m) / l_m))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[k, 1.1e+18], N[(2.0 / N[Power[N[(k * N[(N[(k / l$95$m), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.5e+135], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $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[(N[(k * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(k * N[(k * N[(N[Sqrt[t$95$m], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.1e18

   1. Initial program 38.9%

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

      1. *-commutative 38.9%

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

      2. associate-/r* 38.9%

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

   3. Simplified 45.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 27.9%

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

      2. pow2 27.9%

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

      3. sqrt-prod 21.5%

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

      4. sqrt-div 22.1%

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

      5. sqrt-pow1 25.3%

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

      6. metadata-eval 25.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 17.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 31.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 31.3%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

   7. Taylor expanded in k around 0 34.2%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 34.2%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2}}} \]

      2. associate-/l/ 33.8%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 33.8%

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

      4. pow-prod-down 35.8%

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

      5. *-commutative 35.8%

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

   9. Applied egg-rr 35.8%

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

       1. *-lft-identity 35.8%

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

       2. associate-*l* 35.8%

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

       3. times-frac 31.1%

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

       4. associate-/l* 31.1%

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

   11. Simplified 31.1%

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

   12. Taylor expanded in t around 0 44.6%

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

   if 1.1e18 < k < 2.50000000000000015e135

   1. Initial program 23.1%

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

   3. Simplified 23.2%

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

      1. unpow3 23.1%

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

      2. times-frac 27.0%

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

      3. pow2 27.0%

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

   6. Applied egg-rr 27.0%

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

      1. +-commutative 27.0%

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

      2. associate-+l- 49.2%

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

      3. metadata-eval 49.2%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} - \color{blue}{0}\right)} \]

      4. --rgt-identity 49.2%

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

      5. unpow2 49.2%

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

      6. div-inv 49.2%

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

      7. associate-*r* 49.2%

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

   8. Applied egg-rr 49.2%

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

   if 2.50000000000000015e135 < k

   1. Initial program 33.3%

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

      1. *-commutative 33.3%

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

      2. associate-/r* 33.3%

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

   3. Simplified 42.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 26.2%

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

      2. pow2 26.2%

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

      3. sqrt-prod 14.3%

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

      4. sqrt-div 14.3%

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

      5. sqrt-pow1 19.1%

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

      6. metadata-eval 19.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 7.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 21.5%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 21.5%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

   7. Taylor expanded in k around 0 33.7%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 33.7%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2}}} \]

      2. associate-/l/ 33.7%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 33.7%

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

      4. pow-prod-down 33.8%

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

      5. *-commutative 33.8%

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

   9. Applied egg-rr 33.8%

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

       1. *-lft-identity 33.8%

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

       2. associate-*l* 33.8%

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

       3. times-frac 31.4%

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

       4. associate-/l* 31.3%

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

   11. Simplified 31.3%

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

   12. Taylor expanded in t around 0 40.5%

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

       1. associate-*l/ 40.5%

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

       2. associate-/l* 40.5%

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

   14. Simplified 40.5%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{+135}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\left(k \cdot \frac{k}{t}\right) \cdot \frac{1}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(k \cdot \frac{\sqrt{t}}{\ell}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{k}{l\_m} \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t\_m}^{2}}{l\_m} \cdot \frac{t\_m}{l\_m}\right)\right) \cdot \left(\frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(k \cdot \frac{\sqrt{t\_m}}{l\_m}\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.9e+18)
    (/ 2.0 (pow (* k (* (/ k l_m) (sqrt t_m))) 2.0))
    (if (<= k 2.2e+135)
      (/
       2.0
       (*
        (* (* (sin k) (tan k)) (* (/ (pow t_m 2.0) l_m) (/ t_m l_m)))
        (* (/ k t_m) (/ k t_m))))
      (/ 2.0 (pow (* k (* k (/ (sqrt t_m) l_m))) 2.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 1.9e+18) {
		tmp = 2.0 / pow((k * ((k / l_m) * sqrt(t_m))), 2.0);
	} else if (k <= 2.2e+135) {
		tmp = 2.0 / (((sin(k) * tan(k)) * ((pow(t_m, 2.0) / l_m) * (t_m / l_m))) * ((k / t_m) * (k / t_m)));
	} else {
		tmp = 2.0 / pow((k * (k * (sqrt(t_m) / l_m))), 2.0);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.9d+18) then
        tmp = 2.0d0 / ((k * ((k / l_m) * sqrt(t_m))) ** 2.0d0)
    else if (k <= 2.2d+135) then
        tmp = 2.0d0 / (((sin(k) * tan(k)) * (((t_m ** 2.0d0) / l_m) * (t_m / l_m))) * ((k / t_m) * (k / t_m)))
    else
        tmp = 2.0d0 / ((k * (k * (sqrt(t_m) / l_m))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (k <= 1.9e+18) {
		tmp = 2.0 / Math.pow((k * ((k / l_m) * Math.sqrt(t_m))), 2.0);
	} else if (k <= 2.2e+135) {
		tmp = 2.0 / (((Math.sin(k) * Math.tan(k)) * ((Math.pow(t_m, 2.0) / l_m) * (t_m / l_m))) * ((k / t_m) * (k / t_m)));
	} else {
		tmp = 2.0 / Math.pow((k * (k * (Math.sqrt(t_m) / l_m))), 2.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 1.9e+18:
		tmp = 2.0 / math.pow((k * ((k / l_m) * math.sqrt(t_m))), 2.0)
	elif k <= 2.2e+135:
		tmp = 2.0 / (((math.sin(k) * math.tan(k)) * ((math.pow(t_m, 2.0) / l_m) * (t_m / l_m))) * ((k / t_m) * (k / t_m)))
	else:
		tmp = 2.0 / math.pow((k * (k * (math.sqrt(t_m) / l_m))), 2.0)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 1.9e+18)
		tmp = Float64(2.0 / (Float64(k * Float64(Float64(k / l_m) * sqrt(t_m))) ^ 2.0));
	elseif (k <= 2.2e+135)
		tmp = Float64(2.0 / Float64(Float64(Float64(sin(k) * tan(k)) * Float64(Float64((t_m ^ 2.0) / l_m) * Float64(t_m / l_m))) * Float64(Float64(k / t_m) * Float64(k / t_m))));
	else
		tmp = Float64(2.0 / (Float64(k * Float64(k * Float64(sqrt(t_m) / l_m))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 1.9e+18)
		tmp = 2.0 / ((k * ((k / l_m) * sqrt(t_m))) ^ 2.0);
	elseif (k <= 2.2e+135)
		tmp = 2.0 / (((sin(k) * tan(k)) * (((t_m ^ 2.0) / l_m) * (t_m / l_m))) * ((k / t_m) * (k / t_m)));
	else
		tmp = 2.0 / ((k * (k * (sqrt(t_m) / l_m))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[k, 1.9e+18], N[(2.0 / N[Power[N[(k * N[(N[(k / l$95$m), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.2e+135], N[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $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[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(k * N[(k * N[(N[Sqrt[t$95$m], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.9e18

   1. Initial program 38.9%

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

      1. *-commutative 38.9%

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

      2. associate-/r* 38.9%

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

   3. Simplified 45.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 27.9%

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

      2. pow2 27.9%

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

      3. sqrt-prod 21.5%

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

      4. sqrt-div 22.1%

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

      5. sqrt-pow1 25.3%

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

      6. metadata-eval 25.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 17.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 31.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 31.3%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

   7. Taylor expanded in k around 0 34.2%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 34.2%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2}}} \]

      2. associate-/l/ 33.8%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 33.8%

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

      4. pow-prod-down 35.8%

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

      5. *-commutative 35.8%

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

   9. Applied egg-rr 35.8%

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

       1. *-lft-identity 35.8%

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

       2. associate-*l* 35.8%

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

       3. times-frac 31.1%

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

       4. associate-/l* 31.1%

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

   11. Simplified 31.1%

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

   12. Taylor expanded in t around 0 44.6%

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

   if 1.9e18 < k < 2.1999999999999999e135

   1. Initial program 23.1%

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

   3. Simplified 23.2%

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

      1. unpow3 23.1%

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

      2. times-frac 27.0%

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

      3. pow2 27.0%

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

   6. Applied egg-rr 27.0%

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

      1. +-commutative 27.0%

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

      2. associate-+l- 49.2%

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

      3. metadata-eval 49.2%

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} - \color{blue}{0}\right)} \]

      4. --rgt-identity 49.2%

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

      5. unpow2 49.2%

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

   8. Applied egg-rr 49.2%

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

   if 2.1999999999999999e135 < k

   1. Initial program 33.3%

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

      1. *-commutative 33.3%

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

      2. associate-/r* 33.3%

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

   3. Simplified 42.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 26.2%

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

      2. pow2 26.2%

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

      3. sqrt-prod 14.3%

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

      4. sqrt-div 14.3%

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

      5. sqrt-pow1 19.1%

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

      6. metadata-eval 19.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 7.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 21.5%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 21.5%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

   7. Taylor expanded in k around 0 33.7%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 33.7%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2}}} \]

      2. associate-/l/ 33.7%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 33.7%

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

      4. pow-prod-down 33.8%

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

      5. *-commutative 33.8%

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

   9. Applied egg-rr 33.8%

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

       1. *-lft-identity 33.8%

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

       2. associate-*l* 33.8%

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

       3. times-frac 31.4%

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

       4. associate-/l* 31.3%

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

   11. Simplified 31.3%

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

   12. Taylor expanded in t around 0 40.5%

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

       1. associate-*l/ 40.5%

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

       2. associate-/l* 40.5%

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

   14. Simplified 40.5%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{k}{\ell} \cdot \sqrt{t}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(k \cdot \frac{\sqrt{t}}{\ell}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 28500000:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{k}{l\_m} \cdot \sqrt{t\_m}\right)\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.65 \cdot 10^{+107}:\\ \;\;\;\;\frac{-0.11666666666666667}{t\_m} \cdot {\left({\left({l\_m}^{2}\right)}^{3}\right)}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(k \cdot \frac{\sqrt{t\_m}}{l\_m}\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (*
  t_s
  (if (<= k 28500000.0)
    (/ 2.0 (pow (* k (* (/ k l_m) (sqrt t_m))) 2.0))
    (if (<= k 2.65e+107)
      (*
       (/ -0.11666666666666667 t_m)
       (pow (pow (pow l_m 2.0) 3.0) 0.3333333333333333))
      (/ 2.0 (pow (* k (* k (/ (sqrt t_m) l_m))) 2.0))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	double tmp;
	if (k <= 28500000.0) {
		tmp = 2.0 / pow((k * ((k / l_m) * sqrt(t_m))), 2.0);
	} else if (k <= 2.65e+107) {
		tmp = (-0.11666666666666667 / t_m) * pow(pow(pow(l_m, 2.0), 3.0), 0.3333333333333333);
	} else {
		tmp = 2.0 / pow((k * (k * (sqrt(t_m) / l_m))), 2.0);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 28500000.0d0) then
        tmp = 2.0d0 / ((k * ((k / l_m) * sqrt(t_m))) ** 2.0d0)
    else if (k <= 2.65d+107) then
        tmp = ((-0.11666666666666667d0) / t_m) * (((l_m ** 2.0d0) ** 3.0d0) ** 0.3333333333333333d0)
    else
        tmp = 2.0d0 / ((k * (k * (sqrt(t_m) / l_m))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
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) {
	double tmp;
	if (k <= 28500000.0) {
		tmp = 2.0 / Math.pow((k * ((k / l_m) * Math.sqrt(t_m))), 2.0);
	} else if (k <= 2.65e+107) {
		tmp = (-0.11666666666666667 / t_m) * Math.pow(Math.pow(Math.pow(l_m, 2.0), 3.0), 0.3333333333333333);
	} else {
		tmp = 2.0 / Math.pow((k * (k * (Math.sqrt(t_m) / l_m))), 2.0);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	tmp = 0
	if k <= 28500000.0:
		tmp = 2.0 / math.pow((k * ((k / l_m) * math.sqrt(t_m))), 2.0)
	elif k <= 2.65e+107:
		tmp = (-0.11666666666666667 / t_m) * math.pow(math.pow(math.pow(l_m, 2.0), 3.0), 0.3333333333333333)
	else:
		tmp = 2.0 / math.pow((k * (k * (math.sqrt(t_m) / l_m))), 2.0)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	tmp = 0.0
	if (k <= 28500000.0)
		tmp = Float64(2.0 / (Float64(k * Float64(Float64(k / l_m) * sqrt(t_m))) ^ 2.0));
	elseif (k <= 2.65e+107)
		tmp = Float64(Float64(-0.11666666666666667 / t_m) * (((l_m ^ 2.0) ^ 3.0) ^ 0.3333333333333333));
	else
		tmp = Float64(2.0 / (Float64(k * Float64(k * Float64(sqrt(t_m) / l_m))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l_m, k)
	tmp = 0.0;
	if (k <= 28500000.0)
		tmp = 2.0 / ((k * ((k / l_m) * sqrt(t_m))) ^ 2.0);
	elseif (k <= 2.65e+107)
		tmp = (-0.11666666666666667 / t_m) * (((l_m ^ 2.0) ^ 3.0) ^ 0.3333333333333333);
	else
		tmp = 2.0 / ((k * (k * (sqrt(t_m) / l_m))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * If[LessEqual[k, 28500000.0], N[(2.0 / N[Power[N[(k * N[(N[(k / l$95$m), $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.65e+107], N[(N[(-0.11666666666666667 / t$95$m), $MachinePrecision] * N[Power[N[Power[N[Power[l$95$m, 2.0], $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(k * N[(k * N[(N[Sqrt[t$95$m], $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.85e7

   1. Initial program 39.3%

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

      1. *-commutative 39.3%

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

      2. associate-/r* 39.3%

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

   3. Simplified 45.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 28.2%

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

      2. pow2 28.2%

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

      3. sqrt-prod 21.8%

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

      4. sqrt-div 22.3%

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

      5. sqrt-pow1 25.5%

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

      6. metadata-eval 25.5%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 17.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 31.7%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 31.7%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

   7. Taylor expanded in k around 0 34.5%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 34.5%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2}}} \]

      2. associate-/l/ 34.1%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 34.1%

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

      4. pow-prod-down 36.1%

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

      5. *-commutative 36.1%

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

   9. Applied egg-rr 36.1%

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

       1. *-lft-identity 36.1%

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

       2. associate-*l* 36.1%

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

       3. times-frac 31.4%

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

       4. associate-/l* 31.4%

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

   11. Simplified 31.4%

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

   12. Taylor expanded in t around 0 45.1%

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

   if 2.85e7 < k < 2.65e107

   1. Initial program 23.9%

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

   3. Simplified 46.4%

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

   5. Taylor expanded in k around 0 19.8%

      \[\leadsto \color{blue}{\frac{{k}^{2} \cdot \left(-0.11666666666666667 \cdot \frac{{k}^{2}}{t} - 0.3333333333333333 \cdot \frac{1}{t}\right) + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

   6. Taylor expanded in k around inf 25.9%

      \[\leadsto \color{blue}{\frac{-0.11666666666666667}{t}} \cdot \left(\ell \cdot \ell\right) \]
   7. Step-by-step derivation

      1. add-cbrt-cube 25.8%

         \[\leadsto \frac{-0.11666666666666667}{t} \cdot \color{blue}{\sqrt[3]{\left(\left(\ell \cdot \ell\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(\ell \cdot \ell\right)}} \]

      2. pow1/3 25.8%

         \[\leadsto \frac{-0.11666666666666667}{t} \cdot \color{blue}{{\left(\left(\left(\ell \cdot \ell\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(\ell \cdot \ell\right)\right)}^{0.3333333333333333}} \]

      3. pow3 25.8%

         \[\leadsto \frac{-0.11666666666666667}{t} \cdot {\color{blue}{\left({\left(\ell \cdot \ell\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. pow2 25.8%

         \[\leadsto \frac{-0.11666666666666667}{t} \cdot {\left({\color{blue}{\left({\ell}^{2}\right)}}^{3}\right)}^{0.3333333333333333} \]

   8. Applied egg-rr 25.8%

      \[\leadsto \frac{-0.11666666666666667}{t} \cdot \color{blue}{{\left({\left({\ell}^{2}\right)}^{3}\right)}^{0.3333333333333333}} \]

   if 2.65e107 < k

   1. Initial program 31.2%

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

      1. *-commutative 31.2%

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

      2. associate-/r* 31.2%

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

   3. Simplified 40.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 24.4%

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

      2. pow2 24.4%

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

      3. sqrt-prod 13.3%

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

      4. sqrt-div 13.3%

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

      5. sqrt-pow1 17.8%

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

      6. metadata-eval 17.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 6.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 20.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 20.1%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

   7. Taylor expanded in k around 0 31.5%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 31.5%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2}}} \]

      2. associate-/l/ 31.5%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 31.5%

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

      4. pow-prod-down 31.6%

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

      5. *-commutative 31.6%

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

   9. Applied egg-rr 31.6%

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

       1. *-lft-identity 31.6%

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

       2. associate-*l* 31.6%

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

       3. times-frac 31.5%

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

       4. associate-/l* 31.5%

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

   11. Simplified 31.5%

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

   12. Taylor expanded in t around 0 37.9%

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

       1. associate-*l/ 37.9%

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

       2. associate-/l* 37.9%

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

   14. Simplified 37.9%

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

Alternative 13: 75.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.6%

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

   1. *-commutative 36.6%

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

   2. associate-/r* 36.6%

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

3. Simplified 44.1%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 27.5%

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

   2. pow2 27.5%

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

   3. sqrt-prod 19.2%

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

   4. sqrt-div 19.6%

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

   5. sqrt-pow1 22.8%

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

   6. metadata-eval 22.8%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   7. sqrt-prod 14.4%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   8. add-sqr-sqrt 27.7%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

6. Applied egg-rr 27.7%

   \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

7. Taylor expanded in k around 0 31.8%

   \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]
8. Step-by-step derivation

   1. *-un-lft-identity 31.8%

      \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2}}} \]

   2. associate-/l/ 31.5%

      \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

   3. +-rgt-identity 31.5%

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

   4. pow-prod-down 33.0%

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

   5. *-commutative 33.0%

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

9. Applied egg-rr 33.0%

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

    1. *-lft-identity 33.0%

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

    2. associate-*l* 33.0%

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

    3. times-frac 29.5%

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

    4. associate-/l* 29.5%

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

11. Simplified 29.5%

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

12. Taylor expanded in t around 0 40.8%

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

Alternative 14: 74.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.6%

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

   1. *-commutative 36.6%

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

   2. associate-/r* 36.6%

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

3. Simplified 44.1%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 27.5%

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

   2. pow2 27.5%

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

   3. sqrt-prod 19.2%

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

   4. sqrt-div 19.6%

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

   5. sqrt-pow1 22.8%

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

   6. metadata-eval 22.8%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   7. sqrt-prod 14.4%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   8. add-sqr-sqrt 27.7%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

6. Applied egg-rr 27.7%

   \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

7. Taylor expanded in k around 0 31.8%

   \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]
8. Step-by-step derivation

   1. *-un-lft-identity 31.8%

      \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2}}} \]

   2. associate-/l/ 31.5%

      \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

   3. +-rgt-identity 31.5%

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

   4. pow-prod-down 33.0%

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

   5. *-commutative 33.0%

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

9. Applied egg-rr 33.0%

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

    1. *-lft-identity 33.0%

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

    2. associate-*l* 33.0%

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

    3. times-frac 29.5%

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

    4. associate-/l* 29.5%

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

11. Simplified 29.5%

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

12. Taylor expanded in t around 0 40.8%

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

    1. associate-*l/ 40.7%

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

    2. associate-/l* 40.0%

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

14. Simplified 40.0%

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

Alternative 15: 62.0% accurate, 2.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (* 2.0 (/ (/ (pow l_m 2.0) (pow k 4.0)) t_m))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.6%

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

   1. *-commutative 36.6%

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

   2. associate-/r* 36.6%

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

3. Simplified 44.1%

   \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 27.5%

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

   2. pow2 27.5%

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

   3. sqrt-prod 19.2%

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

   4. sqrt-div 19.6%

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

   5. sqrt-pow1 22.8%

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

   6. metadata-eval 22.8%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   7. sqrt-prod 14.4%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   8. add-sqr-sqrt 27.7%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

6. Applied egg-rr 27.7%

   \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

7. Taylor expanded in k around 0 31.8%

   \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \color{blue}{k}\right)}^{2}} \]
8. Step-by-step derivation

   1. *-un-lft-identity 31.8%

      \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2}}} \]

   2. associate-/l/ 31.5%

      \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot k\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

   3. +-rgt-identity 31.5%

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

   4. pow-prod-down 33.0%

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

   5. *-commutative 33.0%

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

9. Applied egg-rr 33.0%

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

    1. *-lft-identity 33.0%

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

    2. associate-*l* 33.0%

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

    3. times-frac 29.5%

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

    4. associate-/l* 29.5%

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

11. Simplified 29.5%

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

12. Taylor expanded in k around 0 59.9%

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

    1. associate-/r* 60.7%

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

14. Simplified 60.7%

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

Alternative 16: 61.9% accurate, 3.8× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (* (* l_m l_m) (/ 2.0 (* t_m (pow k 4.0))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.6%

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

3. Simplified 44.8%

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

5. Taylor expanded in k around 0 60.0%

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

6. Final simplification 60.0%

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

Alternative 17: 19.8% accurate, 60.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(l\_m \cdot l\_m\right) \cdot \frac{-0.11666666666666667}{t\_m}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l_m k)
 :precision binary64
 (* t_s (* (* l_m l_m) (/ -0.11666666666666667 t_m))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l_m, double k) {
	return t_s * ((l_m * l_m) * (-0.11666666666666667 / t_m));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l_m, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: k
    code = t_s * ((l_m * l_m) * ((-0.11666666666666667d0) / t_m))
end function

Java

l_m = Math.abs(l);
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) {
	return t_s * ((l_m * l_m) * (-0.11666666666666667 / t_m));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l_m, k):
	return t_s * ((l_m * l_m) * (-0.11666666666666667 / t_m))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l_m, k)
	return Float64(t_s * Float64(Float64(l_m * l_m) * Float64(-0.11666666666666667 / t_m)))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l_m, k)
	tmp = t_s * ((l_m * l_m) * (-0.11666666666666667 / t_m));
end

Wolfram

l_m = N[Abs[l], $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_] := N[(t$95$s * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(-0.11666666666666667 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.6%

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

3. Simplified 44.8%

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

5. Taylor expanded in k around 0 40.7%

   \[\leadsto \color{blue}{\frac{{k}^{2} \cdot \left(-0.11666666666666667 \cdot \frac{{k}^{2}}{t} - 0.3333333333333333 \cdot \frac{1}{t}\right) + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

6. Taylor expanded in k around inf 19.9%

   \[\leadsto \color{blue}{\frac{-0.11666666666666667}{t}} \cdot \left(\ell \cdot \ell\right) \]

7. Final simplification 19.9%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024123 
(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))))