Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.9% → 81.9%

Time: 21.4s

Alternatives: 11

Speedup: 3.8×

• 39.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 34.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ t_3 := \frac{\sqrt{2}}{k}\\ t_4 := t\_2 \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.8 \cdot 10^{-214}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{t\_m \cdot k} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k}^{2}}{{\sin k}^{4}}}\right) \cdot \left(t\_3 \cdot \left(t\_m \cdot \frac{1}{\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot t\_2}\right)\right)\\ \mathbf{elif}\;t\_m \leq 7 \cdot 10^{+224}:\\ \;\;\;\;\frac{t\_m \cdot t\_3}{{t\_4}^{2}} \cdot \frac{\sqrt{2}}{t\_4 \cdot \frac{k}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\cos k}{t\_m \cdot {\sin k}^{2}}}{{k}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (cbrt (* (sin k) (tan k))))
        (t_3 (/ (sqrt 2.0) k))
        (t_4 (* t_2 (/ t_m (pow (cbrt l) 2.0)))))
   (*
    t_s
    (if (<= t_m 6.8e-214)
      (*
       (*
        (/ (sqrt 2.0) (* t_m k))
        (cbrt (/ (* (pow l 4.0) (pow (cos k) 2.0)) (pow (sin k) 4.0))))
       (* t_3 (* t_m (/ 1.0 (* (* t_m (pow (cbrt l) -2.0)) t_2)))))
      (if (<= t_m 7e+224)
        (* (/ (* t_m t_3) (pow t_4 2.0)) (/ (sqrt 2.0) (* t_4 (/ k t_m))))
        (*
         (* 2.0 (/ (/ (cos k) (* t_m (pow (sin k) 2.0))) (pow k 2.0)))
         (* l l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = cbrt((sin(k) * tan(k)));
	double t_3 = sqrt(2.0) / k;
	double t_4 = t_2 * (t_m / pow(cbrt(l), 2.0));
	double tmp;
	if (t_m <= 6.8e-214) {
		tmp = ((sqrt(2.0) / (t_m * k)) * cbrt(((pow(l, 4.0) * pow(cos(k), 2.0)) / pow(sin(k), 4.0)))) * (t_3 * (t_m * (1.0 / ((t_m * pow(cbrt(l), -2.0)) * t_2))));
	} else if (t_m <= 7e+224) {
		tmp = ((t_m * t_3) / pow(t_4, 2.0)) * (sqrt(2.0) / (t_4 * (k / t_m)));
	} else {
		tmp = (2.0 * ((cos(k) / (t_m * pow(sin(k), 2.0))) / pow(k, 2.0))) * (l * l);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_3 = Math.sqrt(2.0) / k;
	double t_4 = t_2 * (t_m / Math.pow(Math.cbrt(l), 2.0));
	double tmp;
	if (t_m <= 6.8e-214) {
		tmp = ((Math.sqrt(2.0) / (t_m * k)) * Math.cbrt(((Math.pow(l, 4.0) * Math.pow(Math.cos(k), 2.0)) / Math.pow(Math.sin(k), 4.0)))) * (t_3 * (t_m * (1.0 / ((t_m * Math.pow(Math.cbrt(l), -2.0)) * t_2))));
	} else if (t_m <= 7e+224) {
		tmp = ((t_m * t_3) / Math.pow(t_4, 2.0)) * (Math.sqrt(2.0) / (t_4 * (k / t_m)));
	} else {
		tmp = (2.0 * ((Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0))) / Math.pow(k, 2.0))) * (l * l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = cbrt(Float64(sin(k) * tan(k)))
	t_3 = Float64(sqrt(2.0) / k)
	t_4 = Float64(t_2 * Float64(t_m / (cbrt(l) ^ 2.0)))
	tmp = 0.0
	if (t_m <= 6.8e-214)
		tmp = Float64(Float64(Float64(sqrt(2.0) / Float64(t_m * k)) * cbrt(Float64(Float64((l ^ 4.0) * (cos(k) ^ 2.0)) / (sin(k) ^ 4.0)))) * Float64(t_3 * Float64(t_m * Float64(1.0 / Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * t_2)))));
	elseif (t_m <= 7e+224)
		tmp = Float64(Float64(Float64(t_m * t_3) / (t_4 ^ 2.0)) * Float64(sqrt(2.0) / Float64(t_4 * Float64(k / t_m))));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0))) / (k ^ 2.0))) * Float64(l * l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.8e-214], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$m * k), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[Power[l, 4.0], $MachinePrecision] * N[Power[N[Cos[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[(t$95$m * N[(1.0 / N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7e+224], N[(N[(N[(t$95$m * t$95$3), $MachinePrecision] / N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$4 * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 6.7999999999999998e-214

   1. Initial program 34.3%

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

      1. *-commutative 34.3%

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

      2. associate-/r* 34.3%

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

   3. Simplified 40.1%

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

      1. add-sqr-sqrt 40.1%

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

      2. add-cube-cbrt 40.0%

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

      3. times-frac 40.0%

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

   6. Applied egg-rr 77.4%

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

      1. div-inv 77.4%

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

      2. associate-/r/ 77.4%

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

      3. div-inv 77.4%

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

      4. pow-flip 77.4%

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

      5. metadata-eval 77.4%

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

   8. Applied egg-rr 77.4%

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

      1. associate-*l* 77.4%

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

   10. Simplified 77.4%

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

   11. Taylor expanded in k around inf 66.3%

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

   if 6.7999999999999998e-214 < t < 7e224

   1. Initial program 43.6%

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

      1. *-commutative 43.6%

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

      2. associate-/r* 43.6%

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

   3. Simplified 50.7%

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

      1. add-sqr-sqrt 50.7%

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

      2. add-cube-cbrt 50.7%

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

      3. times-frac 50.6%

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

   6. Applied egg-rr 89.3%

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

      1. associate-/r/ 89.3%

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

      2. associate-/l/ 89.3%

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

   8. Simplified 89.3%

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

   if 7e224 < t

   1. Initial program 5.3%

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

   3. Simplified 15.8%

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

   5. Taylor expanded in t around 0 89.5%

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

      1. associate-/l* 89.5%

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

   7. Simplified 89.5%

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

   8. Taylor expanded in k around inf 89.5%

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

      1. *-commutative 89.5%

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

      2. associate-/r* 89.5%

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

   10. Simplified 89.5%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{-214}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{t \cdot k} \cdot \sqrt[3]{\frac{{\ell}^{4} \cdot {\cos k}^{2}}{{\sin k}^{4}}}\right) \cdot \left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot \frac{1}{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}}\right)\right)\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+224}:\\ \;\;\;\;\frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{\sqrt{2}}{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right) \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\cos k}{t \cdot {\sin k}^{2}}}{{k}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ t_3 := \frac{\sqrt{2}}{k}\\ t_4 := t\_2 \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.5 \cdot 10^{-214}:\\ \;\;\;\;\left(t\_3 \cdot \left(t\_m \cdot \frac{1}{\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot t\_2}\right)\right) \cdot \left(\frac{\sqrt{2}}{t\_m} \cdot \frac{\sqrt[3]{{\ell}^{4} \cdot \frac{{\cos k}^{2}}{{\sin k}^{4}}}}{k}\right)\\ \mathbf{elif}\;t\_m \leq 1.1 \cdot 10^{+225}:\\ \;\;\;\;\frac{t\_m \cdot t\_3}{{t\_4}^{2}} \cdot \frac{\sqrt{2}}{t\_4 \cdot \frac{k}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\cos k}{t\_m \cdot {\sin k}^{2}}}{{k}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (cbrt (* (sin k) (tan k))))
        (t_3 (/ (sqrt 2.0) k))
        (t_4 (* t_2 (/ t_m (pow (cbrt l) 2.0)))))
   (*
    t_s
    (if (<= t_m 6.5e-214)
      (*
       (* t_3 (* t_m (/ 1.0 (* (* t_m (pow (cbrt l) -2.0)) t_2))))
       (*
        (/ (sqrt 2.0) t_m)
        (/ (cbrt (* (pow l 4.0) (/ (pow (cos k) 2.0) (pow (sin k) 4.0)))) k)))
      (if (<= t_m 1.1e+225)
        (* (/ (* t_m t_3) (pow t_4 2.0)) (/ (sqrt 2.0) (* t_4 (/ k t_m))))
        (*
         (* 2.0 (/ (/ (cos k) (* t_m (pow (sin k) 2.0))) (pow k 2.0)))
         (* l l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = cbrt((sin(k) * tan(k)));
	double t_3 = sqrt(2.0) / k;
	double t_4 = t_2 * (t_m / pow(cbrt(l), 2.0));
	double tmp;
	if (t_m <= 6.5e-214) {
		tmp = (t_3 * (t_m * (1.0 / ((t_m * pow(cbrt(l), -2.0)) * t_2)))) * ((sqrt(2.0) / t_m) * (cbrt((pow(l, 4.0) * (pow(cos(k), 2.0) / pow(sin(k), 4.0)))) / k));
	} else if (t_m <= 1.1e+225) {
		tmp = ((t_m * t_3) / pow(t_4, 2.0)) * (sqrt(2.0) / (t_4 * (k / t_m)));
	} else {
		tmp = (2.0 * ((cos(k) / (t_m * pow(sin(k), 2.0))) / pow(k, 2.0))) * (l * l);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_3 = Math.sqrt(2.0) / k;
	double t_4 = t_2 * (t_m / Math.pow(Math.cbrt(l), 2.0));
	double tmp;
	if (t_m <= 6.5e-214) {
		tmp = (t_3 * (t_m * (1.0 / ((t_m * Math.pow(Math.cbrt(l), -2.0)) * t_2)))) * ((Math.sqrt(2.0) / t_m) * (Math.cbrt((Math.pow(l, 4.0) * (Math.pow(Math.cos(k), 2.0) / Math.pow(Math.sin(k), 4.0)))) / k));
	} else if (t_m <= 1.1e+225) {
		tmp = ((t_m * t_3) / Math.pow(t_4, 2.0)) * (Math.sqrt(2.0) / (t_4 * (k / t_m)));
	} else {
		tmp = (2.0 * ((Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0))) / Math.pow(k, 2.0))) * (l * l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = cbrt(Float64(sin(k) * tan(k)))
	t_3 = Float64(sqrt(2.0) / k)
	t_4 = Float64(t_2 * Float64(t_m / (cbrt(l) ^ 2.0)))
	tmp = 0.0
	if (t_m <= 6.5e-214)
		tmp = Float64(Float64(t_3 * Float64(t_m * Float64(1.0 / Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * t_2)))) * Float64(Float64(sqrt(2.0) / t_m) * Float64(cbrt(Float64((l ^ 4.0) * Float64((cos(k) ^ 2.0) / (sin(k) ^ 4.0)))) / k)));
	elseif (t_m <= 1.1e+225)
		tmp = Float64(Float64(Float64(t_m * t_3) / (t_4 ^ 2.0)) * Float64(sqrt(2.0) / Float64(t_4 * Float64(k / t_m))));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0))) / (k ^ 2.0))) * Float64(l * l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.5e-214], N[(N[(t$95$3 * N[(t$95$m * N[(1.0 / N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[Power[N[(N[Power[l, 4.0], $MachinePrecision] * N[(N[Power[N[Cos[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.1e+225], N[(N[(N[(t$95$m * t$95$3), $MachinePrecision] / N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$4 * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 6.5000000000000004e-214

   1. Initial program 34.3%

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

      1. *-commutative 34.3%

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

      2. associate-/r* 34.3%

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

   3. Simplified 40.1%

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

      1. add-sqr-sqrt 40.1%

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

      2. add-cube-cbrt 40.0%

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

      3. times-frac 40.0%

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

   6. Applied egg-rr 77.4%

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

      1. div-inv 77.4%

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

      2. associate-/r/ 77.4%

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

      3. div-inv 77.4%

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

      4. pow-flip 77.4%

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

      5. metadata-eval 77.4%

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

   8. Applied egg-rr 77.4%

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

      1. associate-*l* 77.4%

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

   10. Simplified 77.4%

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

   11. Taylor expanded in k around inf 66.3%

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

       1. associate-*l/ 66.3%

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

       2. *-commutative 66.3%

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

       3. times-frac 66.3%

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

       4. associate-/l* 66.3%

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

   13. Simplified 66.3%

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

   if 6.5000000000000004e-214 < t < 1.10000000000000007e225

   1. Initial program 43.6%

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

      1. *-commutative 43.6%

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

      2. associate-/r* 43.6%

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

   3. Simplified 50.7%

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

      1. add-sqr-sqrt 50.7%

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

      2. add-cube-cbrt 50.7%

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

      3. times-frac 50.6%

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

   6. Applied egg-rr 89.3%

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

      1. associate-/r/ 89.3%

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

      2. associate-/l/ 89.3%

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

   8. Simplified 89.3%

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

   if 1.10000000000000007e225 < t

   1. Initial program 5.3%

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

   3. Simplified 15.8%

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

   5. Taylor expanded in t around 0 89.5%

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

      1. associate-/l* 89.5%

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

   7. Simplified 89.5%

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

   8. Taylor expanded in k around inf 89.5%

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

      1. *-commutative 89.5%

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

      2. associate-/r* 89.5%

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

   10. Simplified 89.5%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.5 \cdot 10^{-214}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{k} \cdot \left(t \cdot \frac{1}{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\sin k \cdot \tan k}}\right)\right) \cdot \left(\frac{\sqrt{2}}{t} \cdot \frac{\sqrt[3]{{\ell}^{4} \cdot \frac{{\cos k}^{2}}{{\sin k}^{4}}}}{k}\right)\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+225}:\\ \;\;\;\;\frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{\sqrt{2}}{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right) \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\cos k}{t \cdot {\sin k}^{2}}}{{k}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_3 := \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.8 \cdot 10^{-228}:\\ \;\;\;\;2 \cdot \left(t\_3 \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)\\ \mathbf{elif}\;t\_m \leq 1.4 \cdot 10^{+226}:\\ \;\;\;\;\frac{t\_m \cdot \frac{\sqrt{2}}{k}}{{t\_2}^{2}} \cdot \frac{\sqrt{2}}{t\_2 \cdot \frac{k}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{t\_3}{{k}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (cbrt (* (sin k) (tan k))) (/ t_m (pow (cbrt l) 2.0))))
        (t_3 (/ (cos k) (* t_m (pow (sin k) 2.0)))))
   (*
    t_s
    (if (<= t_m 9.8e-228)
      (* 2.0 (* t_3 (/ (pow l 2.0) (pow k 2.0))))
      (if (<= t_m 1.4e+226)
        (*
         (/ (* t_m (/ (sqrt 2.0) k)) (pow t_2 2.0))
         (/ (sqrt 2.0) (* t_2 (/ k t_m))))
        (* (* 2.0 (/ t_3 (pow k 2.0))) (* l l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = cbrt((sin(k) * tan(k))) * (t_m / pow(cbrt(l), 2.0));
	double t_3 = cos(k) / (t_m * pow(sin(k), 2.0));
	double tmp;
	if (t_m <= 9.8e-228) {
		tmp = 2.0 * (t_3 * (pow(l, 2.0) / pow(k, 2.0)));
	} else if (t_m <= 1.4e+226) {
		tmp = ((t_m * (sqrt(2.0) / k)) / pow(t_2, 2.0)) * (sqrt(2.0) / (t_2 * (k / t_m)));
	} else {
		tmp = (2.0 * (t_3 / pow(k, 2.0))) * (l * l);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k))) * (t_m / Math.pow(Math.cbrt(l), 2.0));
	double t_3 = Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0));
	double tmp;
	if (t_m <= 9.8e-228) {
		tmp = 2.0 * (t_3 * (Math.pow(l, 2.0) / Math.pow(k, 2.0)));
	} else if (t_m <= 1.4e+226) {
		tmp = ((t_m * (Math.sqrt(2.0) / k)) / Math.pow(t_2, 2.0)) * (Math.sqrt(2.0) / (t_2 * (k / t_m)));
	} else {
		tmp = (2.0 * (t_3 / Math.pow(k, 2.0))) * (l * l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(cbrt(Float64(sin(k) * tan(k))) * Float64(t_m / (cbrt(l) ^ 2.0)))
	t_3 = Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0)))
	tmp = 0.0
	if (t_m <= 9.8e-228)
		tmp = Float64(2.0 * Float64(t_3 * Float64((l ^ 2.0) / (k ^ 2.0))));
	elseif (t_m <= 1.4e+226)
		tmp = Float64(Float64(Float64(t_m * Float64(sqrt(2.0) / k)) / (t_2 ^ 2.0)) * Float64(sqrt(2.0) / Float64(t_2 * Float64(k / t_m))));
	else
		tmp = Float64(Float64(2.0 * Float64(t_3 / (k ^ 2.0))) * Float64(l * l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.8e-228], N[(2.0 * N[(t$95$3 * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.4e+226], 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[(t$95$2 * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(t$95$3 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 9.79999999999999976e-228

   1. Initial program 34.9%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in t around 0 75.6%

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

      1. associate-/l* 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in k around inf 75.8%

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

      1. times-frac 76.6%

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

   10. Simplified 76.6%

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

   if 9.79999999999999976e-228 < t < 1.4000000000000001e226

   1. Initial program 42.5%

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

      1. *-commutative 42.5%

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

      2. associate-/r* 42.5%

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

   3. Simplified 49.2%

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

      1. add-sqr-sqrt 49.2%

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

      2. add-cube-cbrt 49.2%

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

      3. times-frac 49.2%

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

   6. Applied egg-rr 87.0%

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

      1. associate-/r/ 87.0%

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

      2. associate-/l/ 87.0%

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

   8. Simplified 87.0%

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

   if 1.4000000000000001e226 < t

   1. Initial program 5.3%

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

   3. Simplified 15.8%

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

   5. Taylor expanded in t around 0 89.5%

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

      1. associate-/l* 89.5%

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

   7. Simplified 89.5%

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

   8. Taylor expanded in k around inf 89.5%

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

      1. *-commutative 89.5%

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

      2. associate-/r* 89.5%

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

   10. Simplified 89.5%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.8 \cdot 10^{-228}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {\sin k}^{2}} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+226}:\\ \;\;\;\;\frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{\sqrt{2}}{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right) \cdot \frac{k}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\cos k}{t \cdot {\sin k}^{2}}}{{k}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{\sin k \cdot \tan k}\\ t_3 := \frac{\cos k}{t\_m \cdot {\sin k}^{2}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.9 \cdot 10^{-226}:\\ \;\;\;\;2 \cdot \left(t\_3 \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)\\ \mathbf{elif}\;t\_m \leq 8 \cdot 10^{+226}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\frac{k}{t\_m}}}{{\left(t\_2 \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2} \cdot t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{t\_3}{{k}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (cbrt (* (sin k) (tan k))))
        (t_3 (/ (cos k) (* t_m (pow (sin k) 2.0)))))
   (*
    t_s
    (if (<= t_m 1.9e-226)
      (* 2.0 (* t_3 (/ (pow l 2.0) (pow k 2.0))))
      (if (<= t_m 8e+226)
        (*
         (/
          (/ (sqrt 2.0) (/ k t_m))
          (pow (* t_2 (/ t_m (pow (cbrt l) 2.0))) 2.0))
         (* (/ (sqrt 2.0) k) (/ 1.0 (* (pow (cbrt l) -2.0) t_2))))
        (* (* 2.0 (/ t_3 (pow k 2.0))) (* l l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = cbrt((sin(k) * tan(k)));
	double t_3 = cos(k) / (t_m * pow(sin(k), 2.0));
	double tmp;
	if (t_m <= 1.9e-226) {
		tmp = 2.0 * (t_3 * (pow(l, 2.0) / pow(k, 2.0)));
	} else if (t_m <= 8e+226) {
		tmp = ((sqrt(2.0) / (k / t_m)) / pow((t_2 * (t_m / pow(cbrt(l), 2.0))), 2.0)) * ((sqrt(2.0) / k) * (1.0 / (pow(cbrt(l), -2.0) * t_2)));
	} else {
		tmp = (2.0 * (t_3 / pow(k, 2.0))) * (l * l);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.cbrt((Math.sin(k) * Math.tan(k)));
	double t_3 = Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0));
	double tmp;
	if (t_m <= 1.9e-226) {
		tmp = 2.0 * (t_3 * (Math.pow(l, 2.0) / Math.pow(k, 2.0)));
	} else if (t_m <= 8e+226) {
		tmp = ((Math.sqrt(2.0) / (k / t_m)) / Math.pow((t_2 * (t_m / Math.pow(Math.cbrt(l), 2.0))), 2.0)) * ((Math.sqrt(2.0) / k) * (1.0 / (Math.pow(Math.cbrt(l), -2.0) * t_2)));
	} else {
		tmp = (2.0 * (t_3 / Math.pow(k, 2.0))) * (l * l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = cbrt(Float64(sin(k) * tan(k)))
	t_3 = Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0)))
	tmp = 0.0
	if (t_m <= 1.9e-226)
		tmp = Float64(2.0 * Float64(t_3 * Float64((l ^ 2.0) / (k ^ 2.0))));
	elseif (t_m <= 8e+226)
		tmp = Float64(Float64(Float64(sqrt(2.0) / Float64(k / t_m)) / (Float64(t_2 * Float64(t_m / (cbrt(l) ^ 2.0))) ^ 2.0)) * Float64(Float64(sqrt(2.0) / k) * Float64(1.0 / Float64((cbrt(l) ^ -2.0) * t_2))));
	else
		tmp = Float64(Float64(2.0 * Float64(t_3 / (k ^ 2.0))) * Float64(l * l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.9e-226], N[(2.0 * N[(t$95$3 * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8e+226], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[N[(t$95$2 * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision] * N[(1.0 / N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(t$95$3 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.89999999999999991e-226

   1. Initial program 34.9%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in t around 0 75.6%

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

      1. associate-/l* 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in k around inf 75.8%

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

      1. times-frac 76.6%

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

   10. Simplified 76.6%

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

   if 1.89999999999999991e-226 < t < 7.99999999999999969e226

   1. Initial program 42.5%

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

      1. *-commutative 42.5%

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

      2. associate-/r* 42.5%

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

   3. Simplified 49.2%

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

      1. add-sqr-sqrt 49.2%

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

      2. add-cube-cbrt 49.2%

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

      3. times-frac 49.2%

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

   6. Applied egg-rr 87.0%

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

      1. div-inv 87.0%

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

      2. associate-/r/ 86.9%

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

      3. div-inv 86.9%

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

      4. pow-flip 86.9%

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

      5. metadata-eval 86.9%

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

   8. Applied egg-rr 86.9%

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

      1. associate-*l* 86.9%

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

   10. Simplified 86.9%

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

       1. un-div-inv 86.9%

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

       2. associate-*l* 86.9%

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

   12. Applied egg-rr 86.9%

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

       1. associate-/r* 86.9%

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

       2. *-inverses 86.9%

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

   14. Simplified 86.9%

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

   if 7.99999999999999969e226 < t

   1. Initial program 5.3%

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

   3. Simplified 15.8%

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

   5. Taylor expanded in t around 0 89.5%

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

      1. associate-/l* 89.5%

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

   7. Simplified 89.5%

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

   8. Taylor expanded in k around inf 89.5%

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

      1. *-commutative 89.5%

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

      2. associate-/r* 89.5%

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

   10. Simplified 89.5%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{-226}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {\sin k}^{2}} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+226}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\frac{k}{t}}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \left(\frac{\sqrt{2}}{k} \cdot \frac{1}{{\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\sin k \cdot \tan k}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\cos k}{t \cdot {\sin k}^{2}}}{{k}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_3 := \sqrt[3]{\sin k \cdot \tan k} \cdot t\_2\\ t_4 := {t\_3}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.62 \cdot 10^{-162}:\\ \;\;\;\;\frac{t\_m \cdot \frac{\sqrt{2}}{k}}{t\_4} \cdot \frac{\sqrt{2}}{\frac{k}{t\_m} \cdot \left(t\_2 \cdot \sqrt[3]{{k}^{2}}\right)}\\ \mathbf{elif}\;\ell \leq 1.28 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t\_m \cdot {\sin k}^{2}} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\sqrt{2}}{\frac{k}{t\_m}}\right)}^{2}}{t\_4}}{t\_3}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ t_m (pow (cbrt l) 2.0)))
        (t_3 (* (cbrt (* (sin k) (tan k))) t_2))
        (t_4 (pow t_3 2.0)))
   (*
    t_s
    (if (<= l 1.62e-162)
      (*
       (/ (* t_m (/ (sqrt 2.0) k)) t_4)
       (/ (sqrt 2.0) (* (/ k t_m) (* t_2 (cbrt (pow k 2.0))))))
      (if (<= l 1.28e+154)
        (*
         2.0
         (* (/ (cos k) (* t_m (pow (sin k) 2.0))) (/ (pow l 2.0) (pow k 2.0))))
        (/ (/ (pow (/ (sqrt 2.0) (/ k t_m)) 2.0) t_4) t_3))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m / pow(cbrt(l), 2.0);
	double t_3 = cbrt((sin(k) * tan(k))) * t_2;
	double t_4 = pow(t_3, 2.0);
	double tmp;
	if (l <= 1.62e-162) {
		tmp = ((t_m * (sqrt(2.0) / k)) / t_4) * (sqrt(2.0) / ((k / t_m) * (t_2 * cbrt(pow(k, 2.0)))));
	} else if (l <= 1.28e+154) {
		tmp = 2.0 * ((cos(k) / (t_m * pow(sin(k), 2.0))) * (pow(l, 2.0) / pow(k, 2.0)));
	} else {
		tmp = (pow((sqrt(2.0) / (k / t_m)), 2.0) / t_4) / t_3;
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m / Math.pow(Math.cbrt(l), 2.0);
	double t_3 = Math.cbrt((Math.sin(k) * Math.tan(k))) * t_2;
	double t_4 = Math.pow(t_3, 2.0);
	double tmp;
	if (l <= 1.62e-162) {
		tmp = ((t_m * (Math.sqrt(2.0) / k)) / t_4) * (Math.sqrt(2.0) / ((k / t_m) * (t_2 * Math.cbrt(Math.pow(k, 2.0)))));
	} else if (l <= 1.28e+154) {
		tmp = 2.0 * ((Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0))) * (Math.pow(l, 2.0) / Math.pow(k, 2.0)));
	} else {
		tmp = (Math.pow((Math.sqrt(2.0) / (k / t_m)), 2.0) / t_4) / t_3;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m / (cbrt(l) ^ 2.0))
	t_3 = Float64(cbrt(Float64(sin(k) * tan(k))) * t_2)
	t_4 = t_3 ^ 2.0
	tmp = 0.0
	if (l <= 1.62e-162)
		tmp = Float64(Float64(Float64(t_m * Float64(sqrt(2.0) / k)) / t_4) * Float64(sqrt(2.0) / Float64(Float64(k / t_m) * Float64(t_2 * cbrt((k ^ 2.0))))));
	elseif (l <= 1.28e+154)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0))) * Float64((l ^ 2.0) / (k ^ 2.0))));
	else
		tmp = Float64(Float64((Float64(sqrt(2.0) / Float64(k / t_m)) ^ 2.0) / t_4) / t_3);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$3, 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[l, 1.62e-162], N[(N[(N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(t$95$2 * N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.28e+154], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[Sqrt[2.0], $MachinePrecision] / N[(k / t$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$4), $MachinePrecision] / t$95$3), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.6199999999999999e-162

   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 39.1%

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

      1. add-sqr-sqrt 39.1%

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

      2. add-cube-cbrt 39.0%

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

      3. times-frac 39.0%

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

   6. Applied egg-rr 83.8%

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

      1. associate-/r/ 83.8%

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

      2. associate-/l/ 83.8%

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

   8. Simplified 83.8%

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

   9. Taylor expanded in k around 0 71.1%

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

   if 1.6199999999999999e-162 < l < 1.2800000000000001e154

   1. Initial program 43.0%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in t around 0 91.6%

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

      1. associate-/l* 91.6%

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

   7. Simplified 91.6%

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

   8. Taylor expanded in k around inf 91.6%

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

      1. times-frac 92.0%

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

   10. Simplified 92.0%

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

   if 1.2800000000000001e154 < l

   1. Initial program 41.0%

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

      1. *-commutative 41.0%

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

      2. associate-/r* 41.0%

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

   3. Simplified 41.0%

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

      1. add-sqr-sqrt 41.0%

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

      2. add-cube-cbrt 41.0%

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

      3. times-frac 41.0%

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

   6. Applied egg-rr 78.1%

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

      1. associate-*r/ 78.0%

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

      2. associate-*l/ 68.5%

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

      3. unpow2 68.5%

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

   8. Simplified 68.5%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.62 \cdot 10^{-162}:\\ \;\;\;\;\frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{\sqrt{2}}{\frac{k}{t} \cdot \left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{{k}^{2}}\right)}\\ \mathbf{elif}\;\ell \leq 1.28 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {\sin k}^{2}} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\sqrt{2}}{\frac{k}{t}}\right)}^{2}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_3 := \sqrt[3]{\sin k \cdot \tan k} \cdot t\_2\\ t_4 := {t\_3}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{t\_m \cdot \frac{\sqrt{2}}{k}}{t\_4} \cdot \frac{\sqrt{2}}{\frac{k}{t\_m} \cdot \left(t\_2 \cdot \sqrt[3]{{k}^{2}}\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+304}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t\_m \cdot {\sin k}^{2}} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_4} \cdot \frac{{\left(\frac{k}{t\_m}\right)}^{-2}}{t\_3}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ t_m (pow (cbrt l) 2.0)))
        (t_3 (* (cbrt (* (sin k) (tan k))) t_2))
        (t_4 (pow t_3 2.0)))
   (*
    t_s
    (if (<= (* l l) 0.0)
      (*
       (/ (* t_m (/ (sqrt 2.0) k)) t_4)
       (/ (sqrt 2.0) (* (/ k t_m) (* t_2 (cbrt (pow k 2.0))))))
      (if (<= (* l l) 5e+304)
        (*
         2.0
         (* (/ (cos k) (* t_m (pow (sin k) 2.0))) (/ (pow l 2.0) (pow k 2.0))))
        (* (/ 2.0 t_4) (/ (pow (/ k t_m) -2.0) t_3)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m / pow(cbrt(l), 2.0);
	double t_3 = cbrt((sin(k) * tan(k))) * t_2;
	double t_4 = pow(t_3, 2.0);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = ((t_m * (sqrt(2.0) / k)) / t_4) * (sqrt(2.0) / ((k / t_m) * (t_2 * cbrt(pow(k, 2.0)))));
	} else if ((l * l) <= 5e+304) {
		tmp = 2.0 * ((cos(k) / (t_m * pow(sin(k), 2.0))) * (pow(l, 2.0) / pow(k, 2.0)));
	} else {
		tmp = (2.0 / t_4) * (pow((k / t_m), -2.0) / t_3);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = t_m / Math.pow(Math.cbrt(l), 2.0);
	double t_3 = Math.cbrt((Math.sin(k) * Math.tan(k))) * t_2;
	double t_4 = Math.pow(t_3, 2.0);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = ((t_m * (Math.sqrt(2.0) / k)) / t_4) * (Math.sqrt(2.0) / ((k / t_m) * (t_2 * Math.cbrt(Math.pow(k, 2.0)))));
	} else if ((l * l) <= 5e+304) {
		tmp = 2.0 * ((Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0))) * (Math.pow(l, 2.0) / Math.pow(k, 2.0)));
	} else {
		tmp = (2.0 / t_4) * (Math.pow((k / t_m), -2.0) / t_3);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(t_m / (cbrt(l) ^ 2.0))
	t_3 = Float64(cbrt(Float64(sin(k) * tan(k))) * t_2)
	t_4 = t_3 ^ 2.0
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(Float64(Float64(t_m * Float64(sqrt(2.0) / k)) / t_4) * Float64(sqrt(2.0) / Float64(Float64(k / t_m) * Float64(t_2 * cbrt((k ^ 2.0))))));
	elseif (Float64(l * l) <= 5e+304)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0))) * Float64((l ^ 2.0) / (k ^ 2.0))));
	else
		tmp = Float64(Float64(2.0 / t_4) * Float64((Float64(k / t_m) ^ -2.0) / t_3));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$3, 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(N[(N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(k / t$95$m), $MachinePrecision] * N[(t$95$2 * N[Power[N[Power[k, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 5e+304], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t$95$4), $MachinePrecision] * N[(N[Power[N[(k / t$95$m), $MachinePrecision], -2.0], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 23.3%

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

      1. *-commutative 23.3%

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

      2. associate-/r* 23.3%

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

   3. Simplified 35.3%

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

      1. add-sqr-sqrt 35.3%

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

      2. add-cube-cbrt 35.3%

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

      3. times-frac 35.3%

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

   6. Applied egg-rr 86.5%

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

      1. associate-/r/ 86.5%

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

      2. associate-/l/ 86.5%

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

   8. Simplified 86.5%

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

   9. Taylor expanded in k around 0 82.2%

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

   if 0.0 < (*.f64 l l) < 4.9999999999999997e304

   1. Initial program 45.3%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in t around 0 88.9%

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

      1. associate-/l* 88.9%

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

   7. Simplified 88.9%

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

   8. Taylor expanded in k around inf 89.2%

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

      1. times-frac 90.6%

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

   10. Simplified 90.6%

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

   if 4.9999999999999997e304 < (*.f64 l l)

   1. Initial program 27.5%

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

      1. *-commutative 27.5%

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

      2. associate-/r* 27.5%

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

   3. Simplified 27.5%

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

      1. add-cube-cbrt 27.5%

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

      2. div-inv 27.5%

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

      3. times-frac 27.5%

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

   6. Applied egg-rr 66.2%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{t \cdot \frac{\sqrt{2}}{k}}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{\sqrt{2}}{\frac{k}{t} \cdot \left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{{k}^{2}}\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+304}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {\sin k}^{2}} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{{\left(\frac{k}{t}\right)}^{-2}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t_3 := \sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.95 \cdot 10^{-117}:\\ \;\;\;\;\frac{2}{t\_m \cdot {k}^{2}} \cdot \frac{\cos k \cdot {\ell}^{2}}{t\_2}\\ \mathbf{elif}\;t\_m \leq 1.08 \cdot 10^{+161}:\\ \;\;\;\;\frac{2}{{t\_3}^{2}} \cdot \frac{{\left(\frac{k}{t\_m}\right)}^{-2}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\cos k}{t\_m \cdot t\_2}}{{k}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (sin k) 2.0))
        (t_3 (* (cbrt (* (sin k) (tan k))) (/ t_m (pow (cbrt l) 2.0)))))
   (*
    t_s
    (if (<= t_m 1.95e-117)
      (* (/ 2.0 (* t_m (pow k 2.0))) (/ (* (cos k) (pow l 2.0)) t_2))
      (if (<= t_m 1.08e+161)
        (* (/ 2.0 (pow t_3 2.0)) (/ (pow (/ k t_m) -2.0) t_3))
        (* (* 2.0 (/ (/ (cos k) (* t_m t_2)) (pow k 2.0))) (* l l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(sin(k), 2.0);
	double t_3 = cbrt((sin(k) * tan(k))) * (t_m / pow(cbrt(l), 2.0));
	double tmp;
	if (t_m <= 1.95e-117) {
		tmp = (2.0 / (t_m * pow(k, 2.0))) * ((cos(k) * pow(l, 2.0)) / t_2);
	} else if (t_m <= 1.08e+161) {
		tmp = (2.0 / pow(t_3, 2.0)) * (pow((k / t_m), -2.0) / t_3);
	} else {
		tmp = (2.0 * ((cos(k) / (t_m * t_2)) / pow(k, 2.0))) * (l * l);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double t_3 = Math.cbrt((Math.sin(k) * Math.tan(k))) * (t_m / Math.pow(Math.cbrt(l), 2.0));
	double tmp;
	if (t_m <= 1.95e-117) {
		tmp = (2.0 / (t_m * Math.pow(k, 2.0))) * ((Math.cos(k) * Math.pow(l, 2.0)) / t_2);
	} else if (t_m <= 1.08e+161) {
		tmp = (2.0 / Math.pow(t_3, 2.0)) * (Math.pow((k / t_m), -2.0) / t_3);
	} else {
		tmp = (2.0 * ((Math.cos(k) / (t_m * t_2)) / Math.pow(k, 2.0))) * (l * l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = sin(k) ^ 2.0
	t_3 = Float64(cbrt(Float64(sin(k) * tan(k))) * Float64(t_m / (cbrt(l) ^ 2.0)))
	tmp = 0.0
	if (t_m <= 1.95e-117)
		tmp = Float64(Float64(2.0 / Float64(t_m * (k ^ 2.0))) * Float64(Float64(cos(k) * (l ^ 2.0)) / t_2));
	elseif (t_m <= 1.08e+161)
		tmp = Float64(Float64(2.0 / (t_3 ^ 2.0)) * Float64((Float64(k / t_m) ^ -2.0) / t_3));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k) / Float64(t_m * t_2)) / (k ^ 2.0))) * Float64(l * l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.95e-117], 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, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.08e+161], N[(N[(2.0 / N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(k / t$95$m), $MachinePrecision], -2.0], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * t$95$2), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.94999999999999996e-117

   1. Initial program 34.4%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in t around 0 74.1%

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

      1. associate-*r/ 74.1%

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

      2. associate-*r* 74.0%

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

      3. times-frac 73.9%

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

      4. *-commutative 73.9%

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

   7. Simplified 73.9%

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

   if 1.94999999999999996e-117 < t < 1.08e161

   1. Initial program 49.4%

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

      1. *-commutative 49.4%

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

      2. associate-/r* 49.4%

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

   3. Simplified 58.7%

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

      1. add-cube-cbrt 58.7%

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

      2. div-inv 58.7%

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

      3. times-frac 58.6%

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

   6. Applied egg-rr 87.5%

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

   if 1.08e161 < t

   1. Initial program 13.3%

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

   3. Simplified 23.3%

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

   5. Taylor expanded in t around 0 83.3%

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

      1. associate-/l* 83.3%

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

   7. Simplified 83.3%

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

   8. Taylor expanded in k around inf 83.3%

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

      1. *-commutative 83.3%

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

      2. associate-/r* 83.3%

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

   10. Simplified 83.3%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-117}:\\ \;\;\;\;\frac{2}{t \cdot {k}^{2}} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\sin k}^{2}}\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{+161}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{{\left(\frac{k}{t}\right)}^{-2}}{\sqrt[3]{\sin k \cdot \tan k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\cos k}{t \cdot {\sin k}^{2}}}{{k}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \left(2 \cdot \frac{{k}^{-4}}{t\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t\_m \cdot {\sin k}^{2}} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 0.0)
    (* l (* l (* 2.0 (/ (pow k -4.0) t_m))))
    (*
     2.0
     (* (/ (cos k) (* t_m (pow (sin k) 2.0))) (/ (pow l 2.0) (pow k 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = l * (l * (2.0 * (pow(k, -4.0) / t_m)));
	} else {
		tmp = 2.0 * ((cos(k) / (t_m * pow(sin(k), 2.0))) * (pow(l, 2.0) / pow(k, 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = l * (l * (2.0d0 * ((k ** (-4.0d0)) / t_m)))
    else
        tmp = 2.0d0 * ((cos(k) / (t_m * (sin(k) ** 2.0d0))) * ((l ** 2.0d0) / (k ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = l * (l * (2.0 * (Math.pow(k, -4.0) / t_m)));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0))) * (Math.pow(l, 2.0) / Math.pow(k, 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = l * (l * (2.0 * (math.pow(k, -4.0) / t_m)))
	else:
		tmp = 2.0 * ((math.cos(k) / (t_m * math.pow(math.sin(k), 2.0))) * (math.pow(l, 2.0) / math.pow(k, 2.0)))
	return t_s * tmp

Julia

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

MATLAB

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

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(l * N[(l * N[(2.0 * N[(N[Power[k, -4.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 23.3%

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

   3. Simplified 35.1%

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

   5. Taylor expanded in k around 0 63.9%

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

      1. add-cube-cbrt 63.9%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)} \cdot \sqrt[3]{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}\right) \cdot \sqrt[3]{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}} \]

      2. pow3 63.9%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}\right)}^{3}} \]

      3. associate-/r* 63.9%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{\frac{2}{{k}^{4}}}{t}} \cdot \left(\ell \cdot \ell\right)}\right)}^{3} \]

      4. pow2 63.9%

         \[\leadsto {\left(\sqrt[3]{\frac{\frac{2}{{k}^{4}}}{t} \cdot \color{blue}{{\ell}^{2}}}\right)}^{3} \]

   7. Applied egg-rr 63.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\frac{2}{{k}^{4}}}{t} \cdot {\ell}^{2}}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 63.9%

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

      2. div-inv 63.9%

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

      3. pow-flip 63.9%

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

      4. metadata-eval 63.9%

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

      5. *-un-lft-identity 63.9%

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

      6. pow2 63.9%

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

      7. associate-*r* 79.5%

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

      8. *-un-lft-identity 79.5%

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

      9. associate-/l* 79.5%

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

   9. Applied egg-rr 79.5%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 39.6%

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

   3. Simplified 46.6%

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

   5. Taylor expanded in t around 0 77.4%

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

      1. associate-/l* 77.4%

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

   7. Simplified 77.4%

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

   8. Taylor expanded in k around inf 77.6%

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

      1. times-frac 78.7%

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

   10. Simplified 78.7%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \left(2 \cdot \frac{{k}^{-4}}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot {\sin k}^{2}} \cdot \frac{{\ell}^{2}}{{k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-320}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \left(2 \cdot \frac{{k}^{-4}}{t\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\cos k}{t\_m \cdot {\sin k}^{2}}}{{k}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 1e-320)
    (* l (* l (* 2.0 (/ (pow k -4.0) t_m))))
    (*
     (* 2.0 (/ (/ (cos k) (* t_m (pow (sin k) 2.0))) (pow k 2.0)))
     (* l l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-320) {
		tmp = l * (l * (2.0 * (pow(k, -4.0) / t_m)));
	} else {
		tmp = (2.0 * ((cos(k) / (t_m * pow(sin(k), 2.0))) / pow(k, 2.0))) * (l * l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 1d-320) then
        tmp = l * (l * (2.0d0 * ((k ** (-4.0d0)) / t_m)))
    else
        tmp = (2.0d0 * ((cos(k) / (t_m * (sin(k) ** 2.0d0))) / (k ** 2.0d0))) * (l * l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 1e-320) {
		tmp = l * (l * (2.0 * (Math.pow(k, -4.0) / t_m)));
	} else {
		tmp = (2.0 * ((Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0))) / Math.pow(k, 2.0))) * (l * l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 1e-320:
		tmp = l * (l * (2.0 * (math.pow(k, -4.0) / t_m)))
	else:
		tmp = (2.0 * ((math.cos(k) / (t_m * math.pow(math.sin(k), 2.0))) / math.pow(k, 2.0))) * (l * l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 1e-320)
		tmp = Float64(l * Float64(l * Float64(2.0 * Float64((k ^ -4.0) / t_m))));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0))) / (k ^ 2.0))) * Float64(l * l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 1e-320)
		tmp = l * (l * (2.0 * ((k ^ -4.0) / t_m)));
	else
		tmp = (2.0 * ((cos(k) / (t_m * (sin(k) ^ 2.0))) / (k ^ 2.0))) * (l * l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 1e-320], N[(l * N[(l * N[(2.0 * N[(N[Power[k, -4.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.99989e-321

   1. Initial program 23.8%

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

   3. Simplified 35.1%

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

   5. Taylor expanded in k around 0 64.2%

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

      1. add-cube-cbrt 64.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)} \cdot \sqrt[3]{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}\right) \cdot \sqrt[3]{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}} \]

      2. pow3 64.2%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}\right)}^{3}} \]

      3. associate-/r* 64.1%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{\frac{2}{{k}^{4}}}{t}} \cdot \left(\ell \cdot \ell\right)}\right)}^{3} \]

      4. pow2 64.1%

         \[\leadsto {\left(\sqrt[3]{\frac{\frac{2}{{k}^{4}}}{t} \cdot \color{blue}{{\ell}^{2}}}\right)}^{3} \]

   7. Applied egg-rr 64.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\frac{2}{{k}^{4}}}{t} \cdot {\ell}^{2}}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 64.1%

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

      2. div-inv 64.1%

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

      3. pow-flip 64.1%

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

      4. metadata-eval 64.1%

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

      5. *-un-lft-identity 64.1%

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

      6. pow2 64.1%

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

      7. associate-*r* 79.0%

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

      8. *-un-lft-identity 79.0%

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

      9. associate-/l* 79.0%

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

   9. Applied egg-rr 79.0%

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

   if 9.99989e-321 < (*.f64 l l)

   1. Initial program 39.7%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in t around 0 77.6%

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

      1. associate-/l* 77.6%

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

   7. Simplified 77.6%

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

   8. Taylor expanded in k around inf 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-/r* 78.2%

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

   10. Simplified 78.2%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-320}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \left(2 \cdot \frac{{k}^{-4}}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\cos k}{t \cdot {\sin k}^{2}}}{{k}^{2}}\right) \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-8}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \left(2 \cdot \frac{{k}^{-4}}{t\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \frac{t\_m \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}{\cos k}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6.2e-8)
    (* l (* l (* 2.0 (/ (pow k -4.0) t_m))))
    (*
     (* l l)
     (/
      2.0
      (* (pow k 2.0) (/ (* t_m (- 0.5 (/ (cos (* 2.0 k)) 2.0))) (cos k))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.2e-8) {
		tmp = l * (l * (2.0 * (pow(k, -4.0) / t_m)));
	} else {
		tmp = (l * l) * (2.0 / (pow(k, 2.0) * ((t_m * (0.5 - (cos((2.0 * k)) / 2.0))) / cos(k))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.2d-8) then
        tmp = l * (l * (2.0d0 * ((k ** (-4.0d0)) / t_m)))
    else
        tmp = (l * l) * (2.0d0 / ((k ** 2.0d0) * ((t_m * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0))) / cos(k))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 6.2e-8) {
		tmp = l * (l * (2.0 * (Math.pow(k, -4.0) / t_m)));
	} else {
		tmp = (l * l) * (2.0 / (Math.pow(k, 2.0) * ((t_m * (0.5 - (Math.cos((2.0 * k)) / 2.0))) / Math.cos(k))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 6.2e-8:
		tmp = l * (l * (2.0 * (math.pow(k, -4.0) / t_m)))
	else:
		tmp = (l * l) * (2.0 / (math.pow(k, 2.0) * ((t_m * (0.5 - (math.cos((2.0 * k)) / 2.0))) / math.cos(k))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 6.2e-8)
		tmp = Float64(l * Float64(l * Float64(2.0 * Float64((k ^ -4.0) / t_m))));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64((k ^ 2.0) * Float64(Float64(t_m * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))) / cos(k)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 6.2e-8)
		tmp = l * (l * (2.0 * ((k ^ -4.0) / t_m)));
	else
		tmp = (l * l) * (2.0 / ((k ^ 2.0) * ((t_m * (0.5 - (cos((2.0 * k)) / 2.0))) / cos(k))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 6.2e-8], N[(l * N[(l * N[(2.0 * N[(N[Power[k, -4.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[(t$95$m * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.2e-8

   1. Initial program 37.0%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in k around 0 71.1%

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

      1. add-cube-cbrt 71.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)} \cdot \sqrt[3]{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}\right) \cdot \sqrt[3]{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}} \]

      2. pow3 71.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}\right)}^{3}} \]

      3. associate-/r* 71.0%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{\frac{2}{{k}^{4}}}{t}} \cdot \left(\ell \cdot \ell\right)}\right)}^{3} \]

      4. pow2 71.0%

         \[\leadsto {\left(\sqrt[3]{\frac{\frac{2}{{k}^{4}}}{t} \cdot \color{blue}{{\ell}^{2}}}\right)}^{3} \]

   7. Applied egg-rr 71.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\frac{2}{{k}^{4}}}{t} \cdot {\ell}^{2}}\right)}^{3}} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 71.1%

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

      2. div-inv 71.1%

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

      3. pow-flip 71.1%

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

      4. metadata-eval 71.1%

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

      5. *-un-lft-identity 71.1%

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

      6. pow2 71.1%

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

      7. associate-*r* 76.2%

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

      8. *-un-lft-identity 76.2%

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

      9. associate-/l* 76.2%

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

   9. Applied egg-rr 76.2%

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

   if 6.2e-8 < k

   1. Initial program 32.0%

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

   3. Simplified 38.1%

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

   5. Taylor expanded in t around 0 65.8%

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

      1. associate-/l* 65.8%

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

   7. Simplified 65.8%

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

      1. unpow2 65.8%

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

      2. sin-mult 65.6%

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

   9. Applied egg-rr 65.6%

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

       1. div-sub 65.6%

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

       2. +-inverses 65.6%

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

       3. cos-0 65.6%

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

       4. metadata-eval 65.6%

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

       5. count-2 65.6%

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

       6. *-commutative 65.6%

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

   11. Simplified 65.6%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-8}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \left(2 \cdot \frac{{k}^{-4}}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{k}^{2} \cdot \frac{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.8%

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

3. Simplified 43.9%

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

5. Taylor expanded in k around 0 66.1%

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

   1. add-cube-cbrt 66.1%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)} \cdot \sqrt[3]{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}\right) \cdot \sqrt[3]{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}} \]

   2. pow3 66.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{2}{{k}^{4} \cdot t} \cdot \left(\ell \cdot \ell\right)}\right)}^{3}} \]

   3. associate-/r* 66.1%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{\frac{2}{{k}^{4}}}{t}} \cdot \left(\ell \cdot \ell\right)}\right)}^{3} \]

   4. pow2 66.1%

      \[\leadsto {\left(\sqrt[3]{\frac{\frac{2}{{k}^{4}}}{t} \cdot \color{blue}{{\ell}^{2}}}\right)}^{3} \]

7. Applied egg-rr 66.1%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\frac{2}{{k}^{4}}}{t} \cdot {\ell}^{2}}\right)}^{3}} \]
8. Step-by-step derivation

   1. rem-cube-cbrt 66.1%

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

   2. div-inv 66.1%

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

   3. pow-flip 66.1%

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

   4. metadata-eval 66.1%

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

   5. *-un-lft-identity 66.1%

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

   6. pow2 66.1%

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

   7. associate-*r* 70.9%

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

   8. *-un-lft-identity 70.9%

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

   9. associate-/l* 70.9%

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

9. Applied egg-rr 70.9%

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

10. Final simplification 70.9%

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

Reproduce

herbie shell --seed 2024086 
(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))))