Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.8% → 89.6%

Time: 25.1s

Alternatives: 20

Speedup: 2.0×

• 27.5% 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: 54.8% 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: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{-77}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}\right)\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.8e-77)
    (* (* 2.0 (/ l (* (pow k 2.0) (* t_m (sin k))))) (/ l (tan k)))
    (/
     2.0
     (pow
      (*
       (/ t_m (pow (cbrt l) 2.0))
       (* (cbrt (sin k)) (cbrt (* (tan k) (+ 2.0 (pow (/ k t_m) 2.0))))))
      3.0)))))

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 (t_m <= 5.8e-77) {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t_m * sin(k))))) * (l / tan(k));
	} else {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * (cbrt(sin(k)) * cbrt((tan(k) * (2.0 + pow((k / t_m), 2.0)))))), 3.0);
	}
	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 tmp;
	if (t_m <= 5.8e-77) {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t_m * Math.sin(k))))) * (l / Math.tan(k));
	} else {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * (Math.cbrt(Math.sin(k)) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t_m), 2.0)))))), 3.0);
	}
	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 (t_m <= 5.8e-77)
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t_m * sin(k))))) * Float64(l / tan(k)));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * Float64(cbrt(sin(k)) * cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))))) ^ 3.0));
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 5.8e-77], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.7999999999999997e-77

   1. Initial program 49.2%

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

   3. Simplified 52.9%

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

      1. associate-/l/ 52.9%

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

      2. associate-/r/ 52.9%

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

      3. times-frac 53.3%

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

      4. div-inv 53.3%

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

      5. clear-num 53.3%

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

   6. Applied egg-rr 53.3%

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

   7. Taylor expanded in k around inf 69.3%

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

      1. *-commutative 69.3%

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

   9. Simplified 69.3%

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

   if 5.7999999999999997e-77 < t

   1. Initial program 66.1%

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

      1. +-commutative 66.1%

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

      2. associate-+r+ 66.1%

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

      3. metadata-eval 66.1%

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

      4. associate-*l* 59.6%

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

      5. associate-/r* 64.7%

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

      6. add-cube-cbrt 64.6%

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

      7. pow3 64.6%

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

   4. Applied egg-rr 76.7%

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

      1. pow1/3 55.7%

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

      2. associate-*l* 55.7%

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

      3. unpow-prod-down 35.3%

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

      4. pow1/3 53.2%

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

   6. Applied egg-rr 53.2%

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

      1. unpow1/3 92.0%

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

      2. *-commutative 92.0%

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

   8. Simplified 92.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-77}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\\ t_3 := t\_2 \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 4 \cdot 10^{+263}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)\right) \cdot t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{{\ell}^{2}}{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0)))
        (t_3 (* t_2 (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l)))))))
   (*
    t_s
    (if (<= t_3 4e+263)
      (/ 2.0 (* (* (tan k) (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l)))) t_2))
      (if (<= t_3 INFINITY)
        (/ (pow l 2.0) (pow (* t_m (pow (cbrt k) 2.0)) 3.0))
        (* (* 2.0 (/ l (* (pow k 2.0) (* t_m (sin k))))) (/ l (tan 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 t_2 = 1.0 + (pow((k / t_m), 2.0) + 1.0);
	double t_3 = t_2 * (tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l))));
	double tmp;
	if (t_3 <= 4e+263) {
		tmp = 2.0 / ((tan(k) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l)))) * t_2);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = pow(l, 2.0) / pow((t_m * pow(cbrt(k), 2.0)), 3.0);
	} else {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t_m * sin(k))))) * (l / tan(k));
	}
	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 = 1.0 + (Math.pow((k / t_m), 2.0) + 1.0);
	double t_3 = t_2 * (Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l))));
	double tmp;
	if (t_3 <= 4e+263) {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l)))) * t_2);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.pow(l, 2.0) / Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0);
	} else {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t_m * Math.sin(k))))) * (l / Math.tan(k));
	}
	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(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0))
	t_3 = Float64(t_2 * Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))))
	tmp = 0.0
	if (t_3 <= 4e+263)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))) * t_2));
	elseif (t_3 <= Inf)
		tmp = Float64((l ^ 2.0) / (Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0));
	else
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t_m * sin(k))))) * Float64(l / tan(k)));
	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[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, 4e+263], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < 4.00000000000000006e263

   1. Initial program 88.1%

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

      1. unpow3 88.1%

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

      2. times-frac 92.8%

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

      3. pow2 92.8%

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

   4. Applied egg-rr 92.8%

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

   if 4.00000000000000006e263 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < +inf.0

   1. Initial program 78.2%

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

   3. Simplified 78.2%

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

   5. Taylor expanded in k around 0 63.6%

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

      1. *-commutative 63.6%

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

   7. Simplified 63.6%

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

      1. add-cube-cbrt 63.6%

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

      2. pow3 63.6%

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

      3. cbrt-prod 63.6%

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

      4. unpow3 63.6%

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

      5. add-cbrt-cube 68.6%

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

      6. unpow2 68.6%

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

      7. cbrt-prod 93.7%

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

      8. pow2 93.7%

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

   9. Applied egg-rr 93.7%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))

   1. Initial program 0.0%

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

   3. Simplified 6.5%

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

      1. associate-/l/ 6.5%

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

      2. associate-/r/ 6.5%

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

      3. times-frac 6.4%

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

      4. div-inv 6.4%

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

      5. clear-num 6.4%

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

   6. Applied egg-rr 6.4%

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

   7. Taylor expanded in k around inf 49.4%

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

      1. *-commutative 49.4%

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

   9. Simplified 49.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \leq 4 \cdot 10^{+263}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \mathbf{elif}\;\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \leq \infty:\\ \;\;\;\;\frac{{\ell}^{2}}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\tan k}\\ t_3 := \frac{2}{\sin k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\right) \cdot t\_2\\ \mathbf{elif}\;t\_m \leq 6.4 \cdot 10^{+195}:\\ \;\;\;\;t\_2 \cdot \frac{{\left(\frac{\sqrt[3]{\ell \cdot t\_3}}{t\_m}\right)}^{3}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{t\_3} \cdot \frac{\frac{1}{t\_m}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3}}{\frac{2 \cdot \sin k}{\cos k}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (tan k))) (t_3 (/ 2.0 (sin k))))
   (*
    t_s
    (if (<= t_m 7.2e-77)
      (* (* 2.0 (/ l (* (pow k 2.0) (* t_m (sin k))))) t_2)
      (if (<= t_m 6.4e+195)
        (*
         t_2
         (/ (pow (/ (cbrt (* l t_3)) t_m) 3.0) (+ 2.0 (pow (/ k t_m) 2.0))))
        (/
         (pow (* (cbrt t_3) (/ (/ 1.0 t_m) (pow (cbrt l) -2.0))) 3.0)
         (/ (* 2.0 (sin k)) (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 t_2 = l / tan(k);
	double t_3 = 2.0 / sin(k);
	double tmp;
	if (t_m <= 7.2e-77) {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t_m * sin(k))))) * t_2;
	} else if (t_m <= 6.4e+195) {
		tmp = t_2 * (pow((cbrt((l * t_3)) / t_m), 3.0) / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = pow((cbrt(t_3) * ((1.0 / t_m) / pow(cbrt(l), -2.0))), 3.0) / ((2.0 * sin(k)) / cos(k));
	}
	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 = l / Math.tan(k);
	double t_3 = 2.0 / Math.sin(k);
	double tmp;
	if (t_m <= 7.2e-77) {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t_m * Math.sin(k))))) * t_2;
	} else if (t_m <= 6.4e+195) {
		tmp = t_2 * (Math.pow((Math.cbrt((l * t_3)) / t_m), 3.0) / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = Math.pow((Math.cbrt(t_3) * ((1.0 / t_m) / Math.pow(Math.cbrt(l), -2.0))), 3.0) / ((2.0 * Math.sin(k)) / 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)
	t_2 = Float64(l / tan(k))
	t_3 = Float64(2.0 / sin(k))
	tmp = 0.0
	if (t_m <= 7.2e-77)
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t_m * sin(k))))) * t_2);
	elseif (t_m <= 6.4e+195)
		tmp = Float64(t_2 * Float64((Float64(cbrt(Float64(l * t_3)) / t_m) ^ 3.0) / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64((Float64(cbrt(t_3) * Float64(Float64(1.0 / t_m) / (cbrt(l) ^ -2.0))) ^ 3.0) / Float64(Float64(2.0 * sin(k)) / cos(k)));
	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[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.2e-77], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$m, 6.4e+195], N[(t$95$2 * N[(N[Power[N[(N[Power[N[(l * t$95$3), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Power[t$95$3, 1/3], $MachinePrecision] * N[(N[(1.0 / t$95$m), $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(N[(2.0 * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.2e-77

   1. Initial program 48.9%

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

   3. Simplified 52.6%

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

      1. associate-/l/ 52.6%

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

      2. associate-/r/ 52.6%

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

      3. times-frac 53.0%

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

      4. div-inv 53.0%

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

      5. clear-num 53.0%

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

   6. Applied egg-rr 53.0%

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

   7. Taylor expanded in k around inf 69.0%

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

      1. *-commutative 69.0%

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

   9. Simplified 69.0%

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

   if 7.2e-77 < t < 6.39999999999999965e195

   1. Initial program 65.1%

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

   3. Simplified 70.7%

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

      1. associate-/l/ 70.7%

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

      2. associate-/r/ 72.5%

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

      3. times-frac 72.6%

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

      4. div-inv 72.5%

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

      5. clear-num 72.5%

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

   6. Applied egg-rr 72.5%

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

      1. add-cube-cbrt 72.3%

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

      2. pow3 72.3%

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

      3. associate-*r/ 72.3%

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

      4. cbrt-div 72.3%

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

      5. unpow3 72.3%

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

      6. add-cbrt-cube 89.2%

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

   8. Applied egg-rr 89.2%

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

   if 6.39999999999999965e195 < t

   1. Initial program 71.0%

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

   3. Simplified 75.9%

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

      1. associate-/l/ 75.9%

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

      2. add-cube-cbrt 75.9%

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

      3. times-frac 75.9%

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

   6. Applied egg-rr 99.0%

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

      1. frac-times 99.0%

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

      2. unpow2 99.0%

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

      3. pow3 99.0%

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

      4. unpow2 99.0%

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

      5. cbrt-prod 79.5%

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

      6. unpow2 79.5%

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

   8. Applied egg-rr 79.5%

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

      1. div-inv 79.5%

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

      2. unpow2 79.5%

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

      3. cbrt-prod 99.0%

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

      4. unpow2 99.0%

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

      5. div-inv 99.0%

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

      6. pow-flip 99.0%

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

      7. metadata-eval 99.0%

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

   10. Applied egg-rr 99.0%

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

       1. associate-/r* 98.8%

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

   12. Simplified 98.8%

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

   13. Taylor expanded in t around inf 94.9%

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

       1. *-commutative 94.9%

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

       2. associate-*l/ 94.9%

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

   15. Simplified 94.9%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+195}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{{\left(\frac{\sqrt[3]{\ell \cdot \frac{2}{\sin k}}}{t}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{\frac{2}{\sin k}} \cdot \frac{\frac{1}{t}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3}}{\frac{2 \cdot \sin k}{\cos k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{\frac{2}{\sin k}} \cdot \frac{\frac{1}{t\_m}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3}}{\tan k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.2e-77)
    (* (* 2.0 (/ l (* (pow k 2.0) (* t_m (sin k))))) (/ l (tan k)))
    (/
     (pow (* (cbrt (/ 2.0 (sin k))) (/ (/ 1.0 t_m) (pow (cbrt l) -2.0))) 3.0)
     (* (tan k) (+ 2.0 (pow (/ k t_m) 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 (t_m <= 7.2e-77) {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t_m * sin(k))))) * (l / tan(k));
	} else {
		tmp = pow((cbrt((2.0 / sin(k))) * ((1.0 / t_m) / pow(cbrt(l), -2.0))), 3.0) / (tan(k) * (2.0 + pow((k / t_m), 2.0)));
	}
	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 tmp;
	if (t_m <= 7.2e-77) {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t_m * Math.sin(k))))) * (l / Math.tan(k));
	} else {
		tmp = Math.pow((Math.cbrt((2.0 / Math.sin(k))) * ((1.0 / t_m) / Math.pow(Math.cbrt(l), -2.0))), 3.0) / (Math.tan(k) * (2.0 + Math.pow((k / t_m), 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 (t_m <= 7.2e-77)
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t_m * sin(k))))) * Float64(l / tan(k)));
	else
		tmp = Float64((Float64(cbrt(Float64(2.0 / sin(k))) * Float64(Float64(1.0 / t_m) / (cbrt(l) ^ -2.0))) ^ 3.0) / Float64(tan(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 7.2e-77], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Power[N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[(1.0 / t$95$m), $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.2e-77

   1. Initial program 48.9%

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

   3. Simplified 52.6%

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

      1. associate-/l/ 52.6%

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

      2. associate-/r/ 52.6%

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

      3. times-frac 53.0%

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

      4. div-inv 53.0%

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

      5. clear-num 53.0%

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

   6. Applied egg-rr 53.0%

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

   7. Taylor expanded in k around inf 69.0%

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

      1. *-commutative 69.0%

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

   9. Simplified 69.0%

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

   if 7.2e-77 < t

   1. Initial program 66.9%

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

   3. Simplified 72.3%

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

      1. associate-/l/ 72.3%

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

      2. add-cube-cbrt 72.1%

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

      3. times-frac 72.1%

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

   6. Applied egg-rr 93.2%

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

      1. frac-times 90.8%

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

      2. unpow2 90.8%

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

      3. pow3 90.8%

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

      4. unpow2 90.8%

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

      5. cbrt-prod 75.6%

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

      6. unpow2 75.6%

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

   8. Applied egg-rr 75.6%

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

      1. div-inv 75.6%

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

      2. unpow2 75.6%

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

      3. cbrt-prod 90.9%

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

      4. unpow2 90.9%

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

      5. div-inv 90.9%

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

      6. pow-flip 90.9%

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

      7. metadata-eval 90.9%

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

   10. Applied egg-rr 90.9%

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

       1. associate-/r* 90.9%

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

   12. Simplified 90.9%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{\frac{2}{\sin k}} \cdot \frac{\frac{1}{t}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3}}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\tan k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\right) \cdot t\_2\\ \mathbf{elif}\;t\_m \leq 2.5 \cdot 10^{+181}:\\ \;\;\;\;t\_2 \cdot \frac{{\left(\frac{\sqrt[3]{\ell \cdot \frac{2}{\sin k}}}{t\_m}\right)}^{3}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k} \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (tan k))))
   (*
    t_s
    (if (<= t_m 7.2e-77)
      (* (* 2.0 (/ l (* (pow k 2.0) (* t_m (sin k))))) t_2)
      (if (<= t_m 2.5e+181)
        (*
         t_2
         (/
          (pow (/ (cbrt (* l (/ 2.0 (sin k)))) t_m) 3.0)
          (+ 2.0 (pow (/ k t_m) 2.0))))
        (/
         2.0
         (pow
          (*
           (/ t_m (pow (cbrt l) 2.0))
           (* (cbrt (sin k)) (* (cbrt k) (cbrt 2.0))))
          3.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / tan(k);
	double tmp;
	if (t_m <= 7.2e-77) {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t_m * sin(k))))) * t_2;
	} else if (t_m <= 2.5e+181) {
		tmp = t_2 * (pow((cbrt((l * (2.0 / sin(k)))) / t_m), 3.0) / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * (cbrt(sin(k)) * (cbrt(k) * cbrt(2.0)))), 3.0);
	}
	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 = l / Math.tan(k);
	double tmp;
	if (t_m <= 7.2e-77) {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t_m * Math.sin(k))))) * t_2;
	} else if (t_m <= 2.5e+181) {
		tmp = t_2 * (Math.pow((Math.cbrt((l * (2.0 / Math.sin(k)))) / t_m), 3.0) / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * (Math.cbrt(Math.sin(k)) * (Math.cbrt(k) * Math.cbrt(2.0)))), 3.0);
	}
	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(l / tan(k))
	tmp = 0.0
	if (t_m <= 7.2e-77)
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t_m * sin(k))))) * t_2);
	elseif (t_m <= 2.5e+181)
		tmp = Float64(t_2 * Float64((Float64(cbrt(Float64(l * Float64(2.0 / sin(k)))) / t_m) ^ 3.0) / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * Float64(cbrt(sin(k)) * Float64(cbrt(k) * cbrt(2.0)))) ^ 3.0));
	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[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.2e-77], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$m, 2.5e+181], N[(t$95$2 * N[(N[Power[N[(N[Power[N[(l * N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[k, 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 7.2e-77

   1. Initial program 48.9%

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

   3. Simplified 52.6%

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

      1. associate-/l/ 52.6%

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

      2. associate-/r/ 52.6%

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

      3. times-frac 53.0%

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

      4. div-inv 53.0%

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

      5. clear-num 53.0%

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

   6. Applied egg-rr 53.0%

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

   7. Taylor expanded in k around inf 69.0%

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

      1. *-commutative 69.0%

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

   9. Simplified 69.0%

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

   if 7.2e-77 < t < 2.5000000000000002e181

   1. Initial program 65.7%

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

   3. Simplified 71.4%

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

      1. associate-/l/ 71.4%

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

      2. associate-/r/ 73.3%

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

      3. times-frac 73.4%

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

      4. div-inv 73.3%

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

      5. clear-num 73.3%

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

   6. Applied egg-rr 73.3%

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

      1. add-cube-cbrt 73.1%

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

      2. pow3 73.1%

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

      3. associate-*r/ 73.1%

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

      4. cbrt-div 73.1%

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

      5. unpow3 73.1%

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

      6. add-cbrt-cube 89.1%

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

   8. Applied egg-rr 89.1%

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

   if 2.5000000000000002e181 < t

   1. Initial program 69.5%

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

      1. +-commutative 69.5%

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

      2. associate-+r+ 69.5%

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

      3. metadata-eval 69.5%

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

      4. associate-*l* 57.7%

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

      5. associate-/r* 62.0%

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

      6. add-cube-cbrt 62.0%

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

      7. pow3 62.0%

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

   4. Applied egg-rr 69.7%

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

      1. pow1/3 35.0%

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

      2. associate-*l* 35.0%

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

      3. unpow-prod-down 37.4%

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

      4. pow1/3 60.6%

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

   6. Applied egg-rr 60.6%

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

      1. unpow1/3 98.7%

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

      2. *-commutative 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in k around 0 95.4%

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

       1. *-commutative 91.9%

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

   11. Simplified 95.4%

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

       1. cbrt-prod 95.4%

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

   13. Applied egg-rr 95.4%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+181}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{{\left(\frac{\sqrt[3]{\ell \cdot \frac{2}{\sin k}}}{t}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k} \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{2}\right)\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3}}{\tan k}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.2e-77)
    (* (* 2.0 (/ l (* (pow k 2.0) (* t_m (sin k))))) (/ l (tan k)))
    (/
     (/
      (pow (/ (cbrt (/ 2.0 (sin k))) (/ t_m (pow (cbrt l) 2.0))) 3.0)
      (tan k))
     (+ 2.0 (pow (/ k t_m) 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 (t_m <= 7.2e-77) {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t_m * sin(k))))) * (l / tan(k));
	} else {
		tmp = (pow((cbrt((2.0 / sin(k))) / (t_m / pow(cbrt(l), 2.0))), 3.0) / tan(k)) / (2.0 + pow((k / t_m), 2.0));
	}
	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 tmp;
	if (t_m <= 7.2e-77) {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t_m * Math.sin(k))))) * (l / Math.tan(k));
	} else {
		tmp = (Math.pow((Math.cbrt((2.0 / Math.sin(k))) / (t_m / Math.pow(Math.cbrt(l), 2.0))), 3.0) / Math.tan(k)) / (2.0 + Math.pow((k / t_m), 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 (t_m <= 7.2e-77)
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t_m * sin(k))))) * Float64(l / tan(k)));
	else
		tmp = Float64(Float64((Float64(cbrt(Float64(2.0 / sin(k))) / Float64(t_m / (cbrt(l) ^ 2.0))) ^ 3.0) / tan(k)) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 7.2e-77], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[Power[N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.2e-77

   1. Initial program 48.9%

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

   3. Simplified 52.6%

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

      1. associate-/l/ 52.6%

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

      2. associate-/r/ 52.6%

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

      3. times-frac 53.0%

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

      4. div-inv 53.0%

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

      5. clear-num 53.0%

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

   6. Applied egg-rr 53.0%

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

   7. Taylor expanded in k around inf 69.0%

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

      1. *-commutative 69.0%

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

   9. Simplified 69.0%

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

   if 7.2e-77 < t

   1. Initial program 66.9%

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

   3. Simplified 72.3%

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

      1. add-cube-cbrt 72.1%

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

      2. pow2 72.1%

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

      3. cbrt-div 72.1%

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

      4. associate-/r* 66.9%

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

      5. cbrt-div 66.9%

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

      6. rem-cbrt-cube 66.8%

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

      7. cbrt-prod 72.1%

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

      8. pow2 72.1%

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

      9. cbrt-div 72.1%

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

      10. associate-/r* 66.9%

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

      11. cbrt-div 68.0%

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

      12. rem-cbrt-cube 75.6%

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

      13. cbrt-prod 90.8%

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

      14. pow2 90.8%

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

   6. Applied egg-rr 90.8%

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

      1. unpow2 90.8%

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

      2. unpow3 90.8%

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

   8. Simplified 90.8%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\tan k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\right) \cdot t\_2\\ \mathbf{elif}\;t\_m \leq 9 \cdot 10^{+172}:\\ \;\;\;\;t\_2 \cdot \frac{{\left(\frac{\sqrt[3]{\ell \cdot \frac{2}{\sin k}}}{t\_m}\right)}^{3}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{2 \cdot k}\right)\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (tan k))))
   (*
    t_s
    (if (<= t_m 7.2e-77)
      (* (* 2.0 (/ l (* (pow k 2.0) (* t_m (sin k))))) t_2)
      (if (<= t_m 9e+172)
        (*
         t_2
         (/
          (pow (/ (cbrt (* l (/ 2.0 (sin k)))) t_m) 3.0)
          (+ 2.0 (pow (/ k t_m) 2.0))))
        (/
         2.0
         (pow
          (* (/ t_m (pow (cbrt l) 2.0)) (* (cbrt (sin k)) (cbrt (* 2.0 k))))
          3.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / tan(k);
	double tmp;
	if (t_m <= 7.2e-77) {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t_m * sin(k))))) * t_2;
	} else if (t_m <= 9e+172) {
		tmp = t_2 * (pow((cbrt((l * (2.0 / sin(k)))) / t_m), 3.0) / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / pow(((t_m / pow(cbrt(l), 2.0)) * (cbrt(sin(k)) * cbrt((2.0 * k)))), 3.0);
	}
	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 = l / Math.tan(k);
	double tmp;
	if (t_m <= 7.2e-77) {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t_m * Math.sin(k))))) * t_2;
	} else if (t_m <= 9e+172) {
		tmp = t_2 * (Math.pow((Math.cbrt((l * (2.0 / Math.sin(k)))) / t_m), 3.0) / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = 2.0 / Math.pow(((t_m / Math.pow(Math.cbrt(l), 2.0)) * (Math.cbrt(Math.sin(k)) * Math.cbrt((2.0 * k)))), 3.0);
	}
	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(l / tan(k))
	tmp = 0.0
	if (t_m <= 7.2e-77)
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t_m * sin(k))))) * t_2);
	elseif (t_m <= 9e+172)
		tmp = Float64(t_2 * Float64((Float64(cbrt(Float64(l * Float64(2.0 / sin(k)))) / t_m) ^ 3.0) / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_m / (cbrt(l) ^ 2.0)) * Float64(cbrt(sin(k)) * cbrt(Float64(2.0 * k)))) ^ 3.0));
	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[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.2e-77], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$m, 9e+172], N[(t$95$2 * N[(N[Power[N[(N[Power[N[(l * N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(2.0 * k), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 7.2e-77

   1. Initial program 48.9%

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

   3. Simplified 52.6%

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

      1. associate-/l/ 52.6%

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

      2. associate-/r/ 52.6%

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

      3. times-frac 53.0%

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

      4. div-inv 53.0%

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

      5. clear-num 53.0%

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

   6. Applied egg-rr 53.0%

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

   7. Taylor expanded in k around inf 69.0%

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

      1. *-commutative 69.0%

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

   9. Simplified 69.0%

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

   if 7.2e-77 < t < 9.0000000000000004e172

   1. Initial program 65.0%

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

   3. Simplified 70.8%

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

      1. associate-/l/ 70.8%

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

      2. associate-/r/ 72.7%

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

      3. times-frac 72.9%

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

      4. div-inv 72.8%

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

      5. clear-num 72.8%

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

   6. Applied egg-rr 72.8%

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

      1. add-cube-cbrt 72.6%

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

      2. pow3 72.6%

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

      3. associate-*r/ 72.6%

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

      4. cbrt-div 72.6%

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

      5. unpow3 72.6%

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

      6. add-cbrt-cube 88.9%

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

   8. Applied egg-rr 88.9%

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

   if 9.0000000000000004e172 < t

   1. Initial program 70.7%

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

      1. +-commutative 70.7%

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

      2. associate-+r+ 70.7%

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

      3. metadata-eval 70.7%

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

      4. associate-*l* 55.6%

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

      5. associate-/r* 59.7%

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

      6. add-cube-cbrt 59.7%

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

      7. pow3 59.7%

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

   4. Applied egg-rr 67.1%

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

      1. pow1/3 33.7%

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

      2. associate-*l* 33.7%

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

      3. unpow-prod-down 36.1%

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

      4. pow1/3 58.4%

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

   6. Applied egg-rr 58.4%

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

      1. unpow1/3 98.7%

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

      2. *-commutative 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in k around 0 95.6%

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

       1. *-commutative 92.2%

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

   11. Simplified 95.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+172}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{{\left(\frac{\sqrt[3]{\ell \cdot \frac{2}{\sin k}}}{t}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{2 \cdot k}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\tan k}\\ t_3 := \frac{2}{\sin k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\right) \cdot t\_2\\ \mathbf{elif}\;t\_m \leq 6.5 \cdot 10^{+195}:\\ \;\;\;\;t\_2 \cdot \frac{{\left(\frac{\sqrt[3]{\ell \cdot t\_3}}{t\_m}\right)}^{3}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{t\_3} \cdot \frac{\frac{1}{t\_m}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3}}{2 \cdot k}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (tan k))) (t_3 (/ 2.0 (sin k))))
   (*
    t_s
    (if (<= t_m 7.2e-77)
      (* (* 2.0 (/ l (* (pow k 2.0) (* t_m (sin k))))) t_2)
      (if (<= t_m 6.5e+195)
        (*
         t_2
         (/ (pow (/ (cbrt (* l t_3)) t_m) 3.0) (+ 2.0 (pow (/ k t_m) 2.0))))
        (/
         (pow (* (cbrt t_3) (/ (/ 1.0 t_m) (pow (cbrt l) -2.0))) 3.0)
         (* 2.0 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 t_2 = l / tan(k);
	double t_3 = 2.0 / sin(k);
	double tmp;
	if (t_m <= 7.2e-77) {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t_m * sin(k))))) * t_2;
	} else if (t_m <= 6.5e+195) {
		tmp = t_2 * (pow((cbrt((l * t_3)) / t_m), 3.0) / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = pow((cbrt(t_3) * ((1.0 / t_m) / pow(cbrt(l), -2.0))), 3.0) / (2.0 * k);
	}
	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 = l / Math.tan(k);
	double t_3 = 2.0 / Math.sin(k);
	double tmp;
	if (t_m <= 7.2e-77) {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t_m * Math.sin(k))))) * t_2;
	} else if (t_m <= 6.5e+195) {
		tmp = t_2 * (Math.pow((Math.cbrt((l * t_3)) / t_m), 3.0) / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = Math.pow((Math.cbrt(t_3) * ((1.0 / t_m) / Math.pow(Math.cbrt(l), -2.0))), 3.0) / (2.0 * k);
	}
	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(l / tan(k))
	t_3 = Float64(2.0 / sin(k))
	tmp = 0.0
	if (t_m <= 7.2e-77)
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t_m * sin(k))))) * t_2);
	elseif (t_m <= 6.5e+195)
		tmp = Float64(t_2 * Float64((Float64(cbrt(Float64(l * t_3)) / t_m) ^ 3.0) / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64((Float64(cbrt(t_3) * Float64(Float64(1.0 / t_m) / (cbrt(l) ^ -2.0))) ^ 3.0) / Float64(2.0 * k));
	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[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.2e-77], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$m, 6.5e+195], N[(t$95$2 * N[(N[Power[N[(N[Power[N[(l * t$95$3), $MachinePrecision], 1/3], $MachinePrecision] / t$95$m), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Power[t$95$3, 1/3], $MachinePrecision] * N[(N[(1.0 / t$95$m), $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.2e-77

   1. Initial program 48.9%

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

   3. Simplified 52.6%

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

      1. associate-/l/ 52.6%

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

      2. associate-/r/ 52.6%

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

      3. times-frac 53.0%

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

      4. div-inv 53.0%

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

      5. clear-num 53.0%

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

   6. Applied egg-rr 53.0%

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

   7. Taylor expanded in k around inf 69.0%

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

      1. *-commutative 69.0%

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

   9. Simplified 69.0%

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

   if 7.2e-77 < t < 6.5000000000000003e195

   1. Initial program 65.1%

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

   3. Simplified 70.7%

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

      1. associate-/l/ 70.7%

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

      2. associate-/r/ 72.5%

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

      3. times-frac 72.6%

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

      4. div-inv 72.5%

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

      5. clear-num 72.5%

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

   6. Applied egg-rr 72.5%

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

      1. add-cube-cbrt 72.3%

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

      2. pow3 72.3%

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

      3. associate-*r/ 72.3%

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

      4. cbrt-div 72.3%

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

      5. unpow3 72.3%

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

      6. add-cbrt-cube 89.2%

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

   8. Applied egg-rr 89.2%

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

   if 6.5000000000000003e195 < t

   1. Initial program 71.0%

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

   3. Simplified 75.9%

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

      1. associate-/l/ 75.9%

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

      2. add-cube-cbrt 75.9%

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

      3. times-frac 75.9%

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

   6. Applied egg-rr 99.0%

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

      1. frac-times 99.0%

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

      2. unpow2 99.0%

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

      3. pow3 99.0%

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

      4. unpow2 99.0%

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

      5. cbrt-prod 79.5%

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

      6. unpow2 79.5%

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

   8. Applied egg-rr 79.5%

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

      1. div-inv 79.5%

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

      2. unpow2 79.5%

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

      3. cbrt-prod 99.0%

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

      4. unpow2 99.0%

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

      5. div-inv 99.0%

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

      6. pow-flip 99.0%

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

      7. metadata-eval 99.0%

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

   10. Applied egg-rr 99.0%

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

       1. associate-/r* 98.8%

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

   12. Simplified 98.8%

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

   13. Taylor expanded in k around 0 95.2%

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

       1. *-commutative 95.2%

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

   15. Simplified 95.2%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+195}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{{\left(\frac{\sqrt[3]{\ell \cdot \frac{2}{\sin k}}}{t}\right)}^{3}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{\frac{2}{\sin k}} \cdot \frac{\frac{1}{t}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3}}{2 \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;t\_m \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{\frac{2}{\sin k}} \cdot \frac{\frac{1}{t\_m}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3}}{2 \cdot k}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.2e-77)
    (* (* 2.0 (/ l (* (pow k 2.0) (* t_m (sin k))))) (/ l (tan k)))
    (if (<= t_m 1.35e+154)
      (/
       2.0
       (*
        (* (tan k) (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l))))
        (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0))))
      (/
       (pow (* (cbrt (/ 2.0 (sin k))) (/ (/ 1.0 t_m) (pow (cbrt l) -2.0))) 3.0)
       (* 2.0 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 (t_m <= 7.2e-77) {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t_m * sin(k))))) * (l / tan(k));
	} else if (t_m <= 1.35e+154) {
		tmp = 2.0 / ((tan(k) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l)))) * (1.0 + (pow((k / t_m), 2.0) + 1.0)));
	} else {
		tmp = pow((cbrt((2.0 / sin(k))) * ((1.0 / t_m) / pow(cbrt(l), -2.0))), 3.0) / (2.0 * k);
	}
	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 tmp;
	if (t_m <= 7.2e-77) {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t_m * Math.sin(k))))) * (l / Math.tan(k));
	} else if (t_m <= 1.35e+154) {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l)))) * (1.0 + (Math.pow((k / t_m), 2.0) + 1.0)));
	} else {
		tmp = Math.pow((Math.cbrt((2.0 / Math.sin(k))) * ((1.0 / t_m) / Math.pow(Math.cbrt(l), -2.0))), 3.0) / (2.0 * 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 (t_m <= 7.2e-77)
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t_m * sin(k))))) * Float64(l / tan(k)));
	elseif (t_m <= 1.35e+154)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))) * Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0))));
	else
		tmp = Float64((Float64(cbrt(Float64(2.0 / sin(k))) * Float64(Float64(1.0 / t_m) / (cbrt(l) ^ -2.0))) ^ 3.0) / Float64(2.0 * k));
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 7.2e-77], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.35e+154], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Power[N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[(1.0 / t$95$m), $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(2.0 * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.2e-77

   1. Initial program 48.9%

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

   3. Simplified 52.6%

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

      1. associate-/l/ 52.6%

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

      2. associate-/r/ 52.6%

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

      3. times-frac 53.0%

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

      4. div-inv 53.0%

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

      5. clear-num 53.0%

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

   6. Applied egg-rr 53.0%

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

   7. Taylor expanded in k around inf 69.0%

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

      1. *-commutative 69.0%

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

   9. Simplified 69.0%

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

   if 7.2e-77 < t < 1.35000000000000003e154

   1. Initial program 64.2%

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

      1. unpow3 64.2%

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

      2. times-frac 82.4%

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

      3. pow2 82.4%

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

   4. Applied egg-rr 82.4%

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

   if 1.35000000000000003e154 < t

   1. Initial program 71.2%

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

   3. Simplified 75.2%

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

      1. associate-/l/ 75.2%

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

      2. add-cube-cbrt 75.2%

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

      3. times-frac 75.2%

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

   6. Applied egg-rr 98.9%

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

      1. frac-times 95.9%

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

      2. unpow2 95.9%

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

      3. pow3 96.0%

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

      4. unpow2 96.0%

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

      5. cbrt-prod 77.9%

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

      6. unpow2 77.9%

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

   8. Applied egg-rr 77.9%

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

      1. div-inv 77.8%

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

      2. unpow2 77.8%

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

      3. cbrt-prod 96.0%

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

      4. unpow2 96.0%

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

      5. div-inv 96.1%

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

      6. pow-flip 96.0%

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

      7. metadata-eval 96.0%

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

   10. Applied egg-rr 96.0%

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

       1. associate-/r* 95.9%

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

   12. Simplified 95.9%

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

   13. Taylor expanded in k around 0 90.2%

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

       1. *-commutative 90.2%

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

   15. Simplified 90.2%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{\frac{2}{\sin k}} \cdot \frac{\frac{1}{t}}{{\left(\sqrt[3]{\ell}\right)}^{-2}}\right)}^{3}}{2 \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;t\_m \leq 6.2 \cdot 10^{+172}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{{\ell}^{2}}}{t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.2e-77)
    (* (* 2.0 (/ l (* (pow k 2.0) (* t_m (sin k))))) (/ l (tan k)))
    (if (<= t_m 6.2e+172)
      (/
       2.0
       (*
        (* (tan k) (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l))))
        (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0))))
      (pow (/ (cbrt (pow l 2.0)) (* t_m (pow (cbrt k) 2.0))) 3.0)))))

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 (t_m <= 7.2e-77) {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t_m * sin(k))))) * (l / tan(k));
	} else if (t_m <= 6.2e+172) {
		tmp = 2.0 / ((tan(k) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l)))) * (1.0 + (pow((k / t_m), 2.0) + 1.0)));
	} else {
		tmp = pow((cbrt(pow(l, 2.0)) / (t_m * pow(cbrt(k), 2.0))), 3.0);
	}
	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 tmp;
	if (t_m <= 7.2e-77) {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t_m * Math.sin(k))))) * (l / Math.tan(k));
	} else if (t_m <= 6.2e+172) {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l)))) * (1.0 + (Math.pow((k / t_m), 2.0) + 1.0)));
	} else {
		tmp = Math.pow((Math.cbrt(Math.pow(l, 2.0)) / (t_m * Math.pow(Math.cbrt(k), 2.0))), 3.0);
	}
	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 (t_m <= 7.2e-77)
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t_m * sin(k))))) * Float64(l / tan(k)));
	elseif (t_m <= 6.2e+172)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l)))) * Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0))));
	else
		tmp = Float64(cbrt((l ^ 2.0)) / Float64(t_m * (cbrt(k) ^ 2.0))) ^ 3.0;
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 7.2e-77], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.2e+172], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[Power[l, 2.0], $MachinePrecision], 1/3], $MachinePrecision] / N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 7.2e-77

   1. Initial program 48.9%

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

   3. Simplified 52.6%

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

      1. associate-/l/ 52.6%

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

      2. associate-/r/ 52.6%

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

      3. times-frac 53.0%

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

      4. div-inv 53.0%

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

      5. clear-num 53.0%

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

   6. Applied egg-rr 53.0%

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

   7. Taylor expanded in k around inf 69.0%

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

      1. *-commutative 69.0%

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

   9. Simplified 69.0%

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

   if 7.2e-77 < t < 6.19999999999999976e172

   1. Initial program 65.0%

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

      1. unpow3 65.0%

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

      2. times-frac 81.9%

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

      3. pow2 81.9%

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

   4. Applied egg-rr 81.9%

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

   if 6.19999999999999976e172 < t

   1. Initial program 70.7%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in k around 0 55.6%

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

      1. *-commutative 55.6%

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

   7. Simplified 55.6%

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

      1. add-cube-cbrt 55.6%

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

      2. pow2 55.6%

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

      3. cbrt-div 55.6%

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

      4. cbrt-prod 55.6%

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

      5. unpow3 55.6%

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

      6. add-cbrt-cube 55.6%

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

      7. unpow2 55.6%

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

      8. cbrt-prod 55.6%

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

      9. pow2 55.6%

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

      10. cbrt-div 55.6%

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

      11. cbrt-prod 55.6%

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

      12. unpow3 55.6%

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

      13. add-cbrt-cube 56.5%

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

      14. unpow2 56.5%

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

      15. cbrt-prod 81.4%

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

      16. pow2 81.4%

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

   9. Applied egg-rr 81.4%

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

       1. unpow2 81.4%

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

       2. unpow3 81.4%

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

   11. Simplified 81.4%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+172}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{{\ell}^{2}}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.2 \cdot 10^{-25}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;t\_m \leq 6.6 \cdot 10^{+163}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right) \cdot \left(\left(t\_m \cdot \frac{\frac{{t\_m}^{2}}{\ell}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.2e-25)
    (* (* 2.0 (/ l (* (pow k 2.0) (* t_m (sin k))))) (/ l (tan k)))
    (if (<= t_m 6.6e+163)
      (/
       2.0
       (*
        (+ 2.0 (pow (/ k t_m) 2.0))
        (* (* t_m (/ (/ (pow t_m 2.0) l) l)) (* (sin k) (tan k)))))
      (/ (pow l 2.0) (pow (* t_m (pow (cbrt k) 2.0)) 3.0))))))

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 (t_m <= 6.2e-25) {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t_m * sin(k))))) * (l / tan(k));
	} else if (t_m <= 6.6e+163) {
		tmp = 2.0 / ((2.0 + pow((k / t_m), 2.0)) * ((t_m * ((pow(t_m, 2.0) / l) / l)) * (sin(k) * tan(k))));
	} else {
		tmp = pow(l, 2.0) / pow((t_m * pow(cbrt(k), 2.0)), 3.0);
	}
	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 tmp;
	if (t_m <= 6.2e-25) {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t_m * Math.sin(k))))) * (l / Math.tan(k));
	} else if (t_m <= 6.6e+163) {
		tmp = 2.0 / ((2.0 + Math.pow((k / t_m), 2.0)) * ((t_m * ((Math.pow(t_m, 2.0) / l) / l)) * (Math.sin(k) * Math.tan(k))));
	} else {
		tmp = Math.pow(l, 2.0) / Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0);
	}
	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 (t_m <= 6.2e-25)
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t_m * sin(k))))) * Float64(l / tan(k)));
	elseif (t_m <= 6.6e+163)
		tmp = Float64(2.0 / Float64(Float64(2.0 + (Float64(k / t_m) ^ 2.0)) * Float64(Float64(t_m * Float64(Float64((t_m ^ 2.0) / l) / l)) * Float64(sin(k) * tan(k)))));
	else
		tmp = Float64((l ^ 2.0) / (Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0));
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 6.2e-25], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6.6e+163], N[(2.0 / N[(N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.19999999999999989e-25

   1. Initial program 50.0%

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

   3. Simplified 53.5%

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

      1. associate-/l/ 53.5%

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

      2. associate-/r/ 53.5%

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

      3. times-frac 53.9%

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

      4. div-inv 53.9%

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

      5. clear-num 54.0%

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

   6. Applied egg-rr 54.0%

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

   7. Taylor expanded in k around inf 69.1%

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

      1. *-commutative 69.1%

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

   9. Simplified 69.1%

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

   if 6.19999999999999989e-25 < t < 6.5999999999999999e163

   1. Initial program 62.0%

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

   3. Simplified 69.2%

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

      1. cube-mult 69.2%

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

      2. *-un-lft-identity 69.2%

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

      3. times-frac 83.6%

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

      4. pow2 83.6%

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

   6. Applied egg-rr 83.6%

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

      1. /-rgt-identity 83.6%

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

      2. associate-/l* 83.6%

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

   8. Applied egg-rr 83.6%

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

   if 6.5999999999999999e163 < t

   1. Initial program 72.7%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in k around 0 55.2%

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

      1. *-commutative 55.2%

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

   7. Simplified 55.2%

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

      1. add-cube-cbrt 55.2%

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

      2. pow3 55.2%

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

      3. cbrt-prod 55.2%

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

      4. unpow3 55.2%

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

      5. add-cbrt-cube 56.0%

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

      6. unpow2 56.0%

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

      7. cbrt-prod 82.4%

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

      8. pow2 82.4%

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

   9. Applied egg-rr 82.4%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.2 \cdot 10^{-25}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{+163}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(t \cdot \frac{\frac{{t}^{2}}{\ell}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\tan k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\right) \cdot t\_2\\ \mathbf{elif}\;t\_m \leq 5.7 \cdot 10^{+102}:\\ \;\;\;\;t\_2 \cdot \frac{\frac{2}{\sin k} \cdot \frac{\ell}{{t\_m}^{3}}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (tan k))))
   (*
    t_s
    (if (<= t_m 7.2e-77)
      (* (* 2.0 (/ l (* (pow k 2.0) (* t_m (sin k))))) t_2)
      (if (<= t_m 5.7e+102)
        (*
         t_2
         (/
          (* (/ 2.0 (sin k)) (/ l (pow t_m 3.0)))
          (+ 2.0 (pow (/ k t_m) 2.0))))
        (/ (pow l 2.0) (pow (* t_m (pow (cbrt k) 2.0)) 3.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = l / tan(k);
	double tmp;
	if (t_m <= 7.2e-77) {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t_m * sin(k))))) * t_2;
	} else if (t_m <= 5.7e+102) {
		tmp = t_2 * (((2.0 / sin(k)) * (l / pow(t_m, 3.0))) / (2.0 + pow((k / t_m), 2.0)));
	} else {
		tmp = pow(l, 2.0) / pow((t_m * pow(cbrt(k), 2.0)), 3.0);
	}
	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 = l / Math.tan(k);
	double tmp;
	if (t_m <= 7.2e-77) {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t_m * Math.sin(k))))) * t_2;
	} else if (t_m <= 5.7e+102) {
		tmp = t_2 * (((2.0 / Math.sin(k)) * (l / Math.pow(t_m, 3.0))) / (2.0 + Math.pow((k / t_m), 2.0)));
	} else {
		tmp = Math.pow(l, 2.0) / Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0);
	}
	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(l / tan(k))
	tmp = 0.0
	if (t_m <= 7.2e-77)
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t_m * sin(k))))) * t_2);
	elseif (t_m <= 5.7e+102)
		tmp = Float64(t_2 * Float64(Float64(Float64(2.0 / sin(k)) * Float64(l / (t_m ^ 3.0))) / Float64(2.0 + (Float64(k / t_m) ^ 2.0))));
	else
		tmp = Float64((l ^ 2.0) / (Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0));
	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[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 7.2e-77], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$m, 5.7e+102], N[(t$95$2 * N[(N[(N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 7.2e-77

   1. Initial program 48.9%

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

   3. Simplified 52.6%

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

      1. associate-/l/ 52.6%

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

      2. associate-/r/ 52.6%

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

      3. times-frac 53.0%

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

      4. div-inv 53.0%

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

      5. clear-num 53.0%

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

   6. Applied egg-rr 53.0%

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

   7. Taylor expanded in k around inf 69.0%

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

      1. *-commutative 69.0%

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

   9. Simplified 69.0%

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

   if 7.2e-77 < t < 5.6999999999999999e102

   1. Initial program 74.6%

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

   3. Simplified 83.2%

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

      1. associate-/l/ 83.2%

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

      2. associate-/r/ 86.2%

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

      3. times-frac 86.4%

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

      4. div-inv 86.3%

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

      5. clear-num 86.3%

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

   6. Applied egg-rr 86.3%

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

   if 5.6999999999999999e102 < t

   1. Initial program 61.5%

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

   3. Simplified 64.5%

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

   5. Taylor expanded in k around 0 50.2%

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

      1. *-commutative 50.2%

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

   7. Simplified 50.2%

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

      1. add-cube-cbrt 50.2%

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

      2. pow3 50.2%

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

      3. cbrt-prod 50.2%

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

      4. unpow3 50.2%

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

      5. add-cbrt-cube 57.3%

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

      6. unpow2 57.3%

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

      7. cbrt-prod 78.1%

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

      8. pow2 78.1%

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

   9. Applied egg-rr 78.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-77}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;t \leq 5.7 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{2}{\sin k} \cdot \frac{\ell}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.6 \cdot 10^{+40}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{\left(t\_m \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.6e+40)
    (* (* 2.0 (/ l (* (pow k 2.0) (* t_m (sin k))))) (/ l (tan k)))
    (/ (pow l 2.0) (pow (* t_m (pow (cbrt k) 2.0)) 3.0)))))

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 (t_m <= 4.6e+40) {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t_m * sin(k))))) * (l / tan(k));
	} else {
		tmp = pow(l, 2.0) / pow((t_m * pow(cbrt(k), 2.0)), 3.0);
	}
	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 tmp;
	if (t_m <= 4.6e+40) {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t_m * Math.sin(k))))) * (l / Math.tan(k));
	} else {
		tmp = Math.pow(l, 2.0) / Math.pow((t_m * Math.pow(Math.cbrt(k), 2.0)), 3.0);
	}
	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 (t_m <= 4.6e+40)
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t_m * sin(k))))) * Float64(l / tan(k)));
	else
		tmp = Float64((l ^ 2.0) / (Float64(t_m * (cbrt(k) ^ 2.0)) ^ 3.0));
	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_] := N[(t$95$s * If[LessEqual[t$95$m, 4.6e+40], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[N[(t$95$m * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.59999999999999987e40

   1. Initial program 51.0%

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

   3. Simplified 54.4%

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

      1. associate-/l/ 54.4%

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

      2. associate-/r/ 54.4%

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

      3. times-frac 54.8%

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

      4. div-inv 54.8%

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

      5. clear-num 54.8%

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

   6. Applied egg-rr 54.8%

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

   7. Taylor expanded in k around inf 68.6%

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

      1. *-commutative 68.6%

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

   9. Simplified 68.6%

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

   if 4.59999999999999987e40 < t

   1. Initial program 65.2%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in k around 0 55.2%

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

      1. *-commutative 55.2%

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

   7. Simplified 55.2%

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

      1. add-cube-cbrt 55.2%

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

      2. pow3 55.2%

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

      3. cbrt-prod 55.1%

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

      4. unpow3 55.1%

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

      5. add-cbrt-cube 60.4%

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

      6. unpow2 60.4%

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

      7. cbrt-prod 77.3%

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

      8. pow2 77.3%

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

   9. Applied egg-rr 77.3%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{+40}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\tan k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.9 \cdot 10^{-48}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \frac{\frac{2 \cdot \frac{\ell}{k}}{{t\_m}^{3}}}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (tan k))))
   (*
    t_s
    (if (<= t_m 3.9e-48)
      (* (* 2.0 (/ l (* (pow k 2.0) (* t_m (sin k))))) t_2)
      (*
       t_2
       (/ (/ (* 2.0 (/ l k)) (pow t_m 3.0)) (+ 2.0 (pow (/ k t_m) 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 t_2 = l / tan(k);
	double tmp;
	if (t_m <= 3.9e-48) {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t_m * sin(k))))) * t_2;
	} else {
		tmp = t_2 * (((2.0 * (l / k)) / pow(t_m, 3.0)) / (2.0 + pow((k / t_m), 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) :: t_2
    real(8) :: tmp
    t_2 = l / tan(k)
    if (t_m <= 3.9d-48) then
        tmp = (2.0d0 * (l / ((k ** 2.0d0) * (t_m * sin(k))))) * t_2
    else
        tmp = t_2 * (((2.0d0 * (l / k)) / (t_m ** 3.0d0)) / (2.0d0 + ((k / t_m) ** 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 t_2 = l / Math.tan(k);
	double tmp;
	if (t_m <= 3.9e-48) {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t_m * Math.sin(k))))) * t_2;
	} else {
		tmp = t_2 * (((2.0 * (l / k)) / Math.pow(t_m, 3.0)) / (2.0 + Math.pow((k / t_m), 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):
	t_2 = l / math.tan(k)
	tmp = 0
	if t_m <= 3.9e-48:
		tmp = (2.0 * (l / (math.pow(k, 2.0) * (t_m * math.sin(k))))) * t_2
	else:
		tmp = t_2 * (((2.0 * (l / k)) / math.pow(t_m, 3.0)) / (2.0 + math.pow((k / t_m), 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)
	t_2 = Float64(l / tan(k))
	tmp = 0.0
	if (t_m <= 3.9e-48)
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t_m * sin(k))))) * t_2);
	else
		tmp = Float64(t_2 * Float64(Float64(Float64(2.0 * Float64(l / k)) / (t_m ^ 3.0)) / Float64(2.0 + (Float64(k / t_m) ^ 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)
	t_2 = l / tan(k);
	tmp = 0.0;
	if (t_m <= 3.9e-48)
		tmp = (2.0 * (l / ((k ^ 2.0) * (t_m * sin(k))))) * t_2;
	else
		tmp = t_2 * (((2.0 * (l / k)) / (t_m ^ 3.0)) / (2.0 + ((k / t_m) ^ 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_] := Block[{t$95$2 = N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.9e-48], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(t$95$2 * N[(N[(N[(2.0 * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.9e-48

   1. Initial program 50.3%

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

   3. Simplified 53.9%

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

      1. associate-/l/ 53.9%

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

      2. associate-/r/ 53.8%

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

      3. times-frac 54.3%

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

      4. div-inv 54.2%

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

      5. clear-num 54.3%

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

   6. Applied egg-rr 54.3%

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

   7. Taylor expanded in k around inf 69.6%

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

      1. *-commutative 69.6%

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

   9. Simplified 69.6%

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

   if 3.9e-48 < t

   1. Initial program 65.1%

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

   3. Simplified 71.0%

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

      1. associate-/l/ 71.0%

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

      2. associate-/r/ 72.3%

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

      3. times-frac 72.3%

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

      4. div-inv 72.3%

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

      5. clear-num 72.3%

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

   6. Applied egg-rr 72.3%

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

   7. Taylor expanded in k around 0 68.4%

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

      1. associate-/r* 68.5%

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

      2. associate-*r/ 68.5%

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

   9. Simplified 68.5%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.9 \cdot 10^{-48}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{2 \cdot \frac{\ell}{k}}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\ell}{\tan k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-37}:\\ \;\;\;\;t\_2 \cdot \frac{\frac{\ell}{k}}{{t\_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}\right) \cdot t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ l (tan k))))
   (*
    t_s
    (if (<= k 3.7e-37)
      (* t_2 (/ (/ l k) (pow t_m 3.0)))
      (* (* 2.0 (/ l (* (pow k 2.0) (* t_m (sin k))))) t_2)))))

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 = l / tan(k);
	double tmp;
	if (k <= 3.7e-37) {
		tmp = t_2 * ((l / k) / pow(t_m, 3.0));
	} else {
		tmp = (2.0 * (l / (pow(k, 2.0) * (t_m * sin(k))))) * t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_2 = l / tan(k)
    if (k <= 3.7d-37) then
        tmp = t_2 * ((l / k) / (t_m ** 3.0d0))
    else
        tmp = (2.0d0 * (l / ((k ** 2.0d0) * (t_m * sin(k))))) * t_2
    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 t_2 = l / Math.tan(k);
	double tmp;
	if (k <= 3.7e-37) {
		tmp = t_2 * ((l / k) / Math.pow(t_m, 3.0));
	} else {
		tmp = (2.0 * (l / (Math.pow(k, 2.0) * (t_m * Math.sin(k))))) * t_2;
	}
	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):
	t_2 = l / math.tan(k)
	tmp = 0
	if k <= 3.7e-37:
		tmp = t_2 * ((l / k) / math.pow(t_m, 3.0))
	else:
		tmp = (2.0 * (l / (math.pow(k, 2.0) * (t_m * math.sin(k))))) * t_2
	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(l / tan(k))
	tmp = 0.0
	if (k <= 3.7e-37)
		tmp = Float64(t_2 * Float64(Float64(l / k) / (t_m ^ 3.0)));
	else
		tmp = Float64(Float64(2.0 * Float64(l / Float64((k ^ 2.0) * Float64(t_m * sin(k))))) * t_2);
	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)
	t_2 = l / tan(k);
	tmp = 0.0;
	if (k <= 3.7e-37)
		tmp = t_2 * ((l / k) / (t_m ^ 3.0));
	else
		tmp = (2.0 * (l / ((k ^ 2.0) * (t_m * sin(k))))) * t_2;
	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_] := Block[{t$95$2 = N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 3.7e-37], N[(t$95$2 * N[(N[(l / k), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.7e-37

   1. Initial program 54.9%

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

   3. Simplified 59.8%

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

      1. associate-/l/ 59.8%

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

      2. associate-/r/ 60.4%

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

      3. times-frac 60.8%

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

      4. div-inv 60.8%

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

      5. clear-num 60.8%

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

   6. Applied egg-rr 60.8%

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

   7. Taylor expanded in k around 0 58.7%

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

      1. associate-/r* 59.3%

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

   9. Simplified 59.3%

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

   if 3.7e-37 < k

   1. Initial program 53.6%

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

   3. Simplified 56.4%

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

      1. associate-/l/ 56.4%

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

      2. associate-/r/ 56.4%

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

      3. times-frac 56.4%

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

      4. div-inv 56.4%

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

      5. clear-num 56.5%

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

   6. Applied egg-rr 56.5%

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

   7. Taylor expanded in k around inf 73.6%

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

      1. *-commutative 73.6%

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

   9. Simplified 73.6%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-37}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(t \cdot \sin k\right)}\right) \cdot \frac{\ell}{\tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{+63}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\ell}{\sin k \cdot {t\_m}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 6.2e+63)
    (* (/ l (tan k)) (/ l (* (sin k) (pow t_m 3.0))))
    (* 2.0 (/ (pow l 2.0) (* t_m (pow k 4.0)))))))

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 (k <= 6.2e+63) {
		tmp = (l / tan(k)) * (l / (sin(k) * pow(t_m, 3.0)));
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (t_m * pow(k, 4.0)));
	}
	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 (k <= 6.2d+63) then
        tmp = (l / tan(k)) * (l / (sin(k) * (t_m ** 3.0d0)))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) / (t_m * (k ** 4.0d0)))
    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 (k <= 6.2e+63) {
		tmp = (l / Math.tan(k)) * (l / (Math.sin(k) * Math.pow(t_m, 3.0)));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 4.0)));
	}
	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 k <= 6.2e+63:
		tmp = (l / math.tan(k)) * (l / (math.sin(k) * math.pow(t_m, 3.0)))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k, 4.0)))
	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 (k <= 6.2e+63)
		tmp = Float64(Float64(l / tan(k)) * Float64(l / Float64(sin(k) * (t_m ^ 3.0))));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 4.0))));
	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 (k <= 6.2e+63)
		tmp = (l / tan(k)) * (l / (sin(k) * (t_m ^ 3.0)));
	else
		tmp = 2.0 * ((l ^ 2.0) / (t_m * (k ^ 4.0)));
	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[k, 6.2e+63], N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[Sin[k], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.2000000000000001e63

   1. Initial program 55.6%

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

   3. Simplified 59.9%

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

      1. associate-/l/ 59.9%

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

      2. associate-/r/ 60.4%

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

      3. times-frac 60.8%

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

      4. div-inv 60.8%

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

      5. clear-num 60.8%

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

   6. Applied egg-rr 60.8%

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

   7. Taylor expanded in t around inf 60.9%

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

   if 6.2000000000000001e63 < k

   1. Initial program 51.0%

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

   3. Simplified 54.7%

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

      1. cube-mult 54.7%

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

      2. *-un-lft-identity 54.7%

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

      3. times-frac 60.7%

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

      4. pow2 60.7%

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

   6. Applied egg-rr 60.7%

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

   7. Taylor expanded in t around 0 65.5%

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

      1. times-frac 62.7%

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

      2. *-commutative 62.7%

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

   9. Simplified 62.7%

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

   10. Taylor expanded in k around 0 57.8%

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

       1. associate-/l* 57.6%

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

   12. Simplified 57.6%

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

   13. Taylor expanded in k around 0 57.8%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{+63}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\ell}{\sin k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 62.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.16 \cdot 10^{+64}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\ell}{{t\_m}^{3}}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t\_m \cdot {k}^{4}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.16e+64)
    (* (/ l (tan k)) (/ (/ l (pow t_m 3.0)) (sin k)))
    (* 2.0 (/ (pow l 2.0) (* t_m (pow k 4.0)))))))

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 (k <= 1.16e+64) {
		tmp = (l / tan(k)) * ((l / pow(t_m, 3.0)) / sin(k));
	} else {
		tmp = 2.0 * (pow(l, 2.0) / (t_m * pow(k, 4.0)));
	}
	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 (k <= 1.16d+64) then
        tmp = (l / tan(k)) * ((l / (t_m ** 3.0d0)) / sin(k))
    else
        tmp = 2.0d0 * ((l ** 2.0d0) / (t_m * (k ** 4.0d0)))
    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 (k <= 1.16e+64) {
		tmp = (l / Math.tan(k)) * ((l / Math.pow(t_m, 3.0)) / Math.sin(k));
	} else {
		tmp = 2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 4.0)));
	}
	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 k <= 1.16e+64:
		tmp = (l / math.tan(k)) * ((l / math.pow(t_m, 3.0)) / math.sin(k))
	else:
		tmp = 2.0 * (math.pow(l, 2.0) / (t_m * math.pow(k, 4.0)))
	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 (k <= 1.16e+64)
		tmp = Float64(Float64(l / tan(k)) * Float64(Float64(l / (t_m ^ 3.0)) / sin(k)));
	else
		tmp = Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 4.0))));
	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 (k <= 1.16e+64)
		tmp = (l / tan(k)) * ((l / (t_m ^ 3.0)) / sin(k));
	else
		tmp = 2.0 * ((l ^ 2.0) / (t_m * (k ^ 4.0)));
	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[k, 1.16e+64], N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.16e64

   1. Initial program 55.6%

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

   3. Simplified 59.9%

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

      1. associate-/l/ 59.9%

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

      2. associate-/r/ 60.4%

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

      3. times-frac 60.8%

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

      4. div-inv 60.8%

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

      5. clear-num 60.8%

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

   6. Applied egg-rr 60.8%

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

   7. Taylor expanded in t around inf 60.9%

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

      1. associate-/r* 60.9%

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

   9. Simplified 60.9%

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

   if 1.16e64 < k

   1. Initial program 51.0%

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

   3. Simplified 54.7%

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

      1. cube-mult 54.7%

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

      2. *-un-lft-identity 54.7%

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

      3. times-frac 60.7%

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

      4. pow2 60.7%

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

   6. Applied egg-rr 60.7%

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

   7. Taylor expanded in t around 0 65.5%

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

      1. times-frac 62.7%

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

      2. *-commutative 62.7%

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

   9. Simplified 62.7%

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

   10. Taylor expanded in k around 0 57.8%

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

       1. associate-/l* 57.6%

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

   12. Simplified 57.6%

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

   13. Taylor expanded in k around 0 57.8%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.16 \cdot 10^{+64}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\ell}{{t}^{3}}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{t \cdot {k}^{4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 63.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t\_m}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\ell}{k \cdot {t\_m}^{3}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.2)
    (/ 2.0 (* (pow k 4.0) (/ t_m (pow l 2.0))))
    (* (/ l (tan k)) (/ l (* k (pow t_m 3.0)))))))

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 (t_m <= 7.2) {
		tmp = 2.0 / (pow(k, 4.0) * (t_m / pow(l, 2.0)));
	} else {
		tmp = (l / tan(k)) * (l / (k * pow(t_m, 3.0)));
	}
	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 (t_m <= 7.2d0) then
        tmp = 2.0d0 / ((k ** 4.0d0) * (t_m / (l ** 2.0d0)))
    else
        tmp = (l / tan(k)) * (l / (k * (t_m ** 3.0d0)))
    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 (t_m <= 7.2) {
		tmp = 2.0 / (Math.pow(k, 4.0) * (t_m / Math.pow(l, 2.0)));
	} else {
		tmp = (l / Math.tan(k)) * (l / (k * Math.pow(t_m, 3.0)));
	}
	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 t_m <= 7.2:
		tmp = 2.0 / (math.pow(k, 4.0) * (t_m / math.pow(l, 2.0)))
	else:
		tmp = (l / math.tan(k)) * (l / (k * math.pow(t_m, 3.0)))
	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 (t_m <= 7.2)
		tmp = Float64(2.0 / Float64((k ^ 4.0) * Float64(t_m / (l ^ 2.0))));
	else
		tmp = Float64(Float64(l / tan(k)) * Float64(l / Float64(k * (t_m ^ 3.0))));
	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 (t_m <= 7.2)
		tmp = 2.0 / ((k ^ 4.0) * (t_m / (l ^ 2.0)));
	else
		tmp = (l / tan(k)) * (l / (k * (t_m ^ 3.0)));
	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[t$95$m, 7.2], N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.20000000000000018

   1. Initial program 50.5%

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

   3. Simplified 52.2%

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

      1. cube-mult 52.2%

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

      2. *-un-lft-identity 52.2%

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

      3. times-frac 57.8%

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

      4. pow2 57.8%

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

   6. Applied egg-rr 57.8%

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

   7. Taylor expanded in t around 0 65.7%

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

      1. times-frac 65.4%

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

      2. *-commutative 65.4%

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

   9. Simplified 65.4%

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

   10. Taylor expanded in k around 0 56.3%

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

       1. associate-/l* 56.0%

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

   12. Simplified 56.0%

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

   if 7.20000000000000018 < t

   1. Initial program 66.3%

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

   3. Simplified 72.8%

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

      1. associate-/l/ 72.8%

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

      2. associate-/r/ 74.4%

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

      3. times-frac 74.2%

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

      4. div-inv 74.2%

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

      5. clear-num 74.2%

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

   6. Applied egg-rr 74.2%

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

   7. Taylor expanded in k around 0 69.8%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 63.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.2:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t\_m}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\ell}{k}}{{t\_m}^{3}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.2)
    (/ 2.0 (* (pow k 4.0) (/ t_m (pow l 2.0))))
    (* (/ l (tan k)) (/ (/ l k) (pow t_m 3.0))))))

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 (t_m <= 7.2) {
		tmp = 2.0 / (pow(k, 4.0) * (t_m / pow(l, 2.0)));
	} else {
		tmp = (l / tan(k)) * ((l / k) / pow(t_m, 3.0));
	}
	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 (t_m <= 7.2d0) then
        tmp = 2.0d0 / ((k ** 4.0d0) * (t_m / (l ** 2.0d0)))
    else
        tmp = (l / tan(k)) * ((l / k) / (t_m ** 3.0d0))
    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 (t_m <= 7.2) {
		tmp = 2.0 / (Math.pow(k, 4.0) * (t_m / Math.pow(l, 2.0)));
	} else {
		tmp = (l / Math.tan(k)) * ((l / k) / Math.pow(t_m, 3.0));
	}
	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 t_m <= 7.2:
		tmp = 2.0 / (math.pow(k, 4.0) * (t_m / math.pow(l, 2.0)))
	else:
		tmp = (l / math.tan(k)) * ((l / k) / math.pow(t_m, 3.0))
	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 (t_m <= 7.2)
		tmp = Float64(2.0 / Float64((k ^ 4.0) * Float64(t_m / (l ^ 2.0))));
	else
		tmp = Float64(Float64(l / tan(k)) * Float64(Float64(l / k) / (t_m ^ 3.0)));
	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 (t_m <= 7.2)
		tmp = 2.0 / ((k ^ 4.0) * (t_m / (l ^ 2.0)));
	else
		tmp = (l / tan(k)) * ((l / k) / (t_m ^ 3.0));
	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[t$95$m, 7.2], N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.20000000000000018

   1. Initial program 50.5%

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

   3. Simplified 52.2%

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

      1. cube-mult 52.2%

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

      2. *-un-lft-identity 52.2%

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

      3. times-frac 57.8%

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

      4. pow2 57.8%

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

   6. Applied egg-rr 57.8%

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

   7. Taylor expanded in t around 0 65.7%

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

      1. times-frac 65.4%

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

      2. *-commutative 65.4%

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

   9. Simplified 65.4%

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

   10. Taylor expanded in k around 0 56.3%

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

       1. associate-/l* 56.0%

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

   12. Simplified 56.0%

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

   if 7.20000000000000018 < t

   1. Initial program 66.3%

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

   3. Simplified 72.8%

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

      1. associate-/l/ 72.8%

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

      2. associate-/r/ 74.4%

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

      3. times-frac 74.2%

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

      4. div-inv 74.2%

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

      5. clear-num 74.2%

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

   6. Applied egg-rr 74.2%

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

   7. Taylor expanded in k around 0 69.8%

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

      1. associate-/r* 69.9%

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

   9. Simplified 69.9%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \frac{t}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 51.9% accurate, 2.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (/ (pow l 2.0) (* t_m (pow k 4.0))))))

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 * (2.0 * (pow(l, 2.0) / (t_m * pow(k, 4.0))));
}

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 * (2.0d0 * ((l ** 2.0d0) / (t_m * (k ** 4.0d0))))
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 * (2.0 * (Math.pow(l, 2.0) / (t_m * Math.pow(k, 4.0))));
}

Python

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

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64((l ^ 2.0) / Float64(t_m * (k ^ 4.0)))))
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 * (2.0 * ((l ^ 2.0) / (t_m * (k ^ 4.0))));
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[(2.0 * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t$95$m * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.5%

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

3. Simplified 55.3%

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

   1. cube-mult 55.3%

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

   2. *-un-lft-identity 55.3%

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

   3. times-frac 61.7%

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

   4. pow2 61.7%

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

6. Applied egg-rr 61.7%

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

7. Taylor expanded in t around 0 62.1%

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

   1. times-frac 61.2%

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

   2. *-commutative 61.2%

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

9. Simplified 61.2%

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

10. Taylor expanded in k around 0 55.0%

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

    1. associate-/l* 54.4%

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

12. Simplified 54.4%

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

13. Taylor expanded in k around 0 55.0%

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

14. Final simplification 55.0%

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

Reproduce

herbie shell --seed 2024043 
(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))))