Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.1% → 84.8%

Time: 20.4s

Alternatives: 19

Speedup: 2.0×

• 27.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.1% 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: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 7.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \frac{t\_m}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right) \cdot \sqrt[3]{\tan k}\right)}^{3} \cdot \left(1 + \left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 7.8e-28)
    (/
     2.0
     (* (/ (* t_m (pow k 2.0)) (cos k)) (/ (pow (sin k) 2.0) (pow l 2.0))))
    (/
     2.0
     (*
      (pow
       (* (* (cbrt (sin k)) (/ t_m (pow (cbrt l) 2.0))) (cbrt (tan k)))
       3.0)
      (+ 1.0 (+ 1.0 (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 <= 7.8e-28) {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / cos(k)) * (pow(sin(k), 2.0) / pow(l, 2.0)));
	} else {
		tmp = 2.0 / (pow(((cbrt(sin(k)) * (t_m / pow(cbrt(l), 2.0))) * cbrt(tan(k))), 3.0) * (1.0 + (1.0 + 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 <= 7.8e-28) {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.cos(k)) * (Math.pow(Math.sin(k), 2.0) / Math.pow(l, 2.0)));
	} else {
		tmp = 2.0 / (Math.pow(((Math.cbrt(Math.sin(k)) * (t_m / Math.pow(Math.cbrt(l), 2.0))) * Math.cbrt(Math.tan(k))), 3.0) * (1.0 + (1.0 + 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 <= 7.8e-28)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / cos(k)) * Float64((sin(k) ^ 2.0) / (l ^ 2.0))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(cbrt(sin(k)) * Float64(t_m / (cbrt(l) ^ 2.0))) * cbrt(tan(k))) ^ 3.0) * Float64(1.0 + Float64(1.0 + (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, 7.8e-28], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 7.79999999999999998e-28

   1. Initial program 48.3%

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

   3. Taylor expanded in t around 0 62.6%

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

      1. associate-*r* 62.6%

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

      2. *-commutative 62.6%

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

      3. times-frac 63.6%

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

   5. Simplified 63.6%

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

   if 7.79999999999999998e-28 < t

   1. Initial program 65.7%

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

      1. add-cube-cbrt 65.7%

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

      2. pow3 65.7%

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

      3. *-commutative 65.7%

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

      4. cbrt-prod 65.6%

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

      5. cbrt-div 66.9%

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

      6. rem-cbrt-cube 75.1%

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

      7. cbrt-prod 91.7%

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

      8. pow2 91.7%

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

   4. Applied egg-rr 91.7%

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

      1. add-sqr-sqrt 45.7%

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

      2. pow2 45.7%

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

   6. Applied egg-rr 45.7%

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

      1. add-cube-cbrt 45.7%

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

      2. pow3 45.7%

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

   8. Applied egg-rr 97.6%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right) \cdot \sqrt[3]{\tan k}\right)}^{3} \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t_3 := 2 + t\_2\\ t_4 := \left(1 + \left(1 + t\_2\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq 4 \cdot 10^{+270}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \frac{\sin k \cdot \left(\tan k \cdot t\_3\right)}{\ell}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\frac{2}{{\left(k \cdot {t\_m}^{1.5}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)\right)}^{3}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0))
        (t_3 (+ 2.0 t_2))
        (t_4
         (*
          (+ 1.0 (+ 1.0 t_2))
          (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l)))))))
   (*
    t_s
    (if (<= t_4 4e+270)
      (/ 2.0 (* (/ (pow t_m 3.0) l) (/ (* (sin k) (* (tan k) t_3)) l)))
      (if (<= t_4 INFINITY)
        (/ (* (/ 2.0 (pow (* k (pow t_m 1.5)) 2.0)) (* l l)) t_3)
        (/
         2.0
         (pow
          (* t_m (* (pow (cbrt l) -2.0) (cbrt (* 2.0 (pow 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 = pow((k / t_m), 2.0);
	double t_3 = 2.0 + t_2;
	double t_4 = (1.0 + (1.0 + t_2)) * (tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l))));
	double tmp;
	if (t_4 <= 4e+270) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * ((sin(k) * (tan(k) * t_3)) / l));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = ((2.0 / pow((k * pow(t_m, 1.5)), 2.0)) * (l * l)) / t_3;
	} else {
		tmp = 2.0 / pow((t_m * (pow(cbrt(l), -2.0) * cbrt((2.0 * pow(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 = Math.pow((k / t_m), 2.0);
	double t_3 = 2.0 + t_2;
	double t_4 = (1.0 + (1.0 + t_2)) * (Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l))));
	double tmp;
	if (t_4 <= 4e+270) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * ((Math.sin(k) * (Math.tan(k) * t_3)) / l));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = ((2.0 / Math.pow((k * Math.pow(t_m, 1.5)), 2.0)) * (l * l)) / t_3;
	} else {
		tmp = 2.0 / Math.pow((t_m * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt((2.0 * Math.pow(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(k / t_m) ^ 2.0
	t_3 = Float64(2.0 + t_2)
	t_4 = Float64(Float64(1.0 + Float64(1.0 + t_2)) * Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))))
	tmp = 0.0
	if (t_4 <= 4e+270)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(Float64(sin(k) * Float64(tan(k) * t_3)) / l)));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(Float64(2.0 / (Float64(k * (t_m ^ 1.5)) ^ 2.0)) * Float64(l * l)) / t_3);
	else
		tmp = Float64(2.0 / (Float64(t_m * Float64((cbrt(l) ^ -2.0) * cbrt(Float64(2.0 * (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[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(1.0 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] * 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$4, 4e+270], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[(2.0 / N[Power[N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(2.0 / N[Power[N[(t$95$m * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < 4.0000000000000002e270

   1. Initial program 80.3%

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

      1. associate-*l* 76.8%

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

      2. associate-/r* 78.3%

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

      3. +-commutative 78.3%

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

      4. associate-+r+ 78.3%

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

      5. metadata-eval 78.3%

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

      6. associate-*r* 78.3%

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

      7. associate-*l/ 78.4%

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

      8. associate-*l* 78.4%

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

   4. Applied egg-rr 78.4%

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

      1. associate-/l* 79.9%

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

   6. Simplified 79.9%

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

   if 4.0000000000000002e270 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 74.3%

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

   3. Simplified 71.3%

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

      1. add-sqr-sqrt 71.3%

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

      2. pow2 71.3%

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

      3. *-commutative 71.3%

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

      4. sqrt-prod 44.8%

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

      5. sqrt-pow1 45.0%

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

      6. metadata-eval 45.0%

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

   6. Applied egg-rr 45.0%

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

      1. *-commutative 45.0%

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

   8. Simplified 45.0%

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

   9. Taylor expanded in k around 0 58.4%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))

   1. Initial program 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 6.8%

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

   5. Taylor expanded in k around 0 9.7%

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

      1. add-cube-cbrt 9.7%

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

      2. pow3 9.7%

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

      3. cbrt-prod 9.7%

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

      4. associate-/l/ 2.5%

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

      5. cbrt-div 2.6%

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

      6. unpow3 2.6%

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

      7. add-cbrt-cube 33.3%

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

      8. cbrt-prod 40.8%

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

      9. unpow2 40.8%

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

      10. div-inv 40.8%

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

      11. pow-flip 40.8%

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

      12. metadata-eval 40.8%

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

   7. Applied egg-rr 40.8%

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

      1. associate-*l* 41.0%

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

   9. Simplified 41.0%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \leq 4 \cdot 10^{+270}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}\\ \mathbf{elif}\;\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \leq \infty:\\ \;\;\;\;\frac{\frac{2}{{\left(k \cdot {t}^{1.5}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t_3 := 2 + t\_2\\ t_4 := \left(1 + \left(1 + t\_2\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell \cdot \ell}\right)\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq 4 \cdot 10^{+270}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \frac{\sin k \cdot \left(\tan k \cdot t\_3\right)}{\ell}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\frac{2}{{\left(k \cdot {t\_m}^{1.5}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(t\_m \cdot \sqrt[3]{2 \cdot \frac{{k}^{2}}{\ell}}\right)}^{3}}{\ell}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0))
        (t_3 (+ 2.0 t_2))
        (t_4
         (*
          (+ 1.0 (+ 1.0 t_2))
          (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l)))))))
   (*
    t_s
    (if (<= t_4 4e+270)
      (/ 2.0 (* (/ (pow t_m 3.0) l) (/ (* (sin k) (* (tan k) t_3)) l)))
      (if (<= t_4 INFINITY)
        (/ (* (/ 2.0 (pow (* k (pow t_m 1.5)) 2.0)) (* l l)) t_3)
        (/ 2.0 (/ (pow (* t_m (cbrt (* 2.0 (/ (pow k 2.0) l)))) 3.0) l)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double t_3 = 2.0 + t_2;
	double t_4 = (1.0 + (1.0 + t_2)) * (tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l))));
	double tmp;
	if (t_4 <= 4e+270) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * ((sin(k) * (tan(k) * t_3)) / l));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = ((2.0 / pow((k * pow(t_m, 1.5)), 2.0)) * (l * l)) / t_3;
	} else {
		tmp = 2.0 / (pow((t_m * cbrt((2.0 * (pow(k, 2.0) / l)))), 3.0) / l);
	}
	return t_s * tmp;
}

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double t_3 = 2.0 + t_2;
	double t_4 = (1.0 + (1.0 + t_2)) * (Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l))));
	double tmp;
	if (t_4 <= 4e+270) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * ((Math.sin(k) * (Math.tan(k) * t_3)) / l));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = ((2.0 / Math.pow((k * Math.pow(t_m, 1.5)), 2.0)) * (l * l)) / t_3;
	} else {
		tmp = 2.0 / (Math.pow((t_m * Math.cbrt((2.0 * (Math.pow(k, 2.0) / l)))), 3.0) / l);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	t_3 = Float64(2.0 + t_2)
	t_4 = Float64(Float64(1.0 + Float64(1.0 + t_2)) * Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l)))))
	tmp = 0.0
	if (t_4 <= 4e+270)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(Float64(sin(k) * Float64(tan(k) * t_3)) / l)));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(Float64(2.0 / (Float64(k * (t_m ^ 1.5)) ^ 2.0)) * Float64(l * l)) / t_3);
	else
		tmp = Float64(2.0 / Float64((Float64(t_m * cbrt(Float64(2.0 * Float64((k ^ 2.0) / l)))) ^ 3.0) / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(1.0 + N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] * 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$4, 4e+270], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[(2.0 / N[Power[N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$m * N[Power[N[(2.0 * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < 4.0000000000000002e270

   1. Initial program 80.3%

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

      1. associate-*l* 76.8%

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

      2. associate-/r* 78.3%

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

      3. +-commutative 78.3%

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

      4. associate-+r+ 78.3%

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

      5. metadata-eval 78.3%

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

      6. associate-*r* 78.3%

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

      7. associate-*l/ 78.4%

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

      8. associate-*l* 78.4%

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

   4. Applied egg-rr 78.4%

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

      1. associate-/l* 79.9%

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

   6. Simplified 79.9%

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

   if 4.0000000000000002e270 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 74.3%

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

   3. Simplified 71.3%

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

      1. add-sqr-sqrt 71.3%

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

      2. pow2 71.3%

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

      3. *-commutative 71.3%

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

      4. sqrt-prod 44.8%

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

      5. sqrt-pow1 45.0%

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

      6. metadata-eval 45.0%

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

   6. Applied egg-rr 45.0%

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

      1. *-commutative 45.0%

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

   8. Simplified 45.0%

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

   9. Taylor expanded in k around 0 58.4%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64)))

   1. Initial program 0.0%

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

      1. associate-*l* 0.0%

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

      2. associate-/r* 6.8%

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

      3. +-commutative 6.8%

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

      4. associate-+r+ 6.8%

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

      5. metadata-eval 6.8%

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

      6. associate-*r* 6.8%

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

      7. associate-*l/ 6.9%

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

      8. associate-*l* 6.9%

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

   4. Applied egg-rr 6.9%

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

      1. associate-/l* 6.6%

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

   6. Simplified 6.6%

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

   7. Taylor expanded in k around 0 9.3%

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

      1. associate-*r/ 9.3%

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

   9. Simplified 9.3%

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

       1. associate-*l/ 9.3%

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

       2. associate-/l* 9.3%

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

   11. Applied egg-rr 9.3%

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

       1. add-cube-cbrt 9.3%

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

       2. pow3 9.3%

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

       3. cbrt-prod 9.3%

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

       4. unpow3 9.3%

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

       5. add-cbrt-cube 39.4%

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

   13. Applied egg-rr 39.4%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \leq 4 \cdot 10^{+270}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \frac{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\ell}}\\ \mathbf{elif}\;\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right) \leq \infty:\\ \;\;\;\;\frac{\frac{2}{{\left(k \cdot {t}^{1.5}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(t \cdot \sqrt[3]{2 \cdot \frac{{k}^{2}}{\ell}}\right)}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\sqrt[3]{\ell}\right)}^{-2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 7.6 \cdot 10^{-122}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot t\_2\right) \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left(t\_2 \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (cbrt l) -2.0)))
   (*
    t_s
    (if (<= k 7.6e-122)
      (/ 2.0 (pow (* (* t_m t_2) (* (pow (cbrt k) 2.0) (cbrt 2.0))) 3.0))
      (if (<= k 6.2e+49)
        (/
         2.0
         (pow
          (*
           t_m
           (* t_2 (cbrt (* (+ 2.0 (pow (/ k t_m) 2.0)) (* (sin k) (tan k))))))
          3.0))
        (/
         2.0
         (/
          (* (pow k 2.0) (* t_m (pow (sin k) 2.0)))
          (* (cos k) (pow l 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 = pow(cbrt(l), -2.0);
	double tmp;
	if (k <= 7.6e-122) {
		tmp = 2.0 / pow(((t_m * t_2) * (pow(cbrt(k), 2.0) * cbrt(2.0))), 3.0);
	} else if (k <= 6.2e+49) {
		tmp = 2.0 / pow((t_m * (t_2 * cbrt(((2.0 + pow((k / t_m), 2.0)) * (sin(k) * tan(k)))))), 3.0);
	} else {
		tmp = 2.0 / ((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (cos(k) * pow(l, 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 t_2 = Math.pow(Math.cbrt(l), -2.0);
	double tmp;
	if (k <= 7.6e-122) {
		tmp = 2.0 / Math.pow(((t_m * t_2) * (Math.pow(Math.cbrt(k), 2.0) * Math.cbrt(2.0))), 3.0);
	} else if (k <= 6.2e+49) {
		tmp = 2.0 / Math.pow((t_m * (t_2 * Math.cbrt(((2.0 + Math.pow((k / t_m), 2.0)) * (Math.sin(k) * Math.tan(k)))))), 3.0);
	} else {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))) / (Math.cos(k) * Math.pow(l, 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 = cbrt(l) ^ -2.0
	tmp = 0.0
	if (k <= 7.6e-122)
		tmp = Float64(2.0 / (Float64(Float64(t_m * t_2) * Float64((cbrt(k) ^ 2.0) * cbrt(2.0))) ^ 3.0));
	elseif (k <= 6.2e+49)
		tmp = Float64(2.0 / (Float64(t_m * Float64(t_2 * cbrt(Float64(Float64(2.0 + (Float64(k / t_m) ^ 2.0)) * Float64(sin(k) * tan(k)))))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(cos(k) * (l ^ 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_] := Block[{t$95$2 = N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 7.6e-122], N[(2.0 / N[Power[N[(N[(t$95$m * t$95$2), $MachinePrecision] * N[(N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6.2e+49], N[(2.0 / N[Power[N[(t$95$m * N[(t$95$2 * N[Power[N[(N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 7.6000000000000002e-122

   1. Initial program 55.3%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in k around 0 48.4%

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

      1. add-cube-cbrt 48.4%

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

      2. pow3 48.4%

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

      3. cbrt-prod 48.4%

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

      4. associate-/l/ 44.8%

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

      5. cbrt-div 45.5%

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

      6. unpow3 45.5%

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

      7. add-cbrt-cube 54.4%

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

      8. cbrt-prod 58.2%

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

      9. unpow2 58.2%

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

      10. div-inv 58.2%

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

      11. pow-flip 58.2%

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

      12. metadata-eval 58.2%

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

   7. Applied egg-rr 58.2%

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

      1. *-commutative 58.2%

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

      2. cbrt-prod 58.2%

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

      3. unpow2 58.2%

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

      4. cbrt-prod 74.3%

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

      5. pow2 74.3%

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

   9. Applied egg-rr 74.3%

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

   if 7.6000000000000002e-122 < k < 6.19999999999999985e49

   1. Initial program 50.8%

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

      1. associate-*l* 50.8%

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

      2. associate-/r* 54.5%

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

      3. +-commutative 54.5%

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

      4. associate-+r+ 54.5%

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

      5. metadata-eval 54.5%

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

      6. associate-*r* 54.5%

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

      7. associate-*l/ 60.0%

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

      8. associate-*l* 60.0%

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

   4. Applied egg-rr 60.0%

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

      1. associate-/l* 59.6%

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

   6. Simplified 59.6%

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

   8. Applied egg-rr 81.6%

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

      1. associate-*l* 81.9%

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

   10. Simplified 81.9%

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

   if 6.19999999999999985e49 < k

   1. Initial program 50.4%

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

   3. Taylor expanded in t around 0 78.1%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.6 \cdot 10^{-122}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt[3]{k \cdot \sqrt{2}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(t\_2 \cdot t\_2\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 44000000000:\\ \;\;\;\;\frac{2}{{\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t\_m}\right)\right) \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (cbrt (* k (sqrt 2.0)))))
   (*
    t_s
    (if (<= k 2.8e-64)
      (/ 2.0 (pow (* (* t_m (pow (cbrt l) -2.0)) (* t_2 t_2)) 3.0))
      (if (<= k 44000000000.0)
        (/
         2.0
         (pow
          (*
           (/ (pow t_m 1.5) l)
           (* (hypot 1.0 (hypot 1.0 (/ k t_m))) (sqrt (* (sin k) (tan k)))))
          2.0))
        (/
         2.0
         (/
          (* (pow k 2.0) (* t_m (pow (sin k) 2.0)))
          (* (cos k) (pow l 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 = cbrt((k * sqrt(2.0)));
	double tmp;
	if (k <= 2.8e-64) {
		tmp = 2.0 / pow(((t_m * pow(cbrt(l), -2.0)) * (t_2 * t_2)), 3.0);
	} else if (k <= 44000000000.0) {
		tmp = 2.0 / pow(((pow(t_m, 1.5) / l) * (hypot(1.0, hypot(1.0, (k / t_m))) * sqrt((sin(k) * tan(k))))), 2.0);
	} else {
		tmp = 2.0 / ((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (cos(k) * pow(l, 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 t_2 = Math.cbrt((k * Math.sqrt(2.0)));
	double tmp;
	if (k <= 2.8e-64) {
		tmp = 2.0 / Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * (t_2 * t_2)), 3.0);
	} else if (k <= 44000000000.0) {
		tmp = 2.0 / Math.pow(((Math.pow(t_m, 1.5) / l) * (Math.hypot(1.0, Math.hypot(1.0, (k / t_m))) * Math.sqrt((Math.sin(k) * Math.tan(k))))), 2.0);
	} else {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))) / (Math.cos(k) * Math.pow(l, 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 = cbrt(Float64(k * sqrt(2.0)))
	tmp = 0.0
	if (k <= 2.8e-64)
		tmp = Float64(2.0 / (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * Float64(t_2 * t_2)) ^ 3.0));
	elseif (k <= 44000000000.0)
		tmp = Float64(2.0 / (Float64(Float64((t_m ^ 1.5) / l) * Float64(hypot(1.0, hypot(1.0, Float64(k / t_m))) * sqrt(Float64(sin(k) * tan(k))))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(cos(k) * (l ^ 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_] := Block[{t$95$2 = N[Power[N[(k * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 2.8e-64], N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 44000000000.0], N[(2.0 / N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sqrt[1.0 ^ 2 + N[Sqrt[1.0 ^ 2 + N[(k / t$95$m), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 2.80000000000000004e-64

   1. Initial program 56.0%

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

   3. Simplified 55.7%

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

   5. Taylor expanded in k around 0 49.5%

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

      1. add-cube-cbrt 49.5%

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

      2. pow3 49.5%

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

      3. cbrt-prod 49.4%

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

      4. associate-/l/ 46.0%

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

      5. cbrt-div 46.7%

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

      6. unpow3 46.7%

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

      7. add-cbrt-cube 55.7%

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

      8. cbrt-prod 60.5%

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

      9. unpow2 60.5%

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

      10. div-inv 60.5%

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

      11. pow-flip 60.5%

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

      12. metadata-eval 60.5%

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

   7. Applied egg-rr 60.5%

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

      1. pow1/3 59.8%

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

      2. add-sqr-sqrt 59.8%

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

      3. unpow-prod-down 59.8%

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

      4. *-commutative 59.8%

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

      5. sqrt-prod 59.8%

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

      6. sqrt-pow1 16.3%

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

      7. metadata-eval 16.3%

         \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left({\left({k}^{\color{blue}{1}} \cdot \sqrt{2}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{2 \cdot {k}^{2}}\right)}^{0.3333333333333333}\right)\right)}^{3}} \]

      8. pow1 16.3%

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

      9. *-commutative 16.3%

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

      10. sqrt-prod 16.3%

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

      11. sqrt-pow1 22.7%

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

      12. metadata-eval 22.7%

          \[\leadsto \frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left({\left(k \cdot \sqrt{2}\right)}^{0.3333333333333333} \cdot {\left({k}^{\color{blue}{1}} \cdot \sqrt{2}\right)}^{0.3333333333333333}\right)\right)}^{3}} \]

      13. pow1 22.7%

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

   9. Applied egg-rr 22.7%

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

       1. unpow1/3 22.9%

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

       2. unpow1/3 75.6%

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

   11. Simplified 75.6%

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

   if 2.80000000000000004e-64 < k < 4.4e10

   1. Initial program 54.5%

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

   4. Applied egg-rr 60.3%

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

   if 4.4e10 < k

   1. Initial program 47.8%

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

   3. Taylor expanded in t around 0 74.8%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left(\sqrt[3]{k \cdot \sqrt{2}} \cdot \sqrt[3]{k \cdot \sqrt{2}}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 44000000000:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \sqrt{\sin k \cdot \tan k}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 350000000:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 350000000.0)
    (/
     2.0
     (pow
      (* (* t_m (pow (cbrt l) -2.0)) (* (pow (cbrt k) 2.0) (cbrt 2.0)))
      3.0))
    (/
     2.0
     (/ (* (pow k 2.0) (* t_m (pow (sin k) 2.0))) (* (cos k) (pow l 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 (k <= 350000000.0) {
		tmp = 2.0 / pow(((t_m * pow(cbrt(l), -2.0)) * (pow(cbrt(k), 2.0) * cbrt(2.0))), 3.0);
	} else {
		tmp = 2.0 / ((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (cos(k) * pow(l, 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 (k <= 350000000.0) {
		tmp = 2.0 / Math.pow(((t_m * Math.pow(Math.cbrt(l), -2.0)) * (Math.pow(Math.cbrt(k), 2.0) * Math.cbrt(2.0))), 3.0);
	} else {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))) / (Math.cos(k) * Math.pow(l, 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 (k <= 350000000.0)
		tmp = Float64(2.0 / (Float64(Float64(t_m * (cbrt(l) ^ -2.0)) * Float64((cbrt(k) ^ 2.0) * cbrt(2.0))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(cos(k) * (l ^ 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[k, 350000000.0], N[(2.0 / N[Power[N[(N[(t$95$m * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.5e8

   1. Initial program 55.9%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in k around 0 51.2%

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

      1. add-cube-cbrt 51.1%

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

      2. pow3 51.1%

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

      3. cbrt-prod 51.1%

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

      4. associate-/l/ 47.9%

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

      5. cbrt-div 49.5%

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

      6. unpow3 49.5%

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

      7. add-cbrt-cube 57.8%

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

      8. cbrt-prod 63.2%

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

      9. unpow2 63.2%

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

      10. div-inv 63.2%

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

      11. pow-flip 63.2%

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

      12. metadata-eval 63.2%

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

   7. Applied egg-rr 63.2%

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

      1. *-commutative 63.2%

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

      2. cbrt-prod 63.2%

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

      3. unpow2 63.2%

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

      4. cbrt-prod 77.1%

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

      5. pow2 77.1%

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

   9. Applied egg-rr 77.1%

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

   if 3.5e8 < k

   1. Initial program 47.8%

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

   3. Taylor expanded in t around 0 74.8%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 350000000:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot \sqrt[3]{2}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{1}{\sqrt[3]{\ell}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-219}:\\ \;\;\;\;\frac{\frac{2}{{\left(k \cdot {t\_m}^{1.5}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{-167}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3} \cdot {\left(k \cdot \frac{\sqrt{2}}{\sqrt{\ell}}\right)}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq 350000000:\\ \;\;\;\;\frac{2}{{\left(\left(t\_m \cdot \left(t\_2 \cdot t\_2\right)\right) \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ 1.0 (cbrt l))))
   (*
    t_s
    (if (<= k 6e-219)
      (/
       (* (/ 2.0 (pow (* k (pow t_m 1.5)) 2.0)) (* l l))
       (+ 2.0 (pow (/ k t_m) 2.0)))
      (if (<= k 1.2e-167)
        (/ 2.0 (/ (* (pow t_m 3.0) (pow (* k (/ (sqrt 2.0) (sqrt l))) 2.0)) l))
        (if (<= k 350000000.0)
          (/ 2.0 (pow (* (* t_m (* t_2 t_2)) (cbrt (* 2.0 (pow k 2.0)))) 3.0))
          (/
           2.0
           (/
            (* (pow k 2.0) (* t_m (pow (sin k) 2.0)))
            (* (cos k) (pow l 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 = 1.0 / cbrt(l);
	double tmp;
	if (k <= 6e-219) {
		tmp = ((2.0 / pow((k * pow(t_m, 1.5)), 2.0)) * (l * l)) / (2.0 + pow((k / t_m), 2.0));
	} else if (k <= 1.2e-167) {
		tmp = 2.0 / ((pow(t_m, 3.0) * pow((k * (sqrt(2.0) / sqrt(l))), 2.0)) / l);
	} else if (k <= 350000000.0) {
		tmp = 2.0 / pow(((t_m * (t_2 * t_2)) * cbrt((2.0 * pow(k, 2.0)))), 3.0);
	} else {
		tmp = 2.0 / ((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (cos(k) * pow(l, 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 t_2 = 1.0 / Math.cbrt(l);
	double tmp;
	if (k <= 6e-219) {
		tmp = ((2.0 / Math.pow((k * Math.pow(t_m, 1.5)), 2.0)) * (l * l)) / (2.0 + Math.pow((k / t_m), 2.0));
	} else if (k <= 1.2e-167) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) * Math.pow((k * (Math.sqrt(2.0) / Math.sqrt(l))), 2.0)) / l);
	} else if (k <= 350000000.0) {
		tmp = 2.0 / Math.pow(((t_m * (t_2 * t_2)) * Math.cbrt((2.0 * Math.pow(k, 2.0)))), 3.0);
	} else {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))) / (Math.cos(k) * Math.pow(l, 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(1.0 / cbrt(l))
	tmp = 0.0
	if (k <= 6e-219)
		tmp = Float64(Float64(Float64(2.0 / (Float64(k * (t_m ^ 1.5)) ^ 2.0)) * Float64(l * l)) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	elseif (k <= 1.2e-167)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) * (Float64(k * Float64(sqrt(2.0) / sqrt(l))) ^ 2.0)) / l));
	elseif (k <= 350000000.0)
		tmp = Float64(2.0 / (Float64(Float64(t_m * Float64(t_2 * t_2)) * cbrt(Float64(2.0 * (k ^ 2.0)))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(cos(k) * (l ^ 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_] := Block[{t$95$2 = N[(1.0 / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 6e-219], N[(N[(N[(2.0 / N[Power[N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.2e-167], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[Power[N[(k * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 350000000.0], N[(2.0 / N[Power[N[(N[(t$95$m * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if k < 6.0000000000000002e-219

   1. Initial program 56.2%

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

   3. Simplified 52.1%

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

      1. add-sqr-sqrt 22.9%

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

      2. pow2 22.9%

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

      3. *-commutative 22.9%

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

      4. sqrt-prod 17.7%

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

      5. sqrt-pow1 19.2%

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

      6. metadata-eval 19.2%

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

   6. Applied egg-rr 19.2%

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

      1. *-commutative 19.2%

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

   8. Simplified 19.2%

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

   9. Taylor expanded in k around 0 30.7%

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

   if 6.0000000000000002e-219 < k < 1.19999999999999997e-167

   1. Initial program 61.0%

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

      1. associate-*l* 40.0%

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

      2. associate-/r* 40.0%

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

      3. +-commutative 40.0%

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

      4. associate-+r+ 40.0%

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

      5. metadata-eval 40.0%

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

      6. associate-*r* 40.0%

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

      7. associate-*l/ 40.9%

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

      8. associate-*l* 40.9%

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

   4. Applied egg-rr 40.9%

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

      1. associate-/l* 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in k around 0 40.9%

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

      1. associate-*r/ 40.9%

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

   9. Simplified 40.9%

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

       1. add-sqr-sqrt 10.9%

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

       2. pow2 10.9%

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

       3. sqrt-div 10.9%

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

       4. *-commutative 10.9%

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

       5. sqrt-prod 10.9%

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

       6. sqrt-pow1 59.6%

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

       7. metadata-eval 59.6%

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

       8. pow1 59.6%

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

   11. Applied egg-rr 59.6%

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

       1. associate-*l/ 59.7%

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

       2. associate-/l* 59.7%

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

   13. Applied egg-rr 59.7%

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

   if 1.19999999999999997e-167 < k < 3.5e8

   1. Initial program 53.0%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in k around 0 63.4%

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

      1. add-cube-cbrt 63.3%

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

      2. pow3 63.3%

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

      3. cbrt-prod 63.1%

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

      4. associate-/l/ 59.0%

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

      5. cbrt-div 67.9%

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

      6. unpow3 67.9%

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

      7. add-cbrt-cube 73.9%

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

      8. cbrt-prod 92.1%

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

      9. unpow2 92.1%

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

      10. div-inv 92.0%

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

      11. pow-flip 92.1%

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

      12. metadata-eval 92.1%

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

   7. Applied egg-rr 92.1%

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

      1. sqr-pow 92.2%

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

      2. metadata-eval 92.2%

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

      3. unpow-1 92.2%

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

      4. metadata-eval 92.2%

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

      5. unpow-1 92.2%

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

   9. Applied egg-rr 92.2%

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

   if 3.5e8 < k

   1. Initial program 47.8%

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

   3. Taylor expanded in t around 0 74.8%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-219}:\\ \;\;\;\;\frac{\frac{2}{{\left(k \cdot {t}^{1.5}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{-167}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3} \cdot {\left(k \cdot \frac{\sqrt{2}}{\sqrt{\ell}}\right)}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq 350000000:\\ \;\;\;\;\frac{2}{{\left(\left(t \cdot \left(\frac{1}{\sqrt[3]{\ell}} \cdot \frac{1}{\sqrt[3]{\ell}}\right)\right) \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.1 \cdot 10^{-219}:\\ \;\;\;\;\frac{\frac{2}{{\left(k \cdot {t\_m}^{1.5}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{-167}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3} \cdot {\left(k \cdot \frac{\sqrt{2}}{\sqrt{\ell}}\right)}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq 350000000:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t\_m \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 4.1e-219)
    (/
     (* (/ 2.0 (pow (* k (pow t_m 1.5)) 2.0)) (* l l))
     (+ 2.0 (pow (/ k t_m) 2.0)))
    (if (<= k 1.2e-167)
      (/ 2.0 (/ (* (pow t_m 3.0) (pow (* k (/ (sqrt 2.0) (sqrt l))) 2.0)) l))
      (if (<= k 350000000.0)
        (/
         2.0
         (pow (* t_m (* (pow (cbrt l) -2.0) (cbrt (* 2.0 (pow k 2.0))))) 3.0))
        (/
         2.0
         (/
          (* (pow k 2.0) (* t_m (pow (sin k) 2.0)))
          (* (cos k) (pow l 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 (k <= 4.1e-219) {
		tmp = ((2.0 / pow((k * pow(t_m, 1.5)), 2.0)) * (l * l)) / (2.0 + pow((k / t_m), 2.0));
	} else if (k <= 1.2e-167) {
		tmp = 2.0 / ((pow(t_m, 3.0) * pow((k * (sqrt(2.0) / sqrt(l))), 2.0)) / l);
	} else if (k <= 350000000.0) {
		tmp = 2.0 / pow((t_m * (pow(cbrt(l), -2.0) * cbrt((2.0 * pow(k, 2.0))))), 3.0);
	} else {
		tmp = 2.0 / ((pow(k, 2.0) * (t_m * pow(sin(k), 2.0))) / (cos(k) * pow(l, 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 (k <= 4.1e-219) {
		tmp = ((2.0 / Math.pow((k * Math.pow(t_m, 1.5)), 2.0)) * (l * l)) / (2.0 + Math.pow((k / t_m), 2.0));
	} else if (k <= 1.2e-167) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) * Math.pow((k * (Math.sqrt(2.0) / Math.sqrt(l))), 2.0)) / l);
	} else if (k <= 350000000.0) {
		tmp = 2.0 / Math.pow((t_m * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt((2.0 * Math.pow(k, 2.0))))), 3.0);
	} else {
		tmp = 2.0 / ((Math.pow(k, 2.0) * (t_m * Math.pow(Math.sin(k), 2.0))) / (Math.cos(k) * Math.pow(l, 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 (k <= 4.1e-219)
		tmp = Float64(Float64(Float64(2.0 / (Float64(k * (t_m ^ 1.5)) ^ 2.0)) * Float64(l * l)) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	elseif (k <= 1.2e-167)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) * (Float64(k * Float64(sqrt(2.0) / sqrt(l))) ^ 2.0)) / l));
	elseif (k <= 350000000.0)
		tmp = Float64(2.0 / (Float64(t_m * Float64((cbrt(l) ^ -2.0) * cbrt(Float64(2.0 * (k ^ 2.0))))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64((k ^ 2.0) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(cos(k) * (l ^ 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[k, 4.1e-219], N[(N[(N[(2.0 / N[Power[N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.2e-167], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[Power[N[(k * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 350000000.0], N[(2.0 / N[Power[N[(t$95$m * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if k < 4.1e-219

   1. Initial program 56.2%

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

   3. Simplified 52.1%

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

      1. add-sqr-sqrt 22.9%

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

      2. pow2 22.9%

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

      3. *-commutative 22.9%

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

      4. sqrt-prod 17.7%

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

      5. sqrt-pow1 19.2%

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

      6. metadata-eval 19.2%

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

   6. Applied egg-rr 19.2%

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

      1. *-commutative 19.2%

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

   8. Simplified 19.2%

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

   9. Taylor expanded in k around 0 30.7%

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

   if 4.1e-219 < k < 1.19999999999999997e-167

   1. Initial program 61.0%

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

      1. associate-*l* 40.0%

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

      2. associate-/r* 40.0%

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

      3. +-commutative 40.0%

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

      4. associate-+r+ 40.0%

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

      5. metadata-eval 40.0%

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

      6. associate-*r* 40.0%

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

      7. associate-*l/ 40.9%

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

      8. associate-*l* 40.9%

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

   4. Applied egg-rr 40.9%

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

      1. associate-/l* 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in k around 0 40.9%

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

      1. associate-*r/ 40.9%

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

   9. Simplified 40.9%

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

       1. add-sqr-sqrt 10.9%

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

       2. pow2 10.9%

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

       3. sqrt-div 10.9%

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

       4. *-commutative 10.9%

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

       5. sqrt-prod 10.9%

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

       6. sqrt-pow1 59.6%

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

       7. metadata-eval 59.6%

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

       8. pow1 59.6%

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

   11. Applied egg-rr 59.6%

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

       1. associate-*l/ 59.7%

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

       2. associate-/l* 59.7%

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

   13. Applied egg-rr 59.7%

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

   if 1.19999999999999997e-167 < k < 3.5e8

   1. Initial program 53.0%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in k around 0 63.4%

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

      1. add-cube-cbrt 63.3%

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

      2. pow3 63.3%

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

      3. cbrt-prod 63.1%

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

      4. associate-/l/ 59.0%

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

      5. cbrt-div 67.9%

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

      6. unpow3 67.9%

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

      7. add-cbrt-cube 73.9%

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

      8. cbrt-prod 92.1%

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

      9. unpow2 92.1%

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

      10. div-inv 92.0%

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

      11. pow-flip 92.1%

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

      12. metadata-eval 92.1%

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

   7. Applied egg-rr 92.1%

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

      1. associate-*l* 92.2%

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

   9. Simplified 92.2%

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

   if 3.5e8 < k

   1. Initial program 47.8%

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

   3. Taylor expanded in t around 0 74.8%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.1 \cdot 10^{-219}:\\ \;\;\;\;\frac{\frac{2}{{\left(k \cdot {t}^{1.5}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{-167}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3} \cdot {\left(k \cdot \frac{\sqrt{2}}{\sqrt{\ell}}\right)}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq 350000000:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{\frac{2}{{\left(k \cdot {t\_m}^{1.5}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{elif}\;k \leq 8.1 \cdot 10^{-168}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3} \cdot {\left(k \cdot \frac{\sqrt{2}}{\sqrt{\ell}}\right)}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq 430000000:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m \cdot {k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 4.8e-219)
    (/
     (* (/ 2.0 (pow (* k (pow t_m 1.5)) 2.0)) (* l l))
     (+ 2.0 (pow (/ k t_m) 2.0)))
    (if (<= k 8.1e-168)
      (/ 2.0 (/ (* (pow t_m 3.0) (pow (* k (/ (sqrt 2.0) (sqrt l))) 2.0)) l))
      (if (<= k 430000000.0)
        (/
         2.0
         (pow (* t_m (* (pow (cbrt l) -2.0) (cbrt (* 2.0 (pow k 2.0))))) 3.0))
        (/
         2.0
         (*
          (/ (* t_m (pow k 2.0)) (cos k))
          (/ (pow (sin k) 2.0) (pow l 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 (k <= 4.8e-219) {
		tmp = ((2.0 / pow((k * pow(t_m, 1.5)), 2.0)) * (l * l)) / (2.0 + pow((k / t_m), 2.0));
	} else if (k <= 8.1e-168) {
		tmp = 2.0 / ((pow(t_m, 3.0) * pow((k * (sqrt(2.0) / sqrt(l))), 2.0)) / l);
	} else if (k <= 430000000.0) {
		tmp = 2.0 / pow((t_m * (pow(cbrt(l), -2.0) * cbrt((2.0 * pow(k, 2.0))))), 3.0);
	} else {
		tmp = 2.0 / (((t_m * pow(k, 2.0)) / cos(k)) * (pow(sin(k), 2.0) / pow(l, 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 (k <= 4.8e-219) {
		tmp = ((2.0 / Math.pow((k * Math.pow(t_m, 1.5)), 2.0)) * (l * l)) / (2.0 + Math.pow((k / t_m), 2.0));
	} else if (k <= 8.1e-168) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) * Math.pow((k * (Math.sqrt(2.0) / Math.sqrt(l))), 2.0)) / l);
	} else if (k <= 430000000.0) {
		tmp = 2.0 / Math.pow((t_m * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt((2.0 * Math.pow(k, 2.0))))), 3.0);
	} else {
		tmp = 2.0 / (((t_m * Math.pow(k, 2.0)) / Math.cos(k)) * (Math.pow(Math.sin(k), 2.0) / Math.pow(l, 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 (k <= 4.8e-219)
		tmp = Float64(Float64(Float64(2.0 / (Float64(k * (t_m ^ 1.5)) ^ 2.0)) * Float64(l * l)) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	elseif (k <= 8.1e-168)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) * (Float64(k * Float64(sqrt(2.0) / sqrt(l))) ^ 2.0)) / l));
	elseif (k <= 430000000.0)
		tmp = Float64(2.0 / (Float64(t_m * Float64((cbrt(l) ^ -2.0) * cbrt(Float64(2.0 * (k ^ 2.0))))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m * (k ^ 2.0)) / cos(k)) * Float64((sin(k) ^ 2.0) / (l ^ 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[k, 4.8e-219], N[(N[(N[(2.0 / N[Power[N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 8.1e-168], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[Power[N[(k * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 430000000.0], N[(2.0 / N[Power[N[(t$95$m * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if k < 4.80000000000000028e-219

   1. Initial program 56.2%

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

   3. Simplified 52.1%

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

      1. add-sqr-sqrt 22.9%

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

      2. pow2 22.9%

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

      3. *-commutative 22.9%

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

      4. sqrt-prod 17.7%

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

      5. sqrt-pow1 19.2%

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

      6. metadata-eval 19.2%

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

   6. Applied egg-rr 19.2%

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

      1. *-commutative 19.2%

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

   8. Simplified 19.2%

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

   9. Taylor expanded in k around 0 30.7%

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

   if 4.80000000000000028e-219 < k < 8.1e-168

   1. Initial program 61.0%

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

      1. associate-*l* 40.0%

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

      2. associate-/r* 40.0%

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

      3. +-commutative 40.0%

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

      4. associate-+r+ 40.0%

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

      5. metadata-eval 40.0%

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

      6. associate-*r* 40.0%

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

      7. associate-*l/ 40.9%

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

      8. associate-*l* 40.9%

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

   4. Applied egg-rr 40.9%

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

      1. associate-/l* 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in k around 0 40.9%

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

      1. associate-*r/ 40.9%

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

   9. Simplified 40.9%

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

       1. add-sqr-sqrt 10.9%

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

       2. pow2 10.9%

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

       3. sqrt-div 10.9%

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

       4. *-commutative 10.9%

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

       5. sqrt-prod 10.9%

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

       6. sqrt-pow1 59.6%

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

       7. metadata-eval 59.6%

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

       8. pow1 59.6%

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

   11. Applied egg-rr 59.6%

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

       1. associate-*l/ 59.7%

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

       2. associate-/l* 59.7%

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

   13. Applied egg-rr 59.7%

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

   if 8.1e-168 < k < 4.3e8

   1. Initial program 53.0%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in k around 0 63.4%

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

      1. add-cube-cbrt 63.3%

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

      2. pow3 63.3%

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

      3. cbrt-prod 63.1%

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

      4. associate-/l/ 59.0%

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

      5. cbrt-div 67.9%

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

      6. unpow3 67.9%

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

      7. add-cbrt-cube 73.9%

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

      8. cbrt-prod 92.1%

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

      9. unpow2 92.1%

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

      10. div-inv 92.0%

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

      11. pow-flip 92.1%

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

      12. metadata-eval 92.1%

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

   7. Applied egg-rr 92.1%

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

      1. associate-*l* 92.2%

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

   9. Simplified 92.2%

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

   if 4.3e8 < k

   1. Initial program 47.8%

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

   3. Taylor expanded in t around 0 74.8%

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

      1. associate-*r* 74.8%

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

      2. *-commutative 74.8%

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

      3. times-frac 74.8%

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

   5. Simplified 74.8%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{\frac{2}{{\left(k \cdot {t}^{1.5}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 8.1 \cdot 10^{-168}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3} \cdot {\left(k \cdot \frac{\sqrt{2}}{\sqrt{\ell}}\right)}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq 430000000:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.9 \cdot 10^{-219}:\\ \;\;\;\;\frac{\frac{2}{{\left(k \cdot {t\_m}^{1.5}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{-167}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3} \cdot {\left(k \cdot \frac{\sqrt{2}}{\sqrt{\ell}}\right)}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq 10500000000:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{t\_m}}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 4.9e-219)
    (/
     (* (/ 2.0 (pow (* k (pow t_m 1.5)) 2.0)) (* l l))
     (+ 2.0 (pow (/ k t_m) 2.0)))
    (if (<= k 1.2e-167)
      (/ 2.0 (/ (* (pow t_m 3.0) (pow (* k (/ (sqrt 2.0) (sqrt l))) 2.0)) l))
      (if (<= k 10500000000.0)
        (/
         2.0
         (pow (* t_m (* (pow (cbrt l) -2.0) (cbrt (* 2.0 (pow k 2.0))))) 3.0))
        (*
         (/ 2.0 (pow k 2.0))
         (/ (/ (* (cos k) (pow l 2.0)) t_m) (pow (sin k) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 4.9e-219) {
		tmp = ((2.0 / pow((k * pow(t_m, 1.5)), 2.0)) * (l * l)) / (2.0 + pow((k / t_m), 2.0));
	} else if (k <= 1.2e-167) {
		tmp = 2.0 / ((pow(t_m, 3.0) * pow((k * (sqrt(2.0) / sqrt(l))), 2.0)) / l);
	} else if (k <= 10500000000.0) {
		tmp = 2.0 / pow((t_m * (pow(cbrt(l), -2.0) * cbrt((2.0 * pow(k, 2.0))))), 3.0);
	} else {
		tmp = (2.0 / pow(k, 2.0)) * (((cos(k) * pow(l, 2.0)) / t_m) / pow(sin(k), 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 (k <= 4.9e-219) {
		tmp = ((2.0 / Math.pow((k * Math.pow(t_m, 1.5)), 2.0)) * (l * l)) / (2.0 + Math.pow((k / t_m), 2.0));
	} else if (k <= 1.2e-167) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) * Math.pow((k * (Math.sqrt(2.0) / Math.sqrt(l))), 2.0)) / l);
	} else if (k <= 10500000000.0) {
		tmp = 2.0 / Math.pow((t_m * (Math.pow(Math.cbrt(l), -2.0) * Math.cbrt((2.0 * Math.pow(k, 2.0))))), 3.0);
	} else {
		tmp = (2.0 / Math.pow(k, 2.0)) * (((Math.cos(k) * Math.pow(l, 2.0)) / t_m) / Math.pow(Math.sin(k), 2.0));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 4.9e-219)
		tmp = Float64(Float64(Float64(2.0 / (Float64(k * (t_m ^ 1.5)) ^ 2.0)) * Float64(l * l)) / Float64(2.0 + (Float64(k / t_m) ^ 2.0)));
	elseif (k <= 1.2e-167)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) * (Float64(k * Float64(sqrt(2.0) / sqrt(l))) ^ 2.0)) / l));
	elseif (k <= 10500000000.0)
		tmp = Float64(2.0 / (Float64(t_m * Float64((cbrt(l) ^ -2.0) * cbrt(Float64(2.0 * (k ^ 2.0))))) ^ 3.0));
	else
		tmp = Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64(Float64(cos(k) * (l ^ 2.0)) / t_m) / (sin(k) ^ 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[k, 4.9e-219], N[(N[(N[(2.0 / N[Power[N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.2e-167], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[Power[N[(k * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 10500000000.0], N[(2.0 / N[Power[N[(t$95$m * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * N[Power[N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if k < 4.8999999999999999e-219

   1. Initial program 56.2%

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

   3. Simplified 52.1%

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

      1. add-sqr-sqrt 22.9%

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

      2. pow2 22.9%

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

      3. *-commutative 22.9%

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

      4. sqrt-prod 17.7%

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

      5. sqrt-pow1 19.2%

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

      6. metadata-eval 19.2%

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

   6. Applied egg-rr 19.2%

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

      1. *-commutative 19.2%

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

   8. Simplified 19.2%

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

   9. Taylor expanded in k around 0 30.7%

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

   if 4.8999999999999999e-219 < k < 1.19999999999999997e-167

   1. Initial program 61.0%

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

      1. associate-*l* 40.0%

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

      2. associate-/r* 40.0%

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

      3. +-commutative 40.0%

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

      4. associate-+r+ 40.0%

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

      5. metadata-eval 40.0%

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

      6. associate-*r* 40.0%

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

      7. associate-*l/ 40.9%

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

      8. associate-*l* 40.9%

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

   4. Applied egg-rr 40.9%

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

      1. associate-/l* 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in k around 0 40.9%

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

      1. associate-*r/ 40.9%

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

   9. Simplified 40.9%

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

       1. add-sqr-sqrt 10.9%

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

       2. pow2 10.9%

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

       3. sqrt-div 10.9%

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

       4. *-commutative 10.9%

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

       5. sqrt-prod 10.9%

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

       6. sqrt-pow1 59.6%

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

       7. metadata-eval 59.6%

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

       8. pow1 59.6%

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

   11. Applied egg-rr 59.6%

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

       1. associate-*l/ 59.7%

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

       2. associate-/l* 59.7%

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

   13. Applied egg-rr 59.7%

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

   if 1.19999999999999997e-167 < k < 1.05e10

   1. Initial program 53.0%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in k around 0 63.4%

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

      1. add-cube-cbrt 63.3%

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

      2. pow3 63.3%

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

      3. cbrt-prod 63.1%

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

      4. associate-/l/ 59.0%

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

      5. cbrt-div 67.9%

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

      6. unpow3 67.9%

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

      7. add-cbrt-cube 73.9%

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

      8. cbrt-prod 92.1%

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

      9. unpow2 92.1%

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

      10. div-inv 92.0%

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

      11. pow-flip 92.1%

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

      12. metadata-eval 92.1%

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

   7. Applied egg-rr 92.1%

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

      1. associate-*l* 92.2%

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

   9. Simplified 92.2%

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

   if 1.05e10 < k

   1. Initial program 47.8%

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

   3. Simplified 47.8%

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

   5. Taylor expanded in t around 0 74.8%

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

      1. associate-*r/ 74.8%

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

      2. times-frac 73.5%

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

      3. associate-/r* 73.5%

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

   7. Simplified 73.5%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.9 \cdot 10^{-219}:\\ \;\;\;\;\frac{\frac{2}{{\left(k \cdot {t}^{1.5}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{-167}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3} \cdot {\left(k \cdot \frac{\sqrt{2}}{\sqrt{\ell}}\right)}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq 10500000000:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{2 \cdot {k}^{2}}\right)\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{t}}{{\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t\_m}\right)}^{2}\\ t_3 := \left(\ell \cdot \frac{\frac{2}{{t\_m}^{3}}}{\sin k \cdot \tan k}\right) \cdot \frac{\ell}{t\_2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.7 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \sqrt[3]{\frac{2 \cdot \frac{{k}^{2}}{\ell}}{\ell}}\right)}^{3}}\\ \mathbf{elif}\;t\_m \leq 15000000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 4.1 \cdot 10^{+52}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3} \cdot {\left(k \cdot \frac{\sqrt{2}}{\sqrt{\ell}}\right)}^{2}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 5.1 \cdot 10^{+73}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(k \cdot {t\_m}^{1.5}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 2.0 (pow (/ k t_m) 2.0)))
        (t_3
         (* (* l (/ (/ 2.0 (pow t_m 3.0)) (* (sin k) (tan k)))) (/ l t_2))))
   (*
    t_s
    (if (<= t_m 3.7e-79)
      (/ 2.0 (pow (* t_m (cbrt (/ (* 2.0 (/ (pow k 2.0) l)) l))) 3.0))
      (if (<= t_m 15000000000000.0)
        t_3
        (if (<= t_m 4.1e+52)
          (/
           2.0
           (/ (* (pow t_m 3.0) (pow (* k (/ (sqrt 2.0) (sqrt l))) 2.0)) l))
          (if (<= t_m 5.1e+73)
            t_3
            (/ (* (/ 2.0 (pow (* k (pow t_m 1.5)) 2.0)) (* l l)) 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 = 2.0 + pow((k / t_m), 2.0);
	double t_3 = (l * ((2.0 / pow(t_m, 3.0)) / (sin(k) * tan(k)))) * (l / t_2);
	double tmp;
	if (t_m <= 3.7e-79) {
		tmp = 2.0 / pow((t_m * cbrt(((2.0 * (pow(k, 2.0) / l)) / l))), 3.0);
	} else if (t_m <= 15000000000000.0) {
		tmp = t_3;
	} else if (t_m <= 4.1e+52) {
		tmp = 2.0 / ((pow(t_m, 3.0) * pow((k * (sqrt(2.0) / sqrt(l))), 2.0)) / l);
	} else if (t_m <= 5.1e+73) {
		tmp = t_3;
	} else {
		tmp = ((2.0 / pow((k * pow(t_m, 1.5)), 2.0)) * (l * l)) / t_2;
	}
	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 = 2.0 + Math.pow((k / t_m), 2.0);
	double t_3 = (l * ((2.0 / Math.pow(t_m, 3.0)) / (Math.sin(k) * Math.tan(k)))) * (l / t_2);
	double tmp;
	if (t_m <= 3.7e-79) {
		tmp = 2.0 / Math.pow((t_m * Math.cbrt(((2.0 * (Math.pow(k, 2.0) / l)) / l))), 3.0);
	} else if (t_m <= 15000000000000.0) {
		tmp = t_3;
	} else if (t_m <= 4.1e+52) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) * Math.pow((k * (Math.sqrt(2.0) / Math.sqrt(l))), 2.0)) / l);
	} else if (t_m <= 5.1e+73) {
		tmp = t_3;
	} else {
		tmp = ((2.0 / Math.pow((k * Math.pow(t_m, 1.5)), 2.0)) * (l * l)) / 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(2.0 + (Float64(k / t_m) ^ 2.0))
	t_3 = Float64(Float64(l * Float64(Float64(2.0 / (t_m ^ 3.0)) / Float64(sin(k) * tan(k)))) * Float64(l / t_2))
	tmp = 0.0
	if (t_m <= 3.7e-79)
		tmp = Float64(2.0 / (Float64(t_m * cbrt(Float64(Float64(2.0 * Float64((k ^ 2.0) / l)) / l))) ^ 3.0));
	elseif (t_m <= 15000000000000.0)
		tmp = t_3;
	elseif (t_m <= 4.1e+52)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) * (Float64(k * Float64(sqrt(2.0) / sqrt(l))) ^ 2.0)) / l));
	elseif (t_m <= 5.1e+73)
		tmp = t_3;
	else
		tmp = Float64(Float64(Float64(2.0 / (Float64(k * (t_m ^ 1.5)) ^ 2.0)) * Float64(l * l)) / t_2);
	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[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(l * N[(N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t$95$2), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.7e-79], N[(2.0 / N[Power[N[(t$95$m * N[Power[N[(N[(2.0 * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 15000000000000.0], t$95$3, If[LessEqual[t$95$m, 4.1e+52], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[Power[N[(k * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.1e+73], t$95$3, N[(N[(N[(2.0 / N[Power[N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 3.70000000000000018e-79

   1. Initial program 43.7%

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

      1. associate-*l* 41.0%

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

      2. associate-/r* 44.8%

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

      3. +-commutative 44.8%

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

      4. associate-+r+ 44.8%

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

      5. metadata-eval 44.8%

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

      6. associate-*r* 44.8%

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

      7. associate-*l/ 44.3%

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

      8. associate-*l* 44.3%

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

   4. Applied egg-rr 44.3%

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

      1. associate-/l* 45.3%

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

   6. Simplified 45.3%

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

   7. Taylor expanded in k around 0 42.6%

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

      1. associate-*r/ 42.6%

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

   9. Simplified 42.6%

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

       1. associate-*l/ 42.7%

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

       2. associate-/l* 42.7%

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

   11. Applied egg-rr 42.7%

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

       1. add-cube-cbrt 42.6%

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

       2. pow3 42.6%

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

       3. associate-/l* 42.0%

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

       4. cbrt-prod 41.9%

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

       5. unpow3 42.0%

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

       6. add-cbrt-cube 57.3%

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

   13. Applied egg-rr 57.3%

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

   if 3.70000000000000018e-79 < t < 1.5e13 or 4.1e52 < t < 5.10000000000000024e73

   1. Initial program 81.1%

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

   3. Simplified 87.0%

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

      1. associate-*r* 90.3%

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

      2. *-un-lft-identity 90.3%

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

      3. times-frac 93.0%

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

      4. associate-/r* 93.0%

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

   6. Applied egg-rr 93.0%

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

      1. /-rgt-identity 93.0%

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

      2. *-commutative 93.0%

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

   8. Simplified 93.0%

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

   if 1.5e13 < t < 4.1e52

   1. Initial program 86.2%

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

      1. associate-*l* 85.7%

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

      2. associate-/r* 85.7%

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

      3. +-commutative 85.7%

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

      4. associate-+r+ 85.7%

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

      5. metadata-eval 85.7%

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

      6. associate-*r* 85.7%

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

      7. associate-*l/ 86.1%

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

      8. associate-*l* 86.1%

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

   4. Applied egg-rr 86.1%

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

      1. associate-/l* 86.1%

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

   6. Simplified 86.1%

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

   7. Taylor expanded in k around 0 86.1%

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

      1. associate-*r/ 86.1%

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

   9. Simplified 86.1%

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

       1. add-sqr-sqrt 43.2%

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

       2. pow2 43.2%

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

       3. sqrt-div 43.2%

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

       4. *-commutative 43.2%

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

       5. sqrt-prod 43.2%

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

       6. sqrt-pow1 57.1%

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

       7. metadata-eval 57.1%

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

       8. pow1 57.1%

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

   11. Applied egg-rr 57.1%

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

       1. associate-*l/ 56.9%

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

       2. associate-/l* 56.9%

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

   13. Applied egg-rr 56.9%

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

   if 5.10000000000000024e73 < t

   1. Initial program 62.9%

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

   3. Simplified 56.6%

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

      1. add-sqr-sqrt 35.8%

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

      2. pow2 35.8%

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

      3. *-commutative 35.8%

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

      4. sqrt-prod 35.8%

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

      5. sqrt-pow1 39.9%

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

      6. metadata-eval 39.9%

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

   6. Applied egg-rr 39.9%

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

      1. *-commutative 39.9%

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

   8. Simplified 39.9%

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

   9. Taylor expanded in k around 0 73.9%

      \[\leadsto \frac{\frac{2}{{\left({t}^{1.5} \cdot \color{blue}{k}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \sqrt[3]{\frac{2 \cdot \frac{{k}^{2}}{\ell}}{\ell}}\right)}^{3}}\\ \mathbf{elif}\;t \leq 15000000000000:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{2}{{t}^{3}}}{\sin k \cdot \tan k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+52}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3} \cdot {\left(k \cdot \frac{\sqrt{2}}{\sqrt{\ell}}\right)}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+73}:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{2}{{t}^{3}}}{\sin k \cdot \tan k}\right) \cdot \frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(k \cdot {t}^{1.5}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 + {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.9 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \sqrt[3]{\frac{2 \cdot \frac{{k}^{2}}{\ell}}{\ell}}\right)}^{3}}\\ \mathbf{elif}\;t\_m \leq 3.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{{t\_m}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \frac{t\_2}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(k \cdot {t\_m}^{1.5}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{t\_2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 2.0 (pow (/ k t_m) 2.0))))
   (*
    t_s
    (if (<= t_m 2.9e-101)
      (/ 2.0 (pow (* t_m (cbrt (/ (* 2.0 (/ (pow k 2.0) l)) l))) 3.0))
      (if (<= t_m 3.4e+102)
        (/ 2.0 (* (/ (pow t_m 3.0) l) (* (sin k) (* (tan k) (/ t_2 l)))))
        (/ (* (/ 2.0 (pow (* k (pow t_m 1.5)) 2.0)) (* l l)) 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 = 2.0 + pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 2.9e-101) {
		tmp = 2.0 / pow((t_m * cbrt(((2.0 * (pow(k, 2.0) / l)) / l))), 3.0);
	} else if (t_m <= 3.4e+102) {
		tmp = 2.0 / ((pow(t_m, 3.0) / l) * (sin(k) * (tan(k) * (t_2 / l))));
	} else {
		tmp = ((2.0 / pow((k * pow(t_m, 1.5)), 2.0)) * (l * l)) / t_2;
	}
	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 = 2.0 + Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 2.9e-101) {
		tmp = 2.0 / Math.pow((t_m * Math.cbrt(((2.0 * (Math.pow(k, 2.0) / l)) / l))), 3.0);
	} else if (t_m <= 3.4e+102) {
		tmp = 2.0 / ((Math.pow(t_m, 3.0) / l) * (Math.sin(k) * (Math.tan(k) * (t_2 / l))));
	} else {
		tmp = ((2.0 / Math.pow((k * Math.pow(t_m, 1.5)), 2.0)) * (l * l)) / 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(2.0 + (Float64(k / t_m) ^ 2.0))
	tmp = 0.0
	if (t_m <= 2.9e-101)
		tmp = Float64(2.0 / (Float64(t_m * cbrt(Float64(Float64(2.0 * Float64((k ^ 2.0) / l)) / l))) ^ 3.0));
	elseif (t_m <= 3.4e+102)
		tmp = Float64(2.0 / Float64(Float64((t_m ^ 3.0) / l) * Float64(sin(k) * Float64(tan(k) * Float64(t_2 / l)))));
	else
		tmp = Float64(Float64(Float64(2.0 / (Float64(k * (t_m ^ 1.5)) ^ 2.0)) * Float64(l * l)) / t_2);
	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[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.9e-101], N[(2.0 / N[Power[N[(t$95$m * N[Power[N[(N[(2.0 * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.4e+102], N[(2.0 / N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Power[N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.9e-101

   1. Initial program 42.9%

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

      1. associate-*l* 40.2%

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

      2. associate-/r* 43.4%

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

      3. +-commutative 43.4%

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

      4. associate-+r+ 43.4%

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

      5. metadata-eval 43.4%

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

      6. associate-*r* 43.4%

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

      7. associate-*l/ 43.0%

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

      8. associate-*l* 42.9%

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

   4. Applied egg-rr 42.9%

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

      1. associate-/l* 44.0%

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

   6. Simplified 44.0%

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

   7. Taylor expanded in k around 0 41.2%

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

      1. associate-*r/ 41.2%

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

   9. Simplified 41.2%

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

       1. associate-*l/ 41.2%

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

       2. associate-/l* 41.2%

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

   11. Applied egg-rr 41.2%

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

       1. add-cube-cbrt 41.2%

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

       2. pow3 41.2%

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

       3. associate-/l* 40.6%

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

       4. cbrt-prod 40.5%

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

       5. unpow3 40.5%

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

       6. add-cbrt-cube 56.3%

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

   13. Applied egg-rr 56.3%

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

   if 2.9e-101 < t < 3.4e102

   1. Initial program 82.5%

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

      1. associate-*l* 82.2%

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

      2. associate-/r* 86.2%

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

      3. +-commutative 86.2%

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

      4. associate-+r+ 86.2%

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

      5. metadata-eval 86.2%

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

      6. associate-*r* 86.2%

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

      7. associate-*l/ 91.6%

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

      8. associate-*l* 91.6%

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

   4. Applied egg-rr 91.6%

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

      1. associate-/l* 91.7%

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

   6. Simplified 91.7%

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

      1. associate-/l* 95.7%

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

   8. Applied egg-rr 95.7%

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

      1. associate-/l* 95.7%

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

   10. Simplified 95.7%

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

   if 3.4e102 < t

   1. Initial program 58.4%

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

   3. Simplified 51.1%

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

      1. add-sqr-sqrt 28.9%

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

      2. pow2 28.9%

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

      3. *-commutative 28.9%

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

      4. sqrt-prod 28.9%

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

      5. sqrt-pow1 33.7%

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

      6. metadata-eval 33.7%

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

   6. Applied egg-rr 33.7%

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

      1. *-commutative 33.7%

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

   8. Simplified 33.7%

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

   9. Taylor expanded in k around 0 71.4%

      \[\leadsto \frac{\frac{2}{{\left({t}^{1.5} \cdot \color{blue}{k}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \sqrt[3]{\frac{2 \cdot \frac{{k}^{2}}{\ell}}{\ell}}\right)}^{3}}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \frac{2 + {\left(\frac{k}{t}\right)}^{2}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(k \cdot {t}^{1.5}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \sqrt[3]{\frac{2 \cdot \frac{{k}^{2}}{\ell}}{\ell}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(k \cdot {t\_m}^{1.5}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t\_m}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e-79)
    (/ 2.0 (pow (* t_m (cbrt (/ (* 2.0 (/ (pow k 2.0) l)) l))) 3.0))
    (/
     (* (/ 2.0 (pow (* k (pow t_m 1.5)) 2.0)) (* l l))
     (+ 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 <= 8e-79) {
		tmp = 2.0 / pow((t_m * cbrt(((2.0 * (pow(k, 2.0) / l)) / l))), 3.0);
	} else {
		tmp = ((2.0 / pow((k * pow(t_m, 1.5)), 2.0)) * (l * l)) / (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 <= 8e-79) {
		tmp = 2.0 / Math.pow((t_m * Math.cbrt(((2.0 * (Math.pow(k, 2.0) / l)) / l))), 3.0);
	} else {
		tmp = ((2.0 / Math.pow((k * Math.pow(t_m, 1.5)), 2.0)) * (l * l)) / (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 <= 8e-79)
		tmp = Float64(2.0 / (Float64(t_m * cbrt(Float64(Float64(2.0 * Float64((k ^ 2.0) / l)) / l))) ^ 3.0));
	else
		tmp = Float64(Float64(Float64(2.0 / (Float64(k * (t_m ^ 1.5)) ^ 2.0)) * Float64(l * l)) / 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, 8e-79], N[(2.0 / N[Power[N[(t$95$m * N[Power[N[(N[(2.0 * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Power[N[(k * N[Power[t$95$m, 1.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $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 < 8e-79

   1. Initial program 43.7%

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

      1. associate-*l* 41.0%

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

      2. associate-/r* 44.8%

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

      3. +-commutative 44.8%

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

      4. associate-+r+ 44.8%

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

      5. metadata-eval 44.8%

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

      6. associate-*r* 44.8%

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

      7. associate-*l/ 44.3%

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

      8. associate-*l* 44.3%

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

   4. Applied egg-rr 44.3%

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

      1. associate-/l* 45.3%

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

   6. Simplified 45.3%

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

   7. Taylor expanded in k around 0 42.6%

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

      1. associate-*r/ 42.6%

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

   9. Simplified 42.6%

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

       1. associate-*l/ 42.7%

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

       2. associate-/l* 42.7%

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

   11. Applied egg-rr 42.7%

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

       1. add-cube-cbrt 42.6%

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

       2. pow3 42.6%

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

       3. associate-/l* 42.0%

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

       4. cbrt-prod 41.9%

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

       5. unpow3 42.0%

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

       6. add-cbrt-cube 57.3%

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

   13. Applied egg-rr 57.3%

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

   if 8e-79 < t

   1. Initial program 70.9%

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

   3. Simplified 69.2%

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

      1. add-sqr-sqrt 41.6%

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

      2. pow2 41.6%

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

      3. *-commutative 41.6%

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

      4. sqrt-prod 41.6%

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

      5. sqrt-pow1 43.9%

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

      6. metadata-eval 43.9%

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

   6. Applied egg-rr 43.9%

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

      1. *-commutative 43.9%

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

   8. Simplified 43.9%

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

   9. Taylor expanded in k around 0 72.0%

      \[\leadsto \frac{\frac{2}{{\left({t}^{1.5} \cdot \color{blue}{k}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-79}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \sqrt[3]{\frac{2 \cdot \frac{{k}^{2}}{\ell}}{\ell}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{{\left(k \cdot {t}^{1.5}\right)}^{2}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.2 \cdot 10^{+200}:\\ \;\;\;\;\frac{2}{{\left(t\_m \cdot \sqrt[3]{\frac{2 \cdot \frac{{k}^{2}}{\ell}}{\ell}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \sqrt{2 \cdot \frac{\frac{{t\_m}^{3}}{\ell}}{\ell}}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.2e+200)
    (/ 2.0 (pow (* t_m (cbrt (/ (* 2.0 (/ (pow k 2.0) l)) l))) 3.0))
    (/ 2.0 (pow (* k (sqrt (* 2.0 (/ (/ (pow t_m 3.0) l) l)))) 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 <= 5.2e+200) {
		tmp = 2.0 / pow((t_m * cbrt(((2.0 * (pow(k, 2.0) / l)) / l))), 3.0);
	} else {
		tmp = 2.0 / pow((k * sqrt((2.0 * ((pow(t_m, 3.0) / l) / l)))), 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 <= 5.2e+200) {
		tmp = 2.0 / Math.pow((t_m * Math.cbrt(((2.0 * (Math.pow(k, 2.0) / l)) / l))), 3.0);
	} else {
		tmp = 2.0 / Math.pow((k * Math.sqrt((2.0 * ((Math.pow(t_m, 3.0) / l) / l)))), 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 <= 5.2e+200)
		tmp = Float64(2.0 / (Float64(t_m * cbrt(Float64(Float64(2.0 * Float64((k ^ 2.0) / l)) / l))) ^ 3.0));
	else
		tmp = Float64(2.0 / (Float64(k * sqrt(Float64(2.0 * Float64(Float64((t_m ^ 3.0) / l) / l)))) ^ 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, 5.2e+200], N[(2.0 / N[Power[N[(t$95$m * N[Power[N[(N[(2.0 * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(k * N[Sqrt[N[(2.0 * N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.2000000000000003e200

   1. Initial program 53.4%

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

      1. associate-*l* 51.4%

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

      2. associate-/r* 54.5%

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

      3. +-commutative 54.5%

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

      4. associate-+r+ 54.5%

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

      5. metadata-eval 54.5%

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

      6. associate-*r* 54.5%

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

      7. associate-*l/ 55.4%

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

      8. associate-*l* 55.4%

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

   4. Applied egg-rr 55.4%

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

      1. associate-/l* 56.1%

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

   6. Simplified 56.1%

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

   7. Taylor expanded in k around 0 50.2%

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

      1. associate-*r/ 50.2%

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

   9. Simplified 50.2%

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

       1. associate-*l/ 50.3%

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

       2. associate-/l* 50.3%

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

   11. Applied egg-rr 50.3%

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

       1. add-cube-cbrt 50.2%

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

       2. pow3 50.2%

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

       3. associate-/l* 50.1%

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

       4. cbrt-prod 50.0%

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

       5. unpow3 50.0%

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

       6. add-cbrt-cube 61.9%

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

   13. Applied egg-rr 61.9%

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

   if 5.2000000000000003e200 < t

   1. Initial program 52.7%

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

      1. associate-*l* 39.1%

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

      2. associate-/r* 44.9%

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

      3. +-commutative 44.9%

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

      4. associate-+r+ 44.9%

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

      5. metadata-eval 44.9%

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

      6. associate-*r* 44.9%

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

      7. associate-*l/ 44.9%

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

      8. associate-*l* 44.9%

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

   4. Applied egg-rr 44.9%

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

      1. associate-/l* 44.6%

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

   6. Simplified 44.6%

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

   7. Taylor expanded in k around 0 44.6%

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

      1. associate-*r/ 44.6%

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

   9. Simplified 44.6%

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

       1. add-sqr-sqrt 44.6%

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

       2. pow2 44.6%

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

       3. associate-*r/ 44.9%

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

       4. associate-*l/ 44.9%

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

       5. rem-exp-log 22.3%

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

       6. associate-*r* 22.3%

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

       7. sqrt-prod 22.3%

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

       8. rem-exp-log 44.9%

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

       9. sqrt-pow1 58.6%

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

       10. metadata-eval 58.6%

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

       11. pow1 58.6%

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

   11. Applied egg-rr 58.6%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.2 \cdot 10^{+200}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \sqrt[3]{\frac{2 \cdot \frac{{k}^{2}}{\ell}}{\ell}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \sqrt{2 \cdot \frac{\frac{{t}^{3}}{\ell}}{\ell}}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\frac{{\left(t\_m \cdot \sqrt[3]{2 \cdot \frac{{k}^{2}}{\ell}}\right)}^{3}}{\ell}} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (/ (pow (* t_m (cbrt (* 2.0 (/ (pow k 2.0) l)))) 3.0) l))))

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((t_m * cbrt((2.0 * (pow(k, 2.0) / l)))), 3.0) / l));
}

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((t_m * Math.cbrt((2.0 * (Math.pow(k, 2.0) / l)))), 3.0) / l));
}

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((Float64(t_m * cbrt(Float64(2.0 * Float64((k ^ 2.0) / l)))) ^ 3.0) / l)))
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[N[(t$95$m * N[Power[N[(2.0 * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

   1. associate-*l* 50.3%

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

   2. associate-/r* 53.7%

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

   3. +-commutative 53.7%

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

   4. associate-+r+ 53.7%

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

   5. metadata-eval 53.7%

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

   6. associate-*r* 53.7%

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

   7. associate-*l/ 54.4%

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

   8. associate-*l* 54.4%

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

4. Applied egg-rr 54.4%

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

   1. associate-/l* 55.1%

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

6. Simplified 55.1%

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

7. Taylor expanded in k around 0 49.7%

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

   1. associate-*r/ 49.7%

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

9. Simplified 49.7%

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

    1. associate-*l/ 49.8%

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

    2. associate-/l* 49.8%

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

11. Applied egg-rr 49.8%

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

    1. add-cube-cbrt 49.7%

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

    2. pow3 49.7%

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

    3. cbrt-prod 49.7%

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

    4. unpow3 49.7%

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

    5. add-cbrt-cube 60.3%

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

13. Applied egg-rr 60.3%

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

Alternative 16: 58.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{{\left(\frac{t\_m}{\sqrt[3]{\ell}}\right)}^{3} \cdot \frac{2 \cdot {k}^{2}}{\ell}} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (pow (/ t_m (cbrt l)) 3.0) (/ (* 2.0 (pow k 2.0)) l)))))

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((t_m / cbrt(l)), 3.0) * ((2.0 * pow(k, 2.0)) / l)));
}

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((t_m / Math.cbrt(l)), 3.0) * ((2.0 * Math.pow(k, 2.0)) / l)));
}

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((Float64(t_m / cbrt(l)) ^ 3.0) * Float64(Float64(2.0 * (k ^ 2.0)) / l))))
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[N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

   1. associate-*l* 50.3%

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

   2. associate-/r* 53.7%

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

   3. +-commutative 53.7%

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

   4. associate-+r+ 53.7%

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

   5. metadata-eval 53.7%

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

   6. associate-*r* 53.7%

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

   7. associate-*l/ 54.4%

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

   8. associate-*l* 54.4%

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

4. Applied egg-rr 54.4%

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

   1. associate-/l* 55.1%

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

6. Simplified 55.1%

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

7. Taylor expanded in k around 0 49.7%

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

   1. associate-*r/ 49.7%

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

9. Simplified 49.7%

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

    1. add-cube-cbrt 49.7%

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

    2. pow3 49.7%

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

    3. cbrt-div 49.7%

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

    4. rem-cbrt-cube 53.2%

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

11. Applied egg-rr 53.2%

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

Alternative 17: 57.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t\_m}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (* (* 2.0 (pow k 2.0)) (/ (pow (/ t_m (cbrt l)) 3.0) l)))))

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 / ((2.0 * pow(k, 2.0)) * (pow((t_m / cbrt(l)), 3.0) / l)));
}

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 / ((2.0 * Math.pow(k, 2.0)) * (Math.pow((t_m / Math.cbrt(l)), 3.0) / l)));
}

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(Float64(2.0 * (k ^ 2.0)) * Float64((Float64(t_m / cbrt(l)) ^ 3.0) / l))))
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[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

3. Simplified 53.7%

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

5. Taylor expanded in k around 0 48.7%

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

   1. add-cube-cbrt 49.7%

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

   2. pow3 49.7%

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

   3. cbrt-div 49.7%

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

   4. rem-cbrt-cube 53.2%

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

7. Applied egg-rr 52.2%

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

8. Final simplification 52.2%

   \[\leadsto \frac{2}{\left(2 \cdot {k}^{2}\right) \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}} \]
9. Add Preprocessing

Alternative 18: 57.9% accurate, 2.0× speedup

Math

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

FPCore

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

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

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

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 / (((2.0 * math.pow(k, 2.0)) / l) * (t_m * (math.pow(t_m, 2.0) / l))))

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(Float64(Float64(2.0 * (k ^ 2.0)) / l) * Float64(t_m * Float64((t_m ^ 2.0) / l)))))
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 / (((2.0 * (k ^ 2.0)) / l) * (t_m * ((t_m ^ 2.0) / l))));
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[(N[(2.0 * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

   1. associate-*l* 50.3%

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

   2. associate-/r* 53.7%

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

   3. +-commutative 53.7%

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

   4. associate-+r+ 53.7%

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

   5. metadata-eval 53.7%

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

   6. associate-*r* 53.7%

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

   7. associate-*l/ 54.4%

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

   8. associate-*l* 54.4%

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

4. Applied egg-rr 54.4%

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

   1. associate-/l* 55.1%

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

6. Simplified 55.1%

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

7. Taylor expanded in k around 0 49.7%

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

   1. associate-*r/ 49.7%

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

9. Simplified 49.7%

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

    1. cube-mult 49.7%

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

    2. *-un-lft-identity 49.7%

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

    3. times-frac 52.1%

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

    4. pow2 52.1%

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

11. Applied egg-rr 52.1%

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

12. Final simplification 52.1%

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

Alternative 19: 55.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\ell \cdot \frac{2}{\frac{{k}^{2}}{\ell} \cdot \left(2 \cdot {t\_m}^{3}\right)}\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* l (/ 2.0 (* (/ (pow k 2.0) l) (* 2.0 (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) {
	return t_s * (l * (2.0 / ((pow(k, 2.0) / l) * (2.0 * pow(t_m, 3.0)))));
}

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 * (l * (2.0d0 / (((k ** 2.0d0) / l) * (2.0d0 * (t_m ** 3.0d0)))))
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 * (l * (2.0 / ((Math.pow(k, 2.0) / l) * (2.0 * Math.pow(t_m, 3.0)))));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (l * (2.0 / ((math.pow(k, 2.0) / l) * (2.0 * math.pow(t_m, 3.0)))))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(l * Float64(2.0 / Float64(Float64((k ^ 2.0) / l) * Float64(2.0 * (t_m ^ 3.0))))))
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 * (l * (2.0 / (((k ^ 2.0) / l) * (2.0 * (t_m ^ 3.0)))));
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[(l * N[(2.0 / N[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(2.0 * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

   1. associate-*l* 50.3%

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

   2. associate-/r* 53.7%

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

   3. +-commutative 53.7%

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

   4. associate-+r+ 53.7%

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

   5. metadata-eval 53.7%

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

   6. associate-*r* 53.7%

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

   7. associate-*l/ 54.4%

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

   8. associate-*l* 54.4%

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

4. Applied egg-rr 54.4%

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

   1. associate-/l* 55.1%

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

6. Simplified 55.1%

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

7. Taylor expanded in k around 0 49.7%

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

   1. associate-*r/ 49.7%

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

9. Simplified 49.7%

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

    1. associate-*l/ 49.8%

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

    2. associate-/l* 49.8%

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

11. Applied egg-rr 49.8%

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

    1. associate-/r/ 49.8%

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

    2. associate-*r* 49.8%

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

13. Applied egg-rr 49.8%

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

14. Final simplification 49.8%

    \[\leadsto \ell \cdot \frac{2}{\frac{{k}^{2}}{\ell} \cdot \left(2 \cdot {t}^{3}\right)} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024092 
(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))))